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The effect of a magnetic field on transport phenomena in a Bridgman-Stockbarger crystal growth
Journal of Crystal Growth 67 (1984) 405—419 North-Holland, Amsterdam
405
THE EFFECT OF A MAGNETIC FIELD ON TRANSPORT PHENOMENA IN A BRIDGMAN-STOCKBARGER CRYSTAL GROWTH G.M. OREPER and J. SZEKELY Department of Materials Engineering and Center for Materials Processing, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 28 November 1983; manuscript received in final form 29 May 1984
A mathematical formulation and computed results are presented to describe the temperature, velocity and tracer distribution in a Bridgman—Stockbarger crystal growing system, as affected by an externally imposed magnetic field. The computed results have shown that while the temperature profiles will not be significantly affected by the magnetic field, both the concentration fields and the velocity fields may be markedly influenced, especially at high values of the magnetic interaction parameter. In essence the imposition of a magnetic field will tend to both slow down the melt velocities and also modify the nature of the circulation pattern. Additional important effects of the magnetic field include its influence on the local and mean values of the heat and mass transfer rates at the melt solid interface. It was found that intermediate values of the magnetic interaction parameter (i.e. magnetic fields) would tend to provide the most uniform local heat transfer rates. While the velocities and mass transfer rates are damped by the magnetic field, the local value of mass transfer coefficient were the most uniform in the absence of an external magnetic field.
1. Infroduction
in the vicinity of the solid surfaces.
The purpose of this investigation is to develop quantitative understanding of how an externally imposed magnetic field affects the fluid flow field, the temperature field and the concentration profiles in a vertical Bridgman—Stockbarger crystal growing system. The principal motivation for this work is that the use of an externally imposed magnetic field has been suggested by several investigators in order to promote the uniformity of impurity distribution by suppressing convection and flow instabilities [1]. A detailed review of the literature on the application of magnetic fields in the crystal growth has been presented in the previous paper by the present authors, where we examined the role of an externally imposed magnetic field on the transient convection in a rectangular cavity [2], the walls of which were maintained at different temperatures. It was shown that the imposition of a magnetic field reduced the absolute values of the melt velocity and tended to flatten out the velocity profiles
This previous example represented a rather idealized picture of the Bridgman system, both in terms of the (rectangular) geometry selected and of the boundary conditions that were invoked; while this earlier work did provide an improved insight into the effect of an external magnetic field, it did not specifically address the effect of convection, as modified by the magnetic field, on the distribution of impurity or tracer elements. In the present paper we shall consider a more realistic geometry and thermal boundary conditions and will address the problem of the tracer distribution. The fundamental basis for the interaction between magnetic fields and convection has been discussed by Chandrasekhar [3] and Thomson [4]. Other interesting related work has been reported by Lehnert and Little [5], Gershuni and Zhukhovitskii [6,7] and Utech and Early [8]. These previously cited investigations did establish both the fundamental basis for melt and magnetic field interactions on the one hand and the concrete possibility of influencing crystal growth
0022-0248/84/$03.OO © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
406
G.M. Oreper, J. Szekely
/
Effect of magnetic field on transport phenomena
through the imposition of a magnetic field on the other. The first quantitative analysis of how an externally imposed magnetic field affects the behavior of the Czochralski system was presented by Langlois and Walker [9] and Langlois and Ki-Jun Lee [10,11]. However, the quantitative description as to how the imposed field would affect convection and hence crystal quality in geometries of relevance to the growth of single crystals by the Bridgman—Stockbarger technique has yet to be undertaken, The common method of growing single crystals involves the lateral supply of thermal energy, coupled with the axial removal of the latent heat of solidification and other associated heat losses from the system. Problems on thermal natural convection related to the crystal growth in the cylindrical geometry have been tackled by Liang, Vidal and Acrivos [12] and more recently by Brown and co-workers [13—17]. However, the effect of magnetic fields on convection in such geometries has not been undertaken up to the present. The purpose of this paper is to present a formulation and computed results on the effect of a magnetic field on thermal natural convection and on the concentration field in a circular cylinder that is the idealized representation of the Bridgman—Stockbarger system of crystal growth with the melt above the crystal. In general two sets of forces will act on this system: (1) the buoyancy force, due to the temperature gradients; (2) the Lorentz force, which is due to the interac tion between the induced current and the externally imposed magnetic field. It follows that the statement of the problem will have to involve the expression of: — the equation of continuity; — the equation of motion (Navier—Stokes equation) including the buoyancy and the Lorentz terms; — the thermal energy balance equation; — the mass balance equation; — Maxwell’s equations to define the Lorentz forces. These equations are mutually coupled.
2. Formulation for the velocity and temperature distribution Let us consider a circular cylinder of height L and of radius R as sketched in fig. 1. The liquid contained in the cylinder is initially at rest and the initial temperature distribution varies linearly from T= T 1 at the top to T0 at the bottom. At time t = 0 the side walls are subjected to a heat flux from the heater except the small region on the side wall near the center line which is adiabatically separated from the ambient medium, i.e. there is no heat flux through this part of the wall. The length of the adiabatic region is equal to 2R. At the same time the entire system is immersed in a constant magnetic field of strength B0, which acts in the z-direction. The problem is to find the time dependent spatial distribution of the temperature and
r- AXIS
HEATED
I
MELT
REGION B~l
ADIABATIC
-____________
~
Bi~O SOLID
Fig. 1. Sketch of the system.
G.M. Oreper, .1. Szekely
/
Effect of magnetic field on transport phenomena
the velocity and to examine the influence of the magnetic field on these quantities. The quantitative representation of this system has to include the following physical components: (1) Buoyancy driven flow will be established due to temperature gradients in the liquid resulting from the imposed boundary conditions. (2) The motion of an electrically conducting liquid in the presence of a magnetic field will give rise to Lorentz forces, Thus one has to solve the set of equations that govern the fluid flow field together with the energy equation and Maxwell’s equations in the moving medium. These equations are coupled because the fluid flow affects the convective heat transfer and the magnetic and the electric fields, while fluid flow itself is driven by the combined action of the buoyancy and the electromagnetic force field. The following assumptions were made in the statement of the problem: (1) The fluid is Newtonian. (2) The property values, such as viscosity, density, thermal conductivity, etc., are constant and evaluated at the temperature T = T0, the only exception being for the density dependence on the temperature to allow for the effect of buoyancy forces. (3) The flow is considered to be stable and laminar throughout. (4) The walls of the cylinder are electrical insulators. (5) The pulling rate of the crystal has a negligible effect on heat and fluid flow. (6) The viscous negligibly small. dissipation and Joule heating are (7) The melt—solid interface is flat. While these assumptions are somewhat restrictive they permit the formulation of the problem in a manageable form, while retaining the essential features of the interaction between a buoyancy driven flow and a magnetic field. The implication of the simplifying assumptions will be discussed subsequently. The problem may now be stated by expressing: — the conservation of matter, i~e.the equation of continuity; — the conservation of momentum, i.e. the equation of motion; — the conservation of thermal energy, i.e. the dif-
407
ferential thermal energy balance equation; — the electromagnetic force field equations and the associated boundary conditions. The general statement of the equations has been presented previously [2]. Here we shall state the governing equations in cylindrical coordinates. It is convenient to express the governing equations in a dimensionless form. Let us denote L. and L~as the characteristic lengths in the z and the r directions, U~and LJ~as the characteristic velocities in the z and the r directions. Then using the equation of continuity one may write that U, 0 = A L~, where A = L/L~ is the aspect ratio. On defining the dimensionless vorticity/r as: ~ ~
=
-~
— —
rA z r r 2>> 1, ~ c~L~ we note that since A 0A/L~. Now let us denote B0 as the characteristic value of magnetic induction and L~T=T1 — T0 as the temperature scale that equals the temperature drop along the solid and molten part of the medium. The dimensionless melting temperature was so chosen that T~1= 0.5. One may show that the scale for the current densityj0 B0/fleLr and that the proper scale for the electric field strength E0, is U~0B0. Let us define t L~vas the time scale. Now the governing equations may be expressed in the following dimensionless form: 3 O~ 2/ 0 0 r OT Re \ Oz Or 1 a ~ a~ 1 a / ~ — A2 Re ~ r~ — ~ ~~j-,: ~r
1
Gr .~±+ -~- ~ = A Re2 Or A2 Oz 0 (1 0’I’ \ 0 / 1 O’I’ \
~
~
+
~
(1)
~,
+
r~= 0,
(2)
where ur=—, Or
—u~r=——. Oz
Examination of eq. (1) shows that when the convective terms dominate the flow Gr/Re2A
408
G.M. Oreper, J. Szeke/v
/
Effect of magnetic field on transport phenomena
!~tOL~Ucü << I the magnetic field b
0(1) and the velocity scale
= B 0. If there is no outside circuit and walls of the cylinder are not
(v%/Gr/A )/Lc.
=
conductors it follows that the component of the electric field E4 = 0. The dimensionless quantities appearing in these equations are defined as in table 1. Eq. (1) represents the equation of motion, writ-
The differential thermal energy balance equation is written as: Prr~+
Pe(-~-(0ur)+~-(0u Oz Or
OT
) .~L Or I
r))
)
~
drivesside the represents flow, while last terms on the hand thethe “magnetic brake” i.e. left the electromagnetic force which will hinder the flow.
where 0 = (T — 1~)/L%Tis the dimensionless ternperature. The expression for the curl of the electromagnetic force in eq. (1) was derived using the fact that when constant electromagnetic field is applied and the magnetic Reynolds number Re 0 =
Eq. (3) is the thermal energy balance equation, which of course contains the time dependent velocity. It follows that eqs. (1) and (2) are mutually coupled. Eq. (2) is merely a relationship between the stream function and the vorticity.
—
.J.... 2
I .~ Oz ~r Oz
.~.
~ —
~
— 0‘ Or ~—
ten in terms of vorticity transport equation. The term 00/Or represents the buoyancy force, which
A
0
r=0,
The boundary conditions are given as:
~=0, Or
0
r=1,
~=Bi(T—T0),
A—I
r=1,
z=0, z=A,
0r1, 0r1,
—=0, Or 0=1, 0=0.5,
Table I
3/2v2
Gr=g$~TL Re U 0L /~ Pr = c/a Pc = U~0L~/a = p,~c~L~U~0 A L,/L~ 2/Re N Ha= = Ha B~L~/vp
Grashof number Reynolds number Prandtl number Péclet number Magnetic Reynolds number Aspect Parameter ratio of magnetic interaction . Hartmann number
r T
Dimensionless time Temperature
p g ‘s~ z’
Density of the liquid Acceleration due to gravity Magnetic permeability Kinematic viscosity
/3 a a
Coefficient of the thermal expansion Thermal diffusivity Electric conductivity
~=0, (4)
where Bi = 2hL,/k is the Biot number, h is the . . heat . transfer coefficient and k is the thermal conductivity of the melt. These relationships simply express the constraints that the stream function is zero at the solid boundaries and that the boundaries have a specified temperature or heat flux. . The boundary conditions . for the vorticity will be discussed subsequently, in connection with the . . . . . computational technique. The initial conditions are given as: T
.
=
0, 0
z
A, 0
1
=
—
z/2A, ~ = 0, ‘~I’= 0.
In physical terms these represent the stipulations .
.
.
.
that initially the fluid is at rest and that the temperature field is uniform.
G.M. Oreper, J. Szekely
/
Effect of magnetic field on transport phenomena
3. Formulation for the concenfration field The main motivation for studying convection in a crystal growing system is to asses the role of the convection on the impurity distribution. While the exact representation of this problem will require the knowledge of the partition coefficients and the solidification rates, a good preliminary insight may be obtained by considering the dispersion of a tracer (due to the diffusion and convection) which is deposited at the upper boundary and is implanted at the solid—liquid interface. The dispersion of this tracer may be represented by the equation expressing the mass balance
Ot
+(Uvi)C
=
2C,
D v
(6)
or in the dimensionless form Sc r~-~ + Pe (-~-(Cu r) +-~-(Cu r)\ O’r D~ Oz Or I 1 0 / OC \ 0 oc \ — —i ~— ~r-~—-) — ~— r-~—)= 0,
(7)
where Sc = v/D is the Schmidt number, D is the diffusivity and PeD = L,UcO/D is the Péclet number for mass transfer, Let us examine the influence of the magnetic field on the impurity concentration at the time when the temperature and the velocity fields have reached their steady state values and let us assume that the concentration of the impurity does not affect the fluid density. Furthermore, in computing the concentration profiles we shall only present the results corresponding to the steady state conditions. This equation is time dependent but we shall be concerned only with its solution for the time when steady state has been achieved. Hence the velocity distribution used in this equation corresponds to the time 20 s when steady state motion’ has been attained for all the cases considered. The necessary initial and boundary conditions are z = z = r= r=
0,
A, 0,
L,.,
T0,
C C
1, 0, OC/Or = 0, OC/Or 0, = =
0zA,
(8) C1z/A.
409
The first two boundary conditions express the fact that the concentration is constant at the solid—liquid interface and at a long distance from it. The second two boundary conditions specify that there is no mass flux through the axis of symmetry and side walls. The initial condition corresponds to a linear distribution of the concentration along the vertical direction.
4. The technique of solution In order to obtain a numerical solution, the governing equations were put into finite difference form using an 21 X 11 grid. The unsteady state equations for the temperature, concentration and vorticity distribution were solved through the “hopscotch” method, a detailed description of which is readily available in the computational literature [18,19]. The technique proposed by Spalding and co-workers [20] was used for solving the equation for the stream function. The vorticity at the solid walls was calculated in accordance with the method described by Nogotov and Berkovsky [211. The boundary condition for the vorticity invoked at the axis of symmetry specifies zero gradient of ~ at r = 0. 5. Results In the following we shall present a selection of the computed results. Due to the dimensionless form of the governing equations the independent variables entered these calculations in a dimensionless form also. The actual numerical value of the Grashof number was Gr = 6.585 X iO~so that the corresponding Raleigh number Ra = GrPr equalled 9.6 x iO~.The Biot number was unity. One should expect the flow to be stable for such conditions in the light of Oshima’s results [22]. In a practical, physical sense these conditions could correspond to a cylinder with a 7.5 mm radius, 150 mm high, holding molten silicon, with a z2iT value of about 400 K. This means that the temperature drop in the melt is 200 K and Re ~/Gr/A = 0.81 X 102. In the computation of mass transfer phenomena the Schmidt number was taken as 1000.
S—AXIS (MM I .0
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SEC
H—AXIS IMMI .0
2.0
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6.i -
10.0
__________
C
~ 5K 0)0
Fig. 2. Dimensionless temperature distribution in the absence __________________________________ TIME=20. 00 SEC
of a magnetic field at: (a) time 2 s, (b) time 10 s, (c) time 20 s.
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Fig. 3. Velocity field in the absence of a magnetic field at: (a) time 2 s, (b) time 10 s, (c) time 20 s.
A—AXIS 1MM) 2.0
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/
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_______________________
TIME=20. 00 SEC
Fig. 4. Velocity distribution in the presence of a magnetic field after 20 s for: (a) N = 1 (2450 G), (b) N = 00 (7750 G), (c) N
=
69 (20000 G).
H—AXIS (MM) .0
2.0 I
4.0
6.0
I
I
H—AXIS IMMI 8.0
10.0
.0
a
TIME=20.00 SEC
2.12
4.0
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I
TIME=20.00
~.
9—AXISIMMI .0
2.0 I
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8.0
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b
SEC
H—AXISIMMI 8.0 10.0 L......~
.0
2.0 I
4.0 I
6.0 I
8.0
10.0
(I~
TIME=20. 00 SEC
TIME=20. 00 SEC
~.
Fig. 5. Dimensionless stream function distribution (x jQ4) after 20 s for (a) N N = 69 (20000 G).
=
0 (0 G), (b) N
=
1(2450 G). (c) N
=
10 (7750 G). (d)
414
G.M. Oreper, J. S:ekelv / Effect of magnetic field on transport phenomena
Figs. 2a—2c show the transient development of the temperature profiles in the absence of a magnetic field. It is seen that steady state conditions are attained within about 20 s or less and that on reaching of steady state the temperature profiles are largely parallel, indicating that convection plays only a very minor role in affecting heat transfer. This point has been confirmed independently by computing the temperature profiles in the absence of convection, which were almost indistinguishable from the picture seen in fig. 2c. Figs. 3a—3c show the corresponding time dependent velocity fields, again in the absence of a magnetic field. The counterclockwise circulation pattern is readily apparent and the slight overshoot in the velocity field for the intermediate time is a logical consequence of the small Prandtl number. Inspection of fig. 3c shows that there is an intense circulation in the vicinity of the solid—melt interface, while the “far end” of the system is relatively quiescent. Let us now proceed by examining the effect of an externally imposed magnetic field. In these calculation the effect of the magnetic field was characterised by N, the magnetic interaction parameter, the values of which were allowed to range from 0 to 69. For the system chosen, in absolute terms the maximum value of B 11 would correspond to 2 T = 20 kG. The temperature distribution was hardly affected by the magnetic field, therefore the isotherms are not reproduced here. Figs. 4a—4c show the computer velocity fields, on the attainment of the steady state, for various values of N. When viewing these figures, together with fig. 3c, showing the behavior for N = 0, at the first sight the major difference is the marked reduction in the numerical values of the linear velocity for the progressively increasing magnetic fields. Indeed on going from zero to 2 T an approximately thirtyfold reduction in the maximum velocity may be observed. It should be stressed, however, that the imposition of the magnetic field will also cause somewhat more subtle changes in the actual circulation pattern in the system, which are readily discernable on the streamline plots, given in figs. 5a—5d. For a zero magnetic interaction parameter there is an intense circulating ioop in the vicinity of the
melt—solid interface, while the remainder of the melt is relatively quiescent. On increasing the value of the magnetic interaction parameter the domain of the circulation is seen to spread gradually over the whole system. While the absolute value of the melt velocitieis will decrease with the increasing magnetic field, the corresponding spreading of the domain of the circulation will have a marked effect on the solute distribution as will be discussed subsequently. Fig. 6 provides an interesting, alternative insight into the behavior of the system by showing the time dependence of the maximum velocity, with the magnetic interaction as the parameter. It is seen that steady state is attained after about 20 s, and the higher the value of the magnetic interaction parameter, the lower will be the value of the steady state velocity. The velocity “overshoot” for intermediate times is evident on all these curves; in relative terms the overshoot appears to be higher for large values of the magnetic interaction parameter, which may be explained by the consideration that the buoyancy driving force has its highest values during the initial stages and that the time required to aced-
22 20
I
8 6 I
2
4 —
I 2
~uiiJ 0
8
>
6
2
0
2
4
6
~
10
12
14
6
18
20
TIME/5 .
.
.
4 .
Fig. 6. The maximum velocity (X 10 ) as a function of time with N as a parameter: (1) N = 1(2450 G), (2) N = 1(2450 G), (3) N
10 (7750 G), (4) N
=
69 (20000 G).
P AXIS [MM .0
2.0 I
C
4.0
b.r
I
I
P AXIS IMMI • --
-
8.0 I
a
10.1 I
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TIME- 20.20 SEE
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10.0 I
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:C
/
TIME=20. DLI SEC
2.0 I
~.
~//
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Fig. 7. Concentration distribution with N as a parameter after 20 s: (a) N = 0 (0 G). (b) N N = 69 (20000 G).
=
1 (2450 G). (c) N
=
10 (7750 G), (d)
416
G.M. Oreper. J. Szekely
/
Effect of magnetic field on transport phenomena
crate the relevant portion of the melt to the lower absolute values of the velocity attained at higher values of N, will of course be less. This behavior is readily apparent from the curves shown in fig. 6, with the maxima in the curves being displaced toward the origin for higher values of N. Figs. 7a—7d show the concentration isopleths computed for steady state, for progressively increasing applied magnetic fields. It should be stressed that the initial distribution of the solute was specified to be linear and had convection been totally absent, this linear distribution would have remained unaltered. Since the Schmidt number is large (Sc = 1000) one will expect quite a profound effect by the convective motion in the melt, even when convection is substantially reduced, through the application of the magnetic field. Inspection of figs. 7a—7d show, perhaps unexpectedly, that there are much more significant concentration non-uniformities in the absence of a magnetic field than for the cases when a field has been imposed on the system. More specifically, in the absence of an external magnetic field, the vigorous circulation (i.e. the vortex) in the vicinity of the melt—solid interface effectively prevents the transfer of the solute. In contrast, for the higher magnetic fields the circulation, while weak, cxtends over the whole of the domain thus the convective transport of the material is manifested throughout. Finally, for the very high magnetic field convection is suppressed, and more of the linear gradient is being retained. It is of interest to remark that the examination of the extent to which a tracer is dispersed by convection is a much more critical measure of the system behavior than would be the temperature profiles or the velocity fields themselves. Although a flat interface shape has been postulated in the calculations, the calculation of the actual melt—solid profile would be of considerable interest. A first order estimate of the shape of the melt—solid interface may be made by computing the local values of the heat flux at that position. A spatially uniform heat flux would give a flat interface, while a variable heat flux would indicate a correspondingly non-planar melt—solid boundary. The dimensionless local heat flux may be conveniently defined in terms of the local value of the
Nusselt number defined as Nu = —OT/OzI I
=
.1’
Fig. 8 shows a plot of the position dependence of the local Nusselt number at the melt—solid interface, i.e. at the z = A plane, with N, the magnetic interaction as a parameter. It is seen that in the absence of an external magnetic field there is a relatively strong spatial variation in the local heat flux (as reflected by Nusselt number); upon increasing the magnetic interaction parameter the local Nusselt number is made to diminish and will tend to become more uniform. It is of interest to note that the flattest profile of the local Nusselt number will be atained at an intermediate value of the magnetic interaction parameter. This finding has important implications regarding the application of magnetic fields on Bridgman type crystal growing systems and will be discussed subsequently. The fact that the Nusselt number is made to diminish with an increasing magnetic interaction parameter is illustrated further on fig. 9, the effect of N on the mean value of the Nusselt number. Ultimately the purpose of imposing a magnetic field has to be to promote the spatial uniformity of the impurity distribution within the solid phase. An indication of the extent of such non-uniformity
.22 2
~ .215
.2 I
0
2
.6
•~
•B
I
Fig. 8 Nusselt number distribution along the melt—solid interface: (1) N (4) N
=
0 (0 G), (2) N
69 (20000 G).
=
1 (2450 G), (3) N = 10 (7750 G),
G.M. Oreper, J. Szekely
/
Effect of magnetic field on transport phenomena
.22
20
2l5-
~I.2
21 0
lb
20
30
N 40
50
60
70
10
20
30
N
417
40
50
60
70
Fig. 9. Dependence of the average Nusselt number on N, the interaction parameter,
Fig. 11. Dependence of the average Sherwood number on N, the interaction parameter.
may be obtained by computing the local value of the Sherwood number, defined as
number is not immediately obvious. The spatial uniformity of the mass transfer rate
Sh I
appears be the greatest theand absence of an externallytoimposed magneticinfield the highest
=
OC/OzI /A’
Such a plot is presented in fig. 10 showing the spatial variation of the Sherwood number with N as the parameter. It is of interest to note that the effect of the external field on the local value of the Sherwood
2.8
2 4
~
non-uniformity occurs at an intermediate value of N. Finally at very high values of the magnetic interaction parameter one does obtain relatively uniform Sherwood numbers, the absolute values of which are greatly reduced. The important intermediate conclusion that has to be drawn from these results is that the imposition of a magnetic field for a Bridgman type system may not only not provide a more uniform impurity distribution, but in fact might make matters worse. Finally, fig. 11 shows a plot of the mean value of the Sherwood number against the magnetic interaction parameter, which indicates that Sh has
3
2 I
2.0
I 6-
a maximum at an intermediate value of N.
I 2
6. Discussion
4 4
~_~___=___________=~_____0)__=____=___
0 .2
.4
r .6
.8
L
Fig. 10. Sherwood number distribution along the melt—solid interface: (1) N = 0 (0 G), (2) N = 1 (2450 G), (3) N = 10 (7750 G), (4) N = 69 (20000 G).
mathematical formulation has been developed to represent the transient behavior of an electrically conducting liquid in a circular cylinder heated through the side wall and exposed to grayity and magnetic fields. This physical situation provides a somewhat idealized representation of heat and mass transfer and fluid flow in the melt during the crystal growth in an externally imposed magnetic field. Previous related studies were con-
41 8
G. M. Oreper. J. S:ekeli’
/
Effect of magnetic field on transport phenomena
cerned with natural convection between two paralid walls in a magnetic field [6,7] and more recently with the effect of an externally imposed magnetic field on the behavior of the Czochralski system or with experimental observations only (see review in ref. [1]). An efficient numerical technique has been constructed which enabled the generation of time dependent solutions for the temperature and the fluid flow fields and the distribution of the concentration. The principal finding of the work may he summarized as follows: (1) The role played by the magnetic field in affecting the behavior of the buoyancy driven circulation system may be readily quantified through the simultaneous solution of Maxwell’s equations, the Navier—Stokes equations and the differential thermal energy equation. (2) Calculations have shown that steady state has been attained under 20 s or less in real time: this period is very short compared to the duration of typical runs in crystal growth. (3) In a general sense, as expected for a system with a low Prandtl number (0.014) and a large Schmidt number the effect of an externally imposed magnetic field was slight on the temperature field and most marked on the tracer distribution. The effect on the velocity distribution was significant at high values of the magnetic interaction parameter. (4) In examining the detailed effect of an externaily imposed magnetic field on the velocity (or streamline) patterns two phenomena must be stressed. One intuitively obvious finding is that the absolute value of the velocity was markedly reduced by the imposition of the magnetic field, though even very large fields, in excess of several tesla, cannot completely stop the flow. The second perhaps more interesting finding is that the imposition of the field tended to change the overall character of the flow. For the input parameters considered in the absense of a field there existed a strong circulation pattern in the vicinity of the melt—solid interface, whith the rest of the of the melt being relatively quiescent. Upon the imposition of the magnetic field, while the absolute values of the velocity were reduced, the overall circulation became more uniform. This be-
havior has important implications on impurity distrihution. (5) In order to indicate the effect of convection and diffusion on the distribution of impurities, calculation were carried out to study the dispersion of a tracer in the system. It was found that in the absence of a magnetic field the intense circulation in the vicinity of the melt—solid interface resulted in a very marked distortion in the initially linear tracer profiles) more specifically the circulation yielded a “boundary layer” type behavior as far as the concentration was concerned, with steep gradients in the vicinity of the melt—solid interface. In contrast, in the presence of a strong magnetic field, the weaker hut more uniform circulation pattern resulted in a more homogeneous distrihution of the tracer. (6) A key consideration in assessing the role of an externally imposed magnetic field is to examine its effect on both the local and mean values of the heat and mass transfer rates. As far as heat transfer rates are concerned the mean value of the heat transfer coefficient (as expressed by the Nusselt number) is suppressed by the imposition of the magnetic field. The effect of the magnetic field on the local heat transfer coefficient is more interesting. Here it has been shown that the imposition of a magnetic field will tend to provide for a greater spatial uniformity of the heat transfer coefficient, particularly at the intermediate magnetic field levels. This in turn would correspond to a flattening of the melt—solid interface. While this h~s not been considered explicitly, modifications in the interface shape are likely to affect the flow behavior of the system also. (7) Examination of the mass transfer phenomena at the melt—solid interface have shown that the mean value of the Sherwood number (i.e. the dimensionless mass transfer coefficient) will undergo a maximum at intermediate levels of the magnetic interaction parameter. This behavior is consistent with the fluid flow phenomena described in earlier section in the paper. The intense circulation in the absence of magnetic field effects will tend to inhibit mass transfer, while the greatly reduced absolute values of the velocity at very high level of N will also tend to reduce mass transfer; at
G.M. Oreper, J. S:ekell’ / Effect of magnetic field on transport phenomena
intermediate values of N one would expect a maximum for the Sherwood number. The effect of the magnetic field on the local values of the Sherwood number is even more interesting, because it is seen that spatial uniformity of the local mass transfer coefficient may be attained at intermediate values of the magnetic interaction parameter. (8) Overall the most important conclusion which emerges from this work is that the interaction of a magnetic field with the fluid flow, heat flow and mass transfer phenomena in the Bridgman system is subtle, and quite complex. There are no hard and fast rules and the precise effects of an external magnetic field have to be examined on a case to case basis, especially as the optimal field strengths may be at some intermediate level, the absolute values of which would have to depend on the specific system parameters. The formulation and procedure outlined in this paper should provide the quantitative assessment of these phenomena. It should be remarked that the present work has been done under the assumption that the system is stable. The magnetic field usually suppresses turbulence and/or time dependent flows; while these problems are not addressed specifically in the present paper, the general methodology described here should represent a useful starting point in tackling these more complex problems.
Acknowledgements The authors wish to thank Professor A.F. Witt for suggesting the topic of this work, and thanks are also due to the National Aeronautics and Space Administration for partial support of this work under Grant No. NAG—3—365.
419
References [1] JR. Carruthers, in: Preparation and Properties of Solid State Materials, Vol. 3, Ed. W.R. Wilcox (Dekker, New York, 1973) p. 1. [2] G.M. Oreper and J. Szekely. J. Crystal Growth 64 (1983) 505. [3] 5. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability (Clarendon, Oxford, 1961). [4] W.B. Thomson, Phil. Mag. 42 (1951) 1417. [5] B. Lehnert and NC. Little, Tellus 9 (1957) 97. [6] G.Z. Gershuni and EM. Zhukhovitskii, Zh. Eksperim. Teor. Fiz. 34 (1958) 465. [7] G.Z Gershuni and E.M. Zhukhovitskii, Zh. Eksperim. Teor. Fiz. 34 (1958) 470. [8] H.P. Utech and J.G. Early. Acta Met. 15 (1967) 1238. [9] WE. Langlois and J.S. Walker, In: Proc. 2nd Intern. Conf. on Boundary and Interior Layers = Computational and Asymptotic Methods, 1982, Ed. J.J.H. Miller, p. 299. [10] WE. Langlois and Ki-Jun Lee, IBM Research Report, Ri 3677 (42663). [11] W.E. Langlois and Ki-Jun Lee, J. Crystal Growth 62 (1983) 481.
[12] S.F.
Liang, A. Vidal and A. Acrivos, J. Fluid Mech. 36 (1969) 239.
[13] Ci. Chang and R.A. Brown, J. Crystal Growth, submitted. [14] C.J. Chang and R.A. Brown, J. Computational Phys.. submitted. [15] R.A. Brown. Y. Yamagushi and C.J. Chang, in: Proc. 9th US Congr. on Applied Mechanics (ASME, New York, 1982) p. 381 [16] H.M. Ettouney and R.A. Brown, J. Computational Phys. 49 (1983) 118. [17] R.A. Brown, Fluid Flow in Crystal Growth, In: Annual Report of Material Processing Center, MIT (1981) p. 97. [18] AR. Gourlay, J. Inst. Math. AppI. 6 (1970) 375. [19] AR. Gourlay and G.R. McGuire, J. Inst. Math. AppI. 7 (1971) 216. [20] H.D. Gosman, W.M. Pun, A.K. Runchal, D.B. Spalding and M. Wolfshtein, Heat and Mass Transfer in Recirculating Flows (Academic Press, London, 1969). [21] EM. Nogotov, in: Application of Numerical
Heat
Transfer, Ed. B.M. Berkovsky, Consulting Ed. Minkowycz (Hemisphere, Washington, DC, 1978). [22] Y. Oshima, J. Phys. Soc. Japan 30 (1971) 872.
405
THE EFFECT OF A MAGNETIC FIELD ON TRANSPORT PHENOMENA IN A BRIDGMAN-STOCKBARGER CRYSTAL GROWTH G.M. OREPER and J. SZEKELY Department of Materials Engineering and Center for Materials Processing, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 28 November 1983; manuscript received in final form 29 May 1984
A mathematical formulation and computed results are presented to describe the temperature, velocity and tracer distribution in a Bridgman—Stockbarger crystal growing system, as affected by an externally imposed magnetic field. The computed results have shown that while the temperature profiles will not be significantly affected by the magnetic field, both the concentration fields and the velocity fields may be markedly influenced, especially at high values of the magnetic interaction parameter. In essence the imposition of a magnetic field will tend to both slow down the melt velocities and also modify the nature of the circulation pattern. Additional important effects of the magnetic field include its influence on the local and mean values of the heat and mass transfer rates at the melt solid interface. It was found that intermediate values of the magnetic interaction parameter (i.e. magnetic fields) would tend to provide the most uniform local heat transfer rates. While the velocities and mass transfer rates are damped by the magnetic field, the local value of mass transfer coefficient were the most uniform in the absence of an external magnetic field.
1. Infroduction
in the vicinity of the solid surfaces.
The purpose of this investigation is to develop quantitative understanding of how an externally imposed magnetic field affects the fluid flow field, the temperature field and the concentration profiles in a vertical Bridgman—Stockbarger crystal growing system. The principal motivation for this work is that the use of an externally imposed magnetic field has been suggested by several investigators in order to promote the uniformity of impurity distribution by suppressing convection and flow instabilities [1]. A detailed review of the literature on the application of magnetic fields in the crystal growth has been presented in the previous paper by the present authors, where we examined the role of an externally imposed magnetic field on the transient convection in a rectangular cavity [2], the walls of which were maintained at different temperatures. It was shown that the imposition of a magnetic field reduced the absolute values of the melt velocity and tended to flatten out the velocity profiles
This previous example represented a rather idealized picture of the Bridgman system, both in terms of the (rectangular) geometry selected and of the boundary conditions that were invoked; while this earlier work did provide an improved insight into the effect of an external magnetic field, it did not specifically address the effect of convection, as modified by the magnetic field, on the distribution of impurity or tracer elements. In the present paper we shall consider a more realistic geometry and thermal boundary conditions and will address the problem of the tracer distribution. The fundamental basis for the interaction between magnetic fields and convection has been discussed by Chandrasekhar [3] and Thomson [4]. Other interesting related work has been reported by Lehnert and Little [5], Gershuni and Zhukhovitskii [6,7] and Utech and Early [8]. These previously cited investigations did establish both the fundamental basis for melt and magnetic field interactions on the one hand and the concrete possibility of influencing crystal growth
0022-0248/84/$03.OO © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
406
G.M. Oreper, J. Szekely
/
Effect of magnetic field on transport phenomena
through the imposition of a magnetic field on the other. The first quantitative analysis of how an externally imposed magnetic field affects the behavior of the Czochralski system was presented by Langlois and Walker [9] and Langlois and Ki-Jun Lee [10,11]. However, the quantitative description as to how the imposed field would affect convection and hence crystal quality in geometries of relevance to the growth of single crystals by the Bridgman—Stockbarger technique has yet to be undertaken, The common method of growing single crystals involves the lateral supply of thermal energy, coupled with the axial removal of the latent heat of solidification and other associated heat losses from the system. Problems on thermal natural convection related to the crystal growth in the cylindrical geometry have been tackled by Liang, Vidal and Acrivos [12] and more recently by Brown and co-workers [13—17]. However, the effect of magnetic fields on convection in such geometries has not been undertaken up to the present. The purpose of this paper is to present a formulation and computed results on the effect of a magnetic field on thermal natural convection and on the concentration field in a circular cylinder that is the idealized representation of the Bridgman—Stockbarger system of crystal growth with the melt above the crystal. In general two sets of forces will act on this system: (1) the buoyancy force, due to the temperature gradients; (2) the Lorentz force, which is due to the interac tion between the induced current and the externally imposed magnetic field. It follows that the statement of the problem will have to involve the expression of: — the equation of continuity; — the equation of motion (Navier—Stokes equation) including the buoyancy and the Lorentz terms; — the thermal energy balance equation; — the mass balance equation; — Maxwell’s equations to define the Lorentz forces. These equations are mutually coupled.
2. Formulation for the velocity and temperature distribution Let us consider a circular cylinder of height L and of radius R as sketched in fig. 1. The liquid contained in the cylinder is initially at rest and the initial temperature distribution varies linearly from T= T 1 at the top to T0 at the bottom. At time t = 0 the side walls are subjected to a heat flux from the heater except the small region on the side wall near the center line which is adiabatically separated from the ambient medium, i.e. there is no heat flux through this part of the wall. The length of the adiabatic region is equal to 2R. At the same time the entire system is immersed in a constant magnetic field of strength B0, which acts in the z-direction. The problem is to find the time dependent spatial distribution of the temperature and
r- AXIS
HEATED
I
MELT
REGION B~l
ADIABATIC
-____________
~
Bi~O SOLID
Fig. 1. Sketch of the system.
G.M. Oreper, .1. Szekely
/
Effect of magnetic field on transport phenomena
the velocity and to examine the influence of the magnetic field on these quantities. The quantitative representation of this system has to include the following physical components: (1) Buoyancy driven flow will be established due to temperature gradients in the liquid resulting from the imposed boundary conditions. (2) The motion of an electrically conducting liquid in the presence of a magnetic field will give rise to Lorentz forces, Thus one has to solve the set of equations that govern the fluid flow field together with the energy equation and Maxwell’s equations in the moving medium. These equations are coupled because the fluid flow affects the convective heat transfer and the magnetic and the electric fields, while fluid flow itself is driven by the combined action of the buoyancy and the electromagnetic force field. The following assumptions were made in the statement of the problem: (1) The fluid is Newtonian. (2) The property values, such as viscosity, density, thermal conductivity, etc., are constant and evaluated at the temperature T = T0, the only exception being for the density dependence on the temperature to allow for the effect of buoyancy forces. (3) The flow is considered to be stable and laminar throughout. (4) The walls of the cylinder are electrical insulators. (5) The pulling rate of the crystal has a negligible effect on heat and fluid flow. (6) The viscous negligibly small. dissipation and Joule heating are (7) The melt—solid interface is flat. While these assumptions are somewhat restrictive they permit the formulation of the problem in a manageable form, while retaining the essential features of the interaction between a buoyancy driven flow and a magnetic field. The implication of the simplifying assumptions will be discussed subsequently. The problem may now be stated by expressing: — the conservation of matter, i~e.the equation of continuity; — the conservation of momentum, i.e. the equation of motion; — the conservation of thermal energy, i.e. the dif-
407
ferential thermal energy balance equation; — the electromagnetic force field equations and the associated boundary conditions. The general statement of the equations has been presented previously [2]. Here we shall state the governing equations in cylindrical coordinates. It is convenient to express the governing equations in a dimensionless form. Let us denote L. and L~as the characteristic lengths in the z and the r directions, U~and LJ~as the characteristic velocities in the z and the r directions. Then using the equation of continuity one may write that U, 0 = A L~, where A = L/L~ is the aspect ratio. On defining the dimensionless vorticity/r as: ~ ~
=
-~
— —
rA z r r 2>> 1, ~ c~L~ we note that since A 0A/L~. Now let us denote B0 as the characteristic value of magnetic induction and L~T=T1 — T0 as the temperature scale that equals the temperature drop along the solid and molten part of the medium. The dimensionless melting temperature was so chosen that T~1= 0.5. One may show that the scale for the current densityj0 B0/fleLr and that the proper scale for the electric field strength E0, is U~0B0. Let us define t L~vas the time scale. Now the governing equations may be expressed in the following dimensionless form: 3 O~ 2/ 0 0 r OT Re \ Oz Or 1 a ~ a~ 1 a / ~ — A2 Re ~ r~ — ~ ~~j-,: ~r
1
Gr .~±+ -~- ~ = A Re2 Or A2 Oz 0 (1 0’I’ \ 0 / 1 O’I’ \
~
~
+
~
(1)
~,
+
r~= 0,
(2)
where ur=—, Or
—u~r=——. Oz
Examination of eq. (1) shows that when the convective terms dominate the flow Gr/Re2A
408
G.M. Oreper, J. Szeke/v
/
Effect of magnetic field on transport phenomena
!~tOL~Ucü << I the magnetic field b
0(1) and the velocity scale
= B 0. If there is no outside circuit and walls of the cylinder are not
(v%/Gr/A )/Lc.
=
conductors it follows that the component of the electric field E4 = 0. The dimensionless quantities appearing in these equations are defined as in table 1. Eq. (1) represents the equation of motion, writ-
The differential thermal energy balance equation is written as: Prr~+
Pe(-~-(0ur)+~-(0u Oz Or
OT
) .~L Or I
r))
)
~
drivesside the represents flow, while last terms on the hand thethe “magnetic brake” i.e. left the electromagnetic force which will hinder the flow.
where 0 = (T — 1~)/L%Tis the dimensionless ternperature. The expression for the curl of the electromagnetic force in eq. (1) was derived using the fact that when constant electromagnetic field is applied and the magnetic Reynolds number Re 0 =
Eq. (3) is the thermal energy balance equation, which of course contains the time dependent velocity. It follows that eqs. (1) and (2) are mutually coupled. Eq. (2) is merely a relationship between the stream function and the vorticity.
—
.J.... 2
I .~ Oz ~r Oz
.~.
~ —
~
— 0‘ Or ~—
ten in terms of vorticity transport equation. The term 00/Or represents the buoyancy force, which
A
0
r=0,
The boundary conditions are given as:
~=0, Or
0
r=1,
~=Bi(T—T0),
A—I
r=1,
z=0, z=A,
0r1, 0r1,
—=0, Or 0=1, 0=0.5,
Table I
3/2v2
Gr=g$~TL Re U 0L /~ Pr = c/a Pc = U~0L~/a = p,~c~L~U~0 A L,/L~ 2/Re N Ha= = Ha B~L~/vp
Grashof number Reynolds number Prandtl number Péclet number Magnetic Reynolds number Aspect Parameter ratio of magnetic interaction . Hartmann number
r T
Dimensionless time Temperature
p g ‘s~ z’
Density of the liquid Acceleration due to gravity Magnetic permeability Kinematic viscosity
/3 a a
Coefficient of the thermal expansion Thermal diffusivity Electric conductivity
~=0, (4)
where Bi = 2hL,/k is the Biot number, h is the . . heat . transfer coefficient and k is the thermal conductivity of the melt. These relationships simply express the constraints that the stream function is zero at the solid boundaries and that the boundaries have a specified temperature or heat flux. . The boundary conditions . for the vorticity will be discussed subsequently, in connection with the . . . . . computational technique. The initial conditions are given as: T
.
=
0, 0
z
A, 0
1
=
—
z/2A, ~ = 0, ‘~I’= 0.
In physical terms these represent the stipulations .
.
.
.
that initially the fluid is at rest and that the temperature field is uniform.
G.M. Oreper, J. Szekely
/
Effect of magnetic field on transport phenomena
3. Formulation for the concenfration field The main motivation for studying convection in a crystal growing system is to asses the role of the convection on the impurity distribution. While the exact representation of this problem will require the knowledge of the partition coefficients and the solidification rates, a good preliminary insight may be obtained by considering the dispersion of a tracer (due to the diffusion and convection) which is deposited at the upper boundary and is implanted at the solid—liquid interface. The dispersion of this tracer may be represented by the equation expressing the mass balance
Ot
+(Uvi)C
=
2C,
D v
(6)
or in the dimensionless form Sc r~-~ + Pe (-~-(Cu r) +-~-(Cu r)\ O’r D~ Oz Or I 1 0 / OC \ 0 oc \ — —i ~— ~r-~—-) — ~— r-~—)= 0,
(7)
where Sc = v/D is the Schmidt number, D is the diffusivity and PeD = L,UcO/D is the Péclet number for mass transfer, Let us examine the influence of the magnetic field on the impurity concentration at the time when the temperature and the velocity fields have reached their steady state values and let us assume that the concentration of the impurity does not affect the fluid density. Furthermore, in computing the concentration profiles we shall only present the results corresponding to the steady state conditions. This equation is time dependent but we shall be concerned only with its solution for the time when steady state has been achieved. Hence the velocity distribution used in this equation corresponds to the time 20 s when steady state motion’ has been attained for all the cases considered. The necessary initial and boundary conditions are z = z = r= r=
0,
A, 0,
L,.,
T0,
C C
1, 0, OC/Or = 0, OC/Or 0, = =
0zA,
(8) C1z/A.
409
The first two boundary conditions express the fact that the concentration is constant at the solid—liquid interface and at a long distance from it. The second two boundary conditions specify that there is no mass flux through the axis of symmetry and side walls. The initial condition corresponds to a linear distribution of the concentration along the vertical direction.
4. The technique of solution In order to obtain a numerical solution, the governing equations were put into finite difference form using an 21 X 11 grid. The unsteady state equations for the temperature, concentration and vorticity distribution were solved through the “hopscotch” method, a detailed description of which is readily available in the computational literature [18,19]. The technique proposed by Spalding and co-workers [20] was used for solving the equation for the stream function. The vorticity at the solid walls was calculated in accordance with the method described by Nogotov and Berkovsky [211. The boundary condition for the vorticity invoked at the axis of symmetry specifies zero gradient of ~ at r = 0. 5. Results In the following we shall present a selection of the computed results. Due to the dimensionless form of the governing equations the independent variables entered these calculations in a dimensionless form also. The actual numerical value of the Grashof number was Gr = 6.585 X iO~so that the corresponding Raleigh number Ra = GrPr equalled 9.6 x iO~.The Biot number was unity. One should expect the flow to be stable for such conditions in the light of Oshima’s results [22]. In a practical, physical sense these conditions could correspond to a cylinder with a 7.5 mm radius, 150 mm high, holding molten silicon, with a z2iT value of about 400 K. This means that the temperature drop in the melt is 200 K and Re ~/Gr/A = 0.81 X 102. In the computation of mass transfer phenomena the Schmidt number was taken as 1000.
S—AXIS (MM I .0
2.0
_________________
4.0
H —AX IS MM I
5.0 I
__________________ —.
8.0 ~..
0.0
.0
.
________________
a
2.0
4.0
6.0
I
I
I
6.0
10.0
________________________________
___
TIME=2. 02 SEE
TIME=10. 01
SEC
H—AXIS IMMI .0
2.0
4.0
8.0
~0•
6.i -
10.0
__________
C
~ 5K 0)0
Fig. 2. Dimensionless temperature distribution in the absence __________________________________ TIME=20. 00 SEC
of a magnetic field at: (a) time 2 s, (b) time 10 s, (c) time 20 s.
H—AXIS 1MM) .0
2.0
4.0
6.0
I
I
I
Fl—AXIS (MM)
I
I
I
10.0
I
I
a
0
—
8.0
I’’’~
.0 .
2.0
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6.0
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I
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I
I
b
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I
o
I
~‘1’
I
J
Ill
I
I
0
I
VMAX=1.
50
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I
I
1’1 1
I I
I
I
I
II
0
I
lIlt
(
VMPX=1. 99 MM/S
I
I
~o
I!
I
/
0
0
I
/
J
/
/
\
TIME=2,02 SEC
~.
‘\
/
-
I
TIME=10.Ol SEC
H-AXIS IMMI .0
2.0
4.0
6.0
8.0
10.0
I
I
I
I
I
• T
VMAX=l. 75 MM/S
I
I
I
I
I
I
I
/
0
.
Ji
/
/1
/
I
I
I
I
..
I
0
\~ ‘O
C
//I
\ \ ~. — .- __________________________________ TIME=20.OQ SEC
Fig. 3. Velocity field in the absence of a magnetic field at: (a) time 2 s, (b) time 10 s, (c) time 20 s.
A—AXIS 1MM) 2.0
4.0
6.0
I
I
I
H—AXIS (MM) 8.0
a
0
• I I
I I
I
I I
I
I
I
I
P.
I
10.0
.0
~I
•
2.0 ___________________________
• •
•
• .
.
.
.I......... •
I
• •
I •
I
4.0
6.0
I
.
I
/ I /i’.
.
•
.
.
0
/I~
F.
J ]
VMAX=0. 87 MM/S
I
J/I
10.0
I
I
I
I
Il’’) .
6.0
lb
•
I
I
‘
/
‘/ll
/
.
I
I
I
I I
VMAX=O. 21 MM/S
r~.I
/
~.0.
/
/
I
I
‘
I
/
‘
\
/‘
0
I
I
I
~ ~o
1
I
Th
I
/1
/
-\ /
/
0
I
I
II
~.
II
\
TIME=20. 00 SEC
\~=-/
TIME=20. 00
SEC
A—AXIS IMMI .0
2.0 I
4.0
6.0
I
I
-
6.0
10.0
C —
I
I
I
I
/
0
I
I
I
/
I
/
I I
I
I
I
/
I
/ I
I
I
/
I
1 I
/
/
/1
/
/
0
I
/
I
I
I
VMRX=0. OS MM/S
/
I
/1
I o
I
/
\
\ \
\ ~‘
—
/
I
_______________________
TIME=20. 00 SEC
Fig. 4. Velocity distribution in the presence of a magnetic field after 20 s for: (a) N = 1 (2450 G), (b) N = 00 (7750 G), (c) N
=
69 (20000 G).
H—AXIS (MM) .0
2.0 I
4.0
6.0
I
I
H—AXIS IMMI 8.0
10.0
.0
a
TIME=20.00 SEC
2.12
4.0
6.0
I
I
I
TIME=20.00
~.
9—AXISIMMI .0
2.0 I
4.0 I
6.0 I
8.0
10.0
b
SEC
H—AXISIMMI 8.0 10.0 L......~
.0
2.0 I
4.0 I
6.0 I
8.0
10.0
(I~
TIME=20. 00 SEC
TIME=20. 00 SEC
~.
Fig. 5. Dimensionless stream function distribution (x jQ4) after 20 s for (a) N N = 69 (20000 G).
=
0 (0 G), (b) N
=
1(2450 G). (c) N
=
10 (7750 G). (d)
414
G.M. Oreper, J. S:ekelv / Effect of magnetic field on transport phenomena
Figs. 2a—2c show the transient development of the temperature profiles in the absence of a magnetic field. It is seen that steady state conditions are attained within about 20 s or less and that on reaching of steady state the temperature profiles are largely parallel, indicating that convection plays only a very minor role in affecting heat transfer. This point has been confirmed independently by computing the temperature profiles in the absence of convection, which were almost indistinguishable from the picture seen in fig. 2c. Figs. 3a—3c show the corresponding time dependent velocity fields, again in the absence of a magnetic field. The counterclockwise circulation pattern is readily apparent and the slight overshoot in the velocity field for the intermediate time is a logical consequence of the small Prandtl number. Inspection of fig. 3c shows that there is an intense circulation in the vicinity of the solid—melt interface, while the “far end” of the system is relatively quiescent. Let us now proceed by examining the effect of an externally imposed magnetic field. In these calculation the effect of the magnetic field was characterised by N, the magnetic interaction parameter, the values of which were allowed to range from 0 to 69. For the system chosen, in absolute terms the maximum value of B 11 would correspond to 2 T = 20 kG. The temperature distribution was hardly affected by the magnetic field, therefore the isotherms are not reproduced here. Figs. 4a—4c show the computer velocity fields, on the attainment of the steady state, for various values of N. When viewing these figures, together with fig. 3c, showing the behavior for N = 0, at the first sight the major difference is the marked reduction in the numerical values of the linear velocity for the progressively increasing magnetic fields. Indeed on going from zero to 2 T an approximately thirtyfold reduction in the maximum velocity may be observed. It should be stressed, however, that the imposition of the magnetic field will also cause somewhat more subtle changes in the actual circulation pattern in the system, which are readily discernable on the streamline plots, given in figs. 5a—5d. For a zero magnetic interaction parameter there is an intense circulating ioop in the vicinity of the
melt—solid interface, while the remainder of the melt is relatively quiescent. On increasing the value of the magnetic interaction parameter the domain of the circulation is seen to spread gradually over the whole system. While the absolute value of the melt velocitieis will decrease with the increasing magnetic field, the corresponding spreading of the domain of the circulation will have a marked effect on the solute distribution as will be discussed subsequently. Fig. 6 provides an interesting, alternative insight into the behavior of the system by showing the time dependence of the maximum velocity, with the magnetic interaction as the parameter. It is seen that steady state is attained after about 20 s, and the higher the value of the magnetic interaction parameter, the lower will be the value of the steady state velocity. The velocity “overshoot” for intermediate times is evident on all these curves; in relative terms the overshoot appears to be higher for large values of the magnetic interaction parameter, which may be explained by the consideration that the buoyancy driving force has its highest values during the initial stages and that the time required to aced-
22 20
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2
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8
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6
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2
4
6
~
10
12
14
6
18
20
TIME/5 .
.
.
4 .
Fig. 6. The maximum velocity (X 10 ) as a function of time with N as a parameter: (1) N = 1(2450 G), (2) N = 1(2450 G), (3) N
10 (7750 G), (4) N
=
69 (20000 G).
P AXIS [MM .0
2.0 I
C
4.0
b.r
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I
P AXIS IMMI • --
-
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TIME=20. DLI SEC
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Fig. 7. Concentration distribution with N as a parameter after 20 s: (a) N = 0 (0 G). (b) N N = 69 (20000 G).
=
1 (2450 G). (c) N
=
10 (7750 G), (d)
416
G.M. Oreper. J. Szekely
/
Effect of magnetic field on transport phenomena
crate the relevant portion of the melt to the lower absolute values of the velocity attained at higher values of N, will of course be less. This behavior is readily apparent from the curves shown in fig. 6, with the maxima in the curves being displaced toward the origin for higher values of N. Figs. 7a—7d show the concentration isopleths computed for steady state, for progressively increasing applied magnetic fields. It should be stressed that the initial distribution of the solute was specified to be linear and had convection been totally absent, this linear distribution would have remained unaltered. Since the Schmidt number is large (Sc = 1000) one will expect quite a profound effect by the convective motion in the melt, even when convection is substantially reduced, through the application of the magnetic field. Inspection of figs. 7a—7d show, perhaps unexpectedly, that there are much more significant concentration non-uniformities in the absence of a magnetic field than for the cases when a field has been imposed on the system. More specifically, in the absence of an external magnetic field, the vigorous circulation (i.e. the vortex) in the vicinity of the melt—solid interface effectively prevents the transfer of the solute. In contrast, for the higher magnetic fields the circulation, while weak, cxtends over the whole of the domain thus the convective transport of the material is manifested throughout. Finally, for the very high magnetic field convection is suppressed, and more of the linear gradient is being retained. It is of interest to remark that the examination of the extent to which a tracer is dispersed by convection is a much more critical measure of the system behavior than would be the temperature profiles or the velocity fields themselves. Although a flat interface shape has been postulated in the calculations, the calculation of the actual melt—solid profile would be of considerable interest. A first order estimate of the shape of the melt—solid interface may be made by computing the local values of the heat flux at that position. A spatially uniform heat flux would give a flat interface, while a variable heat flux would indicate a correspondingly non-planar melt—solid boundary. The dimensionless local heat flux may be conveniently defined in terms of the local value of the
Nusselt number defined as Nu = —OT/OzI I
=
.1’
Fig. 8 shows a plot of the position dependence of the local Nusselt number at the melt—solid interface, i.e. at the z = A plane, with N, the magnetic interaction as a parameter. It is seen that in the absence of an external magnetic field there is a relatively strong spatial variation in the local heat flux (as reflected by Nusselt number); upon increasing the magnetic interaction parameter the local Nusselt number is made to diminish and will tend to become more uniform. It is of interest to note that the flattest profile of the local Nusselt number will be atained at an intermediate value of the magnetic interaction parameter. This finding has important implications regarding the application of magnetic fields on Bridgman type crystal growing systems and will be discussed subsequently. The fact that the Nusselt number is made to diminish with an increasing magnetic interaction parameter is illustrated further on fig. 9, the effect of N on the mean value of the Nusselt number. Ultimately the purpose of imposing a magnetic field has to be to promote the spatial uniformity of the impurity distribution within the solid phase. An indication of the extent of such non-uniformity
.22 2
~ .215
.2 I
0
2
.6
•~
•B
I
Fig. 8 Nusselt number distribution along the melt—solid interface: (1) N (4) N
=
0 (0 G), (2) N
69 (20000 G).
=
1 (2450 G), (3) N = 10 (7750 G),
G.M. Oreper, J. Szekely
/
Effect of magnetic field on transport phenomena
.22
20
2l5-
~I.2
21 0
lb
20
30
N 40
50
60
70
10
20
30
N
417
40
50
60
70
Fig. 9. Dependence of the average Nusselt number on N, the interaction parameter,
Fig. 11. Dependence of the average Sherwood number on N, the interaction parameter.
may be obtained by computing the local value of the Sherwood number, defined as
number is not immediately obvious. The spatial uniformity of the mass transfer rate
Sh I
appears be the greatest theand absence of an externallytoimposed magneticinfield the highest
=
OC/OzI /A’
Such a plot is presented in fig. 10 showing the spatial variation of the Sherwood number with N as the parameter. It is of interest to note that the effect of the external field on the local value of the Sherwood
2.8
2 4
~
non-uniformity occurs at an intermediate value of N. Finally at very high values of the magnetic interaction parameter one does obtain relatively uniform Sherwood numbers, the absolute values of which are greatly reduced. The important intermediate conclusion that has to be drawn from these results is that the imposition of a magnetic field for a Bridgman type system may not only not provide a more uniform impurity distribution, but in fact might make matters worse. Finally, fig. 11 shows a plot of the mean value of the Sherwood number against the magnetic interaction parameter, which indicates that Sh has
3
2 I
2.0
I 6-
a maximum at an intermediate value of N.
I 2
6. Discussion
4 4
~_~___=___________=~_____0)__=____=___
0 .2
.4
r .6
.8
L
Fig. 10. Sherwood number distribution along the melt—solid interface: (1) N = 0 (0 G), (2) N = 1 (2450 G), (3) N = 10 (7750 G), (4) N = 69 (20000 G).
mathematical formulation has been developed to represent the transient behavior of an electrically conducting liquid in a circular cylinder heated through the side wall and exposed to grayity and magnetic fields. This physical situation provides a somewhat idealized representation of heat and mass transfer and fluid flow in the melt during the crystal growth in an externally imposed magnetic field. Previous related studies were con-
41 8
G. M. Oreper. J. S:ekeli’
/
Effect of magnetic field on transport phenomena
cerned with natural convection between two paralid walls in a magnetic field [6,7] and more recently with the effect of an externally imposed magnetic field on the behavior of the Czochralski system or with experimental observations only (see review in ref. [1]). An efficient numerical technique has been constructed which enabled the generation of time dependent solutions for the temperature and the fluid flow fields and the distribution of the concentration. The principal finding of the work may he summarized as follows: (1) The role played by the magnetic field in affecting the behavior of the buoyancy driven circulation system may be readily quantified through the simultaneous solution of Maxwell’s equations, the Navier—Stokes equations and the differential thermal energy equation. (2) Calculations have shown that steady state has been attained under 20 s or less in real time: this period is very short compared to the duration of typical runs in crystal growth. (3) In a general sense, as expected for a system with a low Prandtl number (0.014) and a large Schmidt number the effect of an externally imposed magnetic field was slight on the temperature field and most marked on the tracer distribution. The effect on the velocity distribution was significant at high values of the magnetic interaction parameter. (4) In examining the detailed effect of an externaily imposed magnetic field on the velocity (or streamline) patterns two phenomena must be stressed. One intuitively obvious finding is that the absolute value of the velocity was markedly reduced by the imposition of the magnetic field, though even very large fields, in excess of several tesla, cannot completely stop the flow. The second perhaps more interesting finding is that the imposition of the field tended to change the overall character of the flow. For the input parameters considered in the absense of a field there existed a strong circulation pattern in the vicinity of the melt—solid interface, whith the rest of the of the melt being relatively quiescent. Upon the imposition of the magnetic field, while the absolute values of the velocity were reduced, the overall circulation became more uniform. This be-
havior has important implications on impurity distrihution. (5) In order to indicate the effect of convection and diffusion on the distribution of impurities, calculation were carried out to study the dispersion of a tracer in the system. It was found that in the absence of a magnetic field the intense circulation in the vicinity of the melt—solid interface resulted in a very marked distortion in the initially linear tracer profiles) more specifically the circulation yielded a “boundary layer” type behavior as far as the concentration was concerned, with steep gradients in the vicinity of the melt—solid interface. In contrast, in the presence of a strong magnetic field, the weaker hut more uniform circulation pattern resulted in a more homogeneous distrihution of the tracer. (6) A key consideration in assessing the role of an externally imposed magnetic field is to examine its effect on both the local and mean values of the heat and mass transfer rates. As far as heat transfer rates are concerned the mean value of the heat transfer coefficient (as expressed by the Nusselt number) is suppressed by the imposition of the magnetic field. The effect of the magnetic field on the local heat transfer coefficient is more interesting. Here it has been shown that the imposition of a magnetic field will tend to provide for a greater spatial uniformity of the heat transfer coefficient, particularly at the intermediate magnetic field levels. This in turn would correspond to a flattening of the melt—solid interface. While this h~s not been considered explicitly, modifications in the interface shape are likely to affect the flow behavior of the system also. (7) Examination of the mass transfer phenomena at the melt—solid interface have shown that the mean value of the Sherwood number (i.e. the dimensionless mass transfer coefficient) will undergo a maximum at intermediate levels of the magnetic interaction parameter. This behavior is consistent with the fluid flow phenomena described in earlier section in the paper. The intense circulation in the absence of magnetic field effects will tend to inhibit mass transfer, while the greatly reduced absolute values of the velocity at very high level of N will also tend to reduce mass transfer; at
G.M. Oreper, J. S:ekell’ / Effect of magnetic field on transport phenomena
intermediate values of N one would expect a maximum for the Sherwood number. The effect of the magnetic field on the local values of the Sherwood number is even more interesting, because it is seen that spatial uniformity of the local mass transfer coefficient may be attained at intermediate values of the magnetic interaction parameter. (8) Overall the most important conclusion which emerges from this work is that the interaction of a magnetic field with the fluid flow, heat flow and mass transfer phenomena in the Bridgman system is subtle, and quite complex. There are no hard and fast rules and the precise effects of an external magnetic field have to be examined on a case to case basis, especially as the optimal field strengths may be at some intermediate level, the absolute values of which would have to depend on the specific system parameters. The formulation and procedure outlined in this paper should provide the quantitative assessment of these phenomena. It should be remarked that the present work has been done under the assumption that the system is stable. The magnetic field usually suppresses turbulence and/or time dependent flows; while these problems are not addressed specifically in the present paper, the general methodology described here should represent a useful starting point in tackling these more complex problems.
Acknowledgements The authors wish to thank Professor A.F. Witt for suggesting the topic of this work, and thanks are also due to the National Aeronautics and Space Administration for partial support of this work under Grant No. NAG—3—365.
419
References [1] JR. Carruthers, in: Preparation and Properties of Solid State Materials, Vol. 3, Ed. W.R. Wilcox (Dekker, New York, 1973) p. 1. [2] G.M. Oreper and J. Szekely. J. Crystal Growth 64 (1983) 505. [3] 5. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability (Clarendon, Oxford, 1961). [4] W.B. Thomson, Phil. Mag. 42 (1951) 1417. [5] B. Lehnert and NC. Little, Tellus 9 (1957) 97. [6] G.Z. Gershuni and EM. Zhukhovitskii, Zh. Eksperim. Teor. Fiz. 34 (1958) 465. [7] G.Z Gershuni and E.M. Zhukhovitskii, Zh. Eksperim. Teor. Fiz. 34 (1958) 470. [8] H.P. Utech and J.G. Early. Acta Met. 15 (1967) 1238. [9] WE. Langlois and J.S. Walker, In: Proc. 2nd Intern. Conf. on Boundary and Interior Layers = Computational and Asymptotic Methods, 1982, Ed. J.J.H. Miller, p. 299. [10] WE. Langlois and Ki-Jun Lee, IBM Research Report, Ri 3677 (42663). [11] W.E. Langlois and Ki-Jun Lee, J. Crystal Growth 62 (1983) 481.
[12] S.F.
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[13] Ci. Chang and R.A. Brown, J. Crystal Growth, submitted. [14] C.J. Chang and R.A. Brown, J. Computational Phys.. submitted. [15] R.A. Brown. Y. Yamagushi and C.J. Chang, in: Proc. 9th US Congr. on Applied Mechanics (ASME, New York, 1982) p. 381 [16] H.M. Ettouney and R.A. Brown, J. Computational Phys. 49 (1983) 118. [17] R.A. Brown, Fluid Flow in Crystal Growth, In: Annual Report of Material Processing Center, MIT (1981) p. 97. [18] AR. Gourlay, J. Inst. Math. AppI. 6 (1970) 375. [19] AR. Gourlay and G.R. McGuire, J. Inst. Math. AppI. 7 (1971) 216. [20] H.D. Gosman, W.M. Pun, A.K. Runchal, D.B. Spalding and M. Wolfshtein, Heat and Mass Transfer in Recirculating Flows (Academic Press, London, 1969). [21] EM. Nogotov, in: Application of Numerical
Heat
Transfer, Ed. B.M. Berkovsky, Consulting Ed. Minkowycz (Hemisphere, Washington, DC, 1978). [22] Y. Oshima, J. Phys. Soc. Japan 30 (1971) 872.