AlAA 93-0467 CENTRIFUGE IN SPACE FLUID FLOW VISUALIZATION EXPERIMENT

William A. Arnold and William R. Wilcox Center for Crystal Growth in Space Clarkson University Potsdam, New York 13699

Liya L. Regel International Center for Gravity Materials Science and Applications Potsdam, New York 13699

Bonnie J. Dunbar Lyndon B. Johnson Space Center Houston, Texas 77058

31 st Aerospace Sciences Meeting & Exhibit

January 11 -1 4, 1993 / Reno, NV For permisslon to copy Or republlsh, contact the American Institute of Aeronautlcs and Astronautics 370 L'Enfant Promenade, S.W., Washlngton, D.C. 20024

CENTRIFUGE IN SPACE FLUID FLOW VISUALIZATION EXPERIMENT

William A. Arnold" and William R. Wilcox* Center for Crystal Growth in Space

Clarkson University Potsdani. New York 13699

W

Liya L. Regel* International Center for Gravity Materials Science and Applications

Potsdam, New York 13699

and

Bonnie J. Dunbar Lyndon B. Johnson Space Center

Houston, Texas 77058

Abstract Aprototype flow visualization system was built

to investigate buoyancy driven flows during centrifuga- tion in space. The flow test cell consisted of a quartz glass ampule with aluminum end caps, filled with water. An axial density gradient was set up by imposing a ther- mal gradient between the two ends of the test cell. Radial density gradients were controlled by machining a radius of curvature into the aluminum end caps. Numerical computations for this geomehy indicated that the Prandtl number plays a weak role in determining the flow. The thermal field of the fluid is governed almost entirely by conduction, so that a difference in Pr number changes only the speed of the flow, but not the characteristic flow patterns, if the Rayleigh number is held constant. Thus, it can be deduced which parameters influence the buoy- ancy driven flow, such as the radial or the axial temperature gradients, aspect ratio, etc. Although the test cell focuses on fluid physics, it has direct relevance to di- rectional solidification of low Prandtl number semiconductor materials.

Ground based fluid flow results of the test cell at l g are presented (here g =9.81m/s2 is the acceleration due to earths gravity). Flow mode patterns are demon- strated by use of a tracer dye. It is shown that this system is capable of detecting buoyancy driven flows as slow as a few micrometerslsecond. Experimentally obtained flow patterns and thermal profiles are compared with nu- merical predictions.

J

Backaround Very little basic materials processing or fluid

dynamics research has been done in the gravitational range 10-6g to lg. Basic research is needed to improve the understanding of the influence of gravity on materials processing and fluid flow processes. A centrifuge in space is needed to accomplish this research over the gravity spectrum up to l g because of the inescapable l g background on earth. Fluid flow research in centrifuges on earth is very complicated, and has a lower accelera- tion bound. In a centrifuge the acceleration field is not homogeneous. Experimental conditions can be tailored such that the buoyancy driven flow is effectively driven by a homogeneous acceleration field. Alternatively, the experimental conditions can be altered so that the buoy- ancy driven flow responds to thc inhomogeneity of the acceleration field. The buoyancy driven flows that occur can be characterized by a new controlling nondimension- al number, Ad'. The Ad number govern the effect of the inhomogeneity of the acceleration field on the buoyancy driven convection.

Rotating fluids can have several unique and counter-intuitive qualities. Examples are geostrophic flows', Taylor columns', the stabilizing effect of the Co- riolis force on buoyancy driven flow3, and the recent experimental evidence on the apparent suppression of convective transport at a well defined acceleration level Ng during solidification of semiconductor materials in a

The last of the listed phenomena is of - Copyright 0 1993 by William A. Arnold. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

most interest to us, as it has direct applicability to the CrYStal growth industry. The apparent convective SUP- pression leads to nearly uniform axial doping in the

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1 * Member AIAA

crystal. To date, no theory can explain this phenomenon. The special acceleration level depended on the ann length of the cenhifuge used to directionally solidify Ag- doped PbTe crystals. It is still not known how the physi- cal driving forces combine to result in uniform axial doping, as would be expected only in the absence of con- vection. It has been suggested that the controlling factor may be the rotation rate and not the average acceleration level.'

In order to investigate the fluid flow phenome- na a flow visualization system was needed. There are no known semiconductor systems that are transparein in the visible spectrum. The test cell presented here uses water, which has a relatively high Prandtl number. This must be taken into account when relating the fluid flow results to low Prandtl semiconductor melts.

Tvoical Crystal Growth Apparatus

Some experimental features of crystal growth in a Bridgman furnace are shown in figure 1. The important feature is the unavoidable radial temperature gradients. These gradients result in convection in the melt." In the idealized case of flat isotherms even a slight misalign- ment of the gravity vector with the ampule causes natural convection in the melt. Although the work presented here focuses primarily on the fluid dynamics of rotating fluids, and in particular a rotating fluid in space, it also applies directly to crystal growth of good thermal con- ductors.

Governina Eauations

For crystal growth of semiconductors by the Bridgman technique, the Boussinesq approximation should be valid for the convective terms and the Coiiolis acceleration, since the temperature gradients are relative- ly low and because of the coefficient of themal expansion is small (1 >> P*AT*). The goveming Navier- Stokes equations, with the Boussinesq approximation applied to the convective terms and the Coriolis acceler- ation, in dimensional form are:

dx *u* = 0 (1)

p,* (4,. +u* .V*&*) = -v*p * at

Y

Perfect Furnace Real Furnace

iA w Solid

Fimre 1. Bridgman crystal growth and temperature variations leading to convection.

PO*% * (F a T* +fi* . V * P ) = V* ( k * V * r )

+ 4: (3)

The equations could be nondimensionalized by choosing the following scalings for the dimensional quantities:

u* = [N~*~*(T*H-T*~),'*]I'

ii = ii*/v*

ii = R*/P p = p*r*/U*K*

t = t*U*/r*

T - (T* -T*c)

(T*H-PC)r

Introducing these scalmgs into the goveming equations, with the material properties constant and in the absence of internal heat generation, yields:

V . 6 - 0 (4)

(6) Pr.JGi ( L T + & VT) = V2T at

where i is a unit directional and

'W4

2

a TOD Aluminum End Cap

UI 4/40 Threaded end rod with nuts

d

-. The prototype flow mode test cell, cross-sectional slice shown.

Gr = p*2Ng*fi*(T*H-T*~)$*3/)t*2 Pr = p*C,*k*

Ta = 4w*2r*4/v*2 ATL = r*/l*

Ar, - (r*H-r*C)r

(Y €I-? c),

* 2 * T* w 1 A a

Ad - The final resulting set of equations in nondmensional form is:

v-ii = o (7)

&($ + f Vf) P - Vp - &(Sg) T + V2ii

- ArTArLAd ( &) T [ (6 - E,) ]

-fi (8, x 6) (8)

t

M. Estimation of the radial temperature gradient near the end cap.

Flow Mode Test Cell

The prototype test cell is shown in figure 2. It consisted of a 12 mm I.D. by 16 mm O.D. quartz tube with aluminum end caps. Water was used for the fluid in- side. When needed, small amounts of deuterated water, glycerin, alcohol or salt were added for density matching the fluid to the tracer particles or dye. Monobasic alumi- num stearate was used initially as the tracer particle material and salt used for density matching. Ivory hand soap was then used to leave a streak enabling visualiza- tion of the flow. The ampule was sealed by an O-ring in the groove in the aluminum end cap, into which the quartz ampule was compressed. Mechanical stress was taken off the quam ampules by stainless steel tubing stops. Heat transfer inhomogeneities, due to the air encir- cling the inner quartz tube, can be eliminated by the use of an outer quartz tube that serves as a wind break. An axial temperature gradient was set up by a heater embed- ded inside the top end cap to yield the so-called thermally stable condition. Radial temperature gradients were in- duced by the curvature in the end caps. In the twt cell, the end caps were actually inserts, thereby allowing the shape of the interface to be changed by machining addi- tional inserts. A 12 inch long, type T, sheathed thermocouple was coiled and used as the heater.

Numerical computations for the prototype test cell geometry indicated that results obtained with the rel- atively high Prandtl water could be related to low Prandtl number flows. In the test cell fluid, the thermal field is

3

w. Modeled portion of the test cell.

governed almost entirely by conduction, thus the thermal field is controlled by the test cell geometry. In the fluid, a difference in Pr number only changes the character of the flow, but not the characteristic flow patterns.

Radial temperature gradients are controlled by the radius ofcurvatwe machined into the end caps. From figure 3 it is seen that the temperature in the center of the cell can be approximated by:

(T*& e (T*& + (T*H-T*c)~Ax* (10)

which leads to an approximation for the radial tempera- ture difference:

(T*H-T*& (T*& + [(T*a-T*&/l*I AX* - (T*& = (T*H-T*& AX*/^* (11)

Thus, the depth of the machined interface determines the radial temperature difference near the interface for a giv- en axial temperature difference and axial length.

Numerical Mode I

In this section are results obtained from a fully nonlinear two-dimensional axisymmetric numerical model for the experimental te$t cell described above. The model included test cell geometry and temperature de- pendent material properties. Both steady-state and transient results are presented. The objectives were to test the feasibility of using a highprandtl number system,

to inspect the impact on the flow of heat loss through the ampule wall, to compute the power required for this sys- tem, and to visualize the expected flow modes. The model was used for the gravity vector perfectly aligned with the ampule axis. Implications of the resulting flow for low Prandtl number semiconductor crystal growth are also discussed.

The model used for this study is shown in figure 4. Numerical work was initially done before the test cell was built with the outer quartz ampule (wind break) in place. The numerical work presented here was done after the test cell was run and does not include the outer quartz ampule. Internal heat transfer in the various materials comprising the test cell was treated as a conjugated prob- lem. Each O-ring groove contains a rubber O-ring, air, and water. Because of the complicated geometry of the various materials filling the O-ring groove and the small volume these materials occupy, the O-ring groove area was approximated as a single material and given a weighted average of the rubber, air and water material properties. Those areas are shown as black squares in fig- ure 4. Similarly, the heater was given a weighted average of the material properties of the thermocouple and air. The bottom face of the bottom end cap was held at room temperature because the test cell was set on a large alu- minum block. Convective boundary conditions were applied to the outer ampule boundary and end caps. As- suming anemissivity of one, the linearized radiative heat transfer coefficient has a value of 5.8 WPC-m' at 295°C and 7.5 WPC-m' at 345°C. Convective heat transfer co- efficients typically have a value of 2-4 WPC-m' when the Grashof number is small. A constant heat transfer co- efficient that approximated the combined radiative and convective effects, having a value of IO (WPC-m'), was used on all outer surfaces. The temperature of the ambi- ent air taken as 22 "C. Because of the low temperatures and complicated geometry involved, exact radiative heat transfer was not incorporated into the model.

ip

- Numerical Methods

The above set of equations in dimensional fonn was solved using a modified version of FIDAP, a finite element basedcode? Nan-slip boundary conditions were unposed on all solid walls. For most of the results pre- sented, the test cell was assumed to be at steady state. The steady state flows here approximate the pseudo steady-state flows present during crystal g r ~ w t h . ' ~ Tran- sient analyses were also used for flow stability verification. In all cases a fixed-grid approach (nodal points spacially fixed) was used.

The results presented hereafter were checked for convergence to within a specified absolute tolerance L, '

4

v' Top Aluminum End Cap

L.

Heater

-n n

Bottom Aluminum End Cap

Egug.5. Calculated isotherms inside the test cell (1 "C between isotherms).

of 0.0001 for both the normalized velocity and the resid- ual error norms. Spatial convergence was ascertained by comparing results obtained with different grid spacings. Typical simulations under normal, high, and micro-grav- ity conditions involved approximately 5600 nodes using 9 node isoparamehic quadrilateral elements for the 2-D simulations. The thennophysical properties of the vari- ous materials are listed in table I.

d

Numerical Results

An axial thermal gradient in the range of 5 'C to about 25 'C/cm was selected to conform to typical crys- tal growth conditions. These temperature gradients along with the depth of the machined end caps were expected to produce radial temperature gradients similar to those encountered during gradient freeze crystal growth. The interface depth used was 1 mm. From the scaling analy- sis, it was expected that the flow velocity would be different and that the flow patterns in the test cell would not be exactly replicated to that ofcrystal growth, due the differences in Grashof and Prandtl numbers. However, the flow character (Le. flow mode) was expected to be alike. With the length of the ampule set at 2.4 cm (2tol aspect ratio), a maximum axial temperature difference of about 60 "C is required. If the bottom cap is near room

Axial position (em)

EgxcJj. Radial temperature difference (OC) as a func- tion of axial position inside the cell.

temperature, the top cap must be near 80°C. The numerical model predicted that 4.3 watts of

power would be required to attain a top cap temperature of 77 "C. The resulting thermal field in 1 g conditions is shown in figure 5. Note that the end caps, especially the bottom, are nearly isothermal. In the immediate vicinity of the heater, temperatures are about 2 degrees higher than the rest of the top cap. The isotherms in the water next to the aluminum end caps follow the curvature of the interface. This result was expected due to the high thermal conductivity of the aluminum in comparison with the water. With 4.3 watts of power applied, the tem- perature of the top cap was 77 "C, while the bottom cap stayed near the ambient temperature of 22 OC. The axial temperature difference is about 55 OC. Figure 6 shows the radial temperature difference between the inner am- pule wall and the center of the water as a function of axial position. The maximum radial temperature difference in

F,@.EJ.. Transient thermal response at a point on the axis near the middle of the test cell.

5

. Top Aluminum End Cap

Bottom Aluminum End Cap

w. Computed streamlines inside the test cell.

the cell is approximately 3 OC at the top and 1.5 OC at the bottom. The average of the calculated maximum radial temperature differences agrees very well with that pre- dicted by use of equation 11,2.25 OC. The effect of machining a radius of curvature into the end caps lo in- duce radial temperature gradients is seen. Experimentally, a steady thermal condition was desired. The transient thermal response was simulated for a con- stant heater power. Initially, the test cell is at a corntaut temperature of 22 OC. The resulting transient tempera- ture for a point located on the axis and near the middle of the test cell is shown in figure 7. The thermal profile of the cell changes little after about 30 minutes. A natural limitation imposed on the test cell is to keep the top cap below the boiling point of water.

testcellareshowninfigure8. Twoprimary toroidal cells arepresent.Eachprimaqcellisveryneartotheinterface machined into the aluminum end cap. In addition, two secondary toroidal cells are seen. The upper secoiidiuy cell transits most of the test cell length. The lower, weak- er secondary cell next to the bottom primary cell is confined to a small region. The maximum fluid velocity here is 116 p d s .

The calculated streamlines in the water of the

i

E~LXE.~. Tracer soap pattern inside the test cell 5 minutes after introduction of the soap.

ExDerimental Results

Only the tracer dye results are presented here. The test cell was run at lg, with the gravitational accel- eration vector aligned with the ampule. The interface depth that was machined into the end caps was about 1.5 mm. This increased interface depth should not have changed the cellular pattern from the 1 mm interface ap- preciably, but has the effect of increasing the flow velocity by a factor of about upon a large aluminum metal block that effectively kept the bottom cap at room temperature of 22 OC.

The cell was bolted together and filled with wa- ter from a hypodermic needle inserted into the water vent hole. No air bubbles were in the cell initially. Tap water was used and density matching was not done. The outer wind break was not used. Five watts of power (12 volts) was applied to the heater and the cell was allowed to sit for an hour, in order to reach steady state. Some of this power was lost in the power leads and the short length of the thermocouple heater outside to the cell, as the portion of the thermocouple heater outside the cell was warm to the touch. The top cap of the cell attained a temperature of 76 'C. The axial temperature difference with this ap- plied power was 54 OC. The 12 volt figure was chosen because we initially had planned to run the test cell on batteries.

'b'

The cell was set

A very small amount of liquid Ivory hand soap 'L-:

6

Fipllre IO. Tracer soap patterns inside the test cell 25 minutes after introduction of the soap.

was then put on the tip of a hypodermic needle and insert- ed into the cell. A small drop of soap initially came off the needle. This soap drop was slightly heavier thanthe water and settled toward the bottom of the cell. As the drop set- tled it left a streak trail, diluted, and eventually diluted to a density nearly the same as the water. Thus, initially the soap tracer appeared as an axial streak line, as shown in the center of figure 9. Bright lights were turned on only when photographing the tracer flow patterns and then turned off. The lights were turned off so as not to intro- duce buoyancy driven flow due to the tracer dye absorbing light and increasing its temperature. The above effect has been observed in similar experiments. The soap streak was not visible to the naked eye, except under bright lights.

cy driven flow and was incorporated into the flow cells. The cellular pattern 5 minutes after introduction of the soap streak is shown in figure 9. The secondary cells at the top of the test cell are visible early in the run. The bot- tom primary cellular pattern is just beginning to form. Figure 10 shows the flow pattern 25 minutes after intro- duction of the soap streak. However, by 25 minutes the soap at the top of the cell bad mixed and the top cells were no longer visible. This was a 3 dimensional pattern, but the centers of the cells and their character are clearly vis- ible. The bottom primary cell flow pattern was nearly the same when viewed from any position around the cell, in-

d

The soap streak began to move with the buoyan-

J

1

Figure 11. Computed streamlines inside the test cell for a low Prandtl, but identical Rayleigh number fluid.

dicathg it was axisymmetric. This is referred to as the axisymmetric flow mode in the crystal growth commnni- ty.” We believe this is the fnst photograph of a slow buoyancy-driven axisymmetric flow. Many in the crystal growth community had even doubted this flow mode’s existence. The lower secondary cell can also be seen in figure IO on the right hand side, but is absent from the left hand side. This asymmetry in the secondary cell may be due to the inability to align the test cell perfectly with the gravity vector. From following the streak as a function of time, flow velocities were determined to be on the order of 70 p / s . This is in very good agreement with the nu- merical model, which predicted maximum velocities of about 116 W s . The small “puffs” of the trace soap seen in the center of the cell may have been due to small heat- ing effects of the illuminating lights. The velocities were very small in that region and any perturbations such as light heating the tracer soap would beexpected to persist, since there are no convective effects. There is good agree- ment between the experimental results and the numerical predictions, as seen by comparing figures 8,9, and 10.

&levance of to Crvstal Growth

As stated earlier, we are interested in using the results obtained from the test cell to understand crystal growth. The numerical model was m again under near identical thermal conditions and at the same Rayleigh

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. number, except that a low Prandtl number was used. Figure 11 shows the streamlines in the fluid. Their patteni is nearly identical to the streamlmes of the water case. The primary to- roidal cells have spread out a little. The secondary cells are still intact and have also spread. The cells have the same ro- tation sense and character as their high Prandtl number counterparts. The similarity exists because the same influ- ence on the buoyancy driven flow is effecting both flows, the radial temperature differences. Also, as stated above both computations were done at the same Rayleigh number. Max- imum velocities here are 956 p d s , as compared to that of water of 116 Ws. The order of magnitude difference in ve- locities is solely due to the difference in Prandtl number, and occurs because the energy equation becomes strongly cou- pled to the momentum equation for high Pr number systems. Differences like these must be realized and accounted for when attempting to compare the high Prandtl number water results to low Prandtl number fluids. Results from other wa- ter based flow visualization systems may not be comelatable to their low Prandtl number counterparts.

Conclusions

A test cell for investigating slow buoyancy driven flows in a centrifuge in space was built and tested in OUT lab. Numerical and experimental results are in good agreement. The cell operated with low power. Axial and radial tempera- ture gradients comparable to that experienced during crystal growth can be obtained. The radial temperature gradient can be adjusted by changing the radii of cufvature machined into the end caps.

A comparison of the fluid flow pattems can be made between the test cell and results obtained with low Prandtl number fluids with similar thermal fields and Ray- leigh numbers. These flow pattems can be correlated to parameters influencing the buoyancy-driven flow. For the re- sults presented here, the flow was driven by the radial temperature gradient. It was also shown that the gravity vec- tor can be aligned close enough with an ampule to produce the axisymmetric flow mode.

In the numerical model, the Boussinesq approxima- tion was employed. A by-product of this research is that the Boussinesq approximation was apparently validated by the good agreement between numerical and experimental re- sults.

Future Directions

If the acceleration vector is not aligned with ampule axis, the buoyancy driven flow begins to be driven off the ax- ial temperature gradient. A flow mode transition occurs within a few degrees of misalignment and is well document- ed for microgravity conditions." We plan to use the cell and additional numerical modeling to correlate alignment sensi- tivity with parameters influencing buoyancy driven flow, i.e. radial and axial temperature differences.

A 5 foot ami, 90 rpm, 13.5 maximum g centrifuge is nearing completion at Clarksou University. Investigation of buoyancy driven flow will be done on this terrestrial cen- wifuge at higher than earth's gravity acceleration levels. Here, the quartz windbreak may be used to lower power con- sumption and reduce heat loss off the inner quartz ampule. The quartz windbreak not only acts to reduce convective losses, but will also act to reduce radiative losses, since quartz is opaque throughout much of the infrared spectrum. This research will lay the groundwork to develop a buoyancy driven fluid flow experiment for use in a space centrifuge.

'd

Acknowledaments

This work was sponsored by the NASA Graduate Student Researchers Program under grant NAGW-976. We also wish to thank Frederick Carlson of Clarkson University, and George Foester and Barry Licht of NASA LeRC for their very helpful suggestions on the design of the test cell.

Nomenclature

Dimensional Ouantities

i* Gravity Vector k* Thermal Conductivity 1* p* . Pressure r*

R* Radius Vector From Axis

EO* T* Temperature t* Time i* Vector Velocity Ax* U* Characteristic velocity Vm, Maximum Velocity Magnitude

Specific Heat cp*

Length of the Fluid Column

Radius of the Fluid Column

of Rotation

Average Radius Vector From Axis of Rotation in the Fluid

Depth of the Machined Interface

P* Thermal Expansion Coefficient P* Viscosity P* Density w* Centrifuge Rotation Rate

Subscriuts a Axially C Centrifuge (centrifugal) C Cold H Hot L Refers to a Length r Radially T Refers to Temperature

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References 1) W. A. Arnold, W. R. Wilcox, F. Carlson, A. Chait, L. L. Regel, “Transport Modes During Crystal Growth in a Centri- fuge,”J. Crystal Growth 119 (1992), pp. 24-40.

2) D.J. Tritton, “Physical Fluid Dynamics,” 2nd edition, Ox- ford University Press, N.Y. (1988)

3) W. Weber, G. Neumann and 0. Muller, “Stabilizing M u - ence of the Coriolis Force During Melt Growth on a Centrifuge,” J . Crystal Growth, 100 (1990), pp. 145- 158.

4) H. Rodot, L.L. Regel, and A.M. Turtchaninov, “Crystal Growth of IV-VI Semiconductors in a Centrifuge,” J. Crysfal Growth, 104 (1990). pp. 280-284.

5) L.L. Regel, “Kosmicheskoye Materialovedeniye,” Part 2, Volume 29 of the series Issledovaniye Kosmicheskovo Pros- transtva, Vwm, Moscow (1987); translated into English as “Materials Science in Space: Theory, Experiments, and Technology,” Plenum Press (1990).

6) L.L. Regel, “Kosmicheskoye Materialovedeniye”, Part 3, Volume 34 and Part 4, Volume 39 of the series Issledovaniye Kosmicheskovo Prostranstva, VINITI, Moscow (1991).

7) First International Workshop on Materials Processing in High Gravity, Dubna USSR, May 20-25.1991.

8) G. T. Neugebauer and W. R. Wilcox, “Experimental Ob- servation of the Influence of Furnace Temperature Profile on Convection and Segregation in the Vertical Bridgman Crys- tal Growth Technique,” Acta Astro/iunfica Vol. 25, No. 7,

v’

&’

pp. 357-362, 1991.

9) M. Engelman, FIDAP Theoretical Manual (1987), Fluid Dynamics International, Inc., 500 Davis Street, Suite 600, Evanston, Illinois 60201

10) P. M. Adornato and R. A. Brown, “The Effect of Am- poule on Convection and Segregation During Vertical Bridgman Growth ofDilute andNondiluteBinary Alloys,”J. Crystal Growth, 80 (1987) 155.

11) Chait, A,, and Arnold, W. A,, “Residual Acceleration Ef- fects in Directional Solidification Experiments Conducted in Various Low-g Environments,” Materials Science Forum, Vol. 50, pp. 13-28, 1989, Trans Tech Publications, Switzer- land.

12) Arnold, W. A,, Jacqmin, D. A,, Gaug, R. L., and Chait, A.,” Three-Dimensional Flow Transport Modes in Direc- tional SolidificationDuring Space Processing,”J. Spacecraft midRockets, Vol. 28, Num. 2, pp.238-243, 1991.

.J’

Table 1. Material Propertie3 Water

3 Density = 1 .O g / cm Specific Heat = 1.0 calf g-C - Thermal Conductivity: k (calf cm-s-K]

0 0.00135 20 0.00144 40 0.00151 60 0.00156 80 0.00160 100 0.00163

0 0.0179 20 0.00982 40 0.00620 60 0.0047 1 80 0.00352 100 0.00297

Viscosity: &Wcm-s)

Volumetric Expansion Coefficient: r n i 2 . L K 2

20 9.82 x IO -5

40 6.21 x IO -5

60 4.71 x 80 3.52 x 10 -5

100 2.97 x 10 -5 Ouartz

3 Density = 2.2 g f cm Specific Heat = 0.15 cal/ g-C

Thermal Conductivity = 3.4 x 10 -4 call cm-s-C

Density = 2.7 g f cm Specific Heat = 0.21 cal / g-C Thermal Conductivity = 0.5 call cm-s-C

Density= 1.15g/cm Specific Heat = 0.5 cal / g-C Thermal Conductivity = 3.5 x

Aluminum 3

O-Rinas 3

cal/ cm-s-C 4%

Density = 1.2 x 10 -3 g / cm 3

Specific Heat = 0.24 call g-C

Thermal Conductivity = 6.3 x 10 -5 cal/ cm-s-C Themial Expansion Coefficient = 3.34 x IO -3 l/C

Density in the melt = 7.0 g f cm3 Specific Heat = 0.0204 cal f g-C Thermal Conductivity =properties of water used Thermal Expansion Coefficient =props. of water used Viscosity = properties of water used

Low Prandtl Number Semiconductgl

9