Melting and Solidification of a Pure Metal on a Vertical Wall Essay

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Melting and Solidification of a Pure Metal on a Vertical Wall

citing and Solidification of a Pure Metal on a Vertical Wall This paper reports on the role of natural convection on solid-liquid interface motion and heat transfer during melting and solidification of a pure metal {gallium) on a vertical wall. The measurements of the position of the phase-change boundary as well as of temperature distributions and temperature fluctuations were used as a qualitative indication of the natural convection flow regimes and structure in the melt during phase transformation taking place in a rectangular test cell heated or cooled from one of the vertical walls. For melting, the measured melt volume and heat transfer coefficients are correlated in terms of relevant dimensionless parameters. For solidification, the measured volume of metal solidified on the wall is compared with predictions based on a one-dimensional model.

Introduction Heat transfer in the processing of materials involving solid-liquid phase transformations (melting and solidification) is commonplace in such fields as metallurgy, crystal growth from melts and solutions, purification of materials, and solidification of metals. The associated density gradients in a gravitational field can induce natural convection flows. Convection in the liquid phase influences the process in two different ways, one of which is beneficial and the other of which can be detrimental. During melting convection increases the overall transport rate and, hence, the growth rate of the new phase, which is desirable. On the other hand, during solidification convection decreases the growth of the new phase and also seems to affect the morphology of the solid-liquid interface adversely [1, 2].

The nature of the solid is largely determined by what occurs in the vicinity of the solid-liquid interface. The heat release (absorption), density change, and other processes that take place in the vicinity of the transformation front result in nonuniformities along the front that cause its shape to change. The resulting density gradients in the liquid generate buoyancy-driven convection that can affect the transport of heat, constituent chemicals, and the growth rate. Melting and solidification heat transfer from a vertical side (wall) has been receiving research attention, and a recent review is available [3]. However, most of the studies have been with materials other than metals. The authors have been able to identify only relatively few studies which were concerned with the effects of buoyancy-induced natural convection in the liquid metal during solidification [4-7]. The effect of natural convection on solidification of metals has been recognized for some time and a review is available [4]; however, much of the information is qualitative in nature and very little effort has been made to give a quantitative interpretation of solidification and of melting in the presence of natural convection.

Szekeley and Chhabra [5] appear to have been the first to demonstrate experimentally and analytically the importance of natural convection on the interface shape in a rectangular tank with the heat source and sink on the two opposite, vertical sidewalls. Solidification experiments from above with lead [6] and with a Lipowitz metal [7] have shown that convection in the melt decreases the rate of solidification. The composition stratification of a eutectic before the initiation of melting from below was found to suppress natural convection and to decrease the melting rate [7]. The heat transfer literature has failed to reveal any melting and solidification studies with pure metals in which the interface shape and motion, the phase-change rate, and convective heat Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division February 10, 1984.

transfer have been measured and reported for the situation where the material was heated or cooled from a vertical wall. No comparisons between analytical predictions and experimental data have been found in the heat transfer or the metallurgy literature. Yet, such information is of considerable practical importance to solidification of castings and other materials processing problems [1,8]. The purpose of this paper is to report on an experimental study of buoyancy-induced flow in the melt and its effect on the solid-liquid interface motion and heat transfer during melting and solidification of a pure metal (gallium) from a vertical wall. Melting and solidification experiments have been performed in a rectangular test cell with the heat source and the heat sink at the two opposite sidewalls of the test cell.

The importance of natural convection in the melt on the shape of the phase-change boundary and its motion as well as on heat transfer is demonstrated from the measurements of temperature distributions. The natural convection flow patterns are deduced indirectly from the measured solidliquid interface positions and temperatures. Experiments Apparatus and Instrumentation. Solidification and melting experiments were performed in a rectangular test cell (Fig. 1) that had inside dimensions of 8.89 cm in height, 6.35 cm in width, and 3.81 cm in depth. The two end walls, which served as the heat source/sink, were made of multipass heat exchanger machined from a copper plate. The copper surfaces were plated with a layer of 0.0127-mm-thick chromium for protection against corrosion. The top, bottom, and sidewalls were made of Plexiglass.

The two vertical sidewalls were 1.27cm-thick plates to support the cell, and, for better insulation, the front and back sidewalls had a 0.318-cm air gap between the plates. One of the plates is 0.318 cm thick and the other is 0.635 cm thick. A feed line connected to a hole in the bottom of the heated plate was used to fill or drain the melt. A long, thin cable electric heater was inserted in the tube to maintain the inside metal in liquid phase. The horizontal temperature distributions in the central region along the top wall, the center line, and the bottom wall of the test cell were measured with copper constantan (type K) thermocouples having wire diameter of 0.127 mm. A total of 26 equally spaced thermocouples were installed on the top and the bottom walls. The precise location of thermocouples along the top and the bottom walls was insured by placing them in small holes which were drilled in the walls. The thermocouples were then inserted close to the surface of the walls and were then sealed. A total of 17 other thermocouples, used to measure the temperature distribution along the center line, were made into a thermocouple rack. Each pair of Transactions of the ASME

174/Vol. 108, FEBRUARY 1986

Copyright © 1986 by ASME

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in which the temperature could be regulated and kept constant by a temperature controller. The purpose of the controlled environment was to reduce the temperature difference between the test cell and environment, thus minimizing the heat loss/gain from the test cell to the ambient surroundings. A heat storage material was also placed in the box to minimize the frequent on-and-off switching of the heater and to keep the inside temperature close to a constant value. Experimental Procedure and Data Reduction. The metal used in the experiments was gallium. It had a purity of 99.6 percent and a fusion temperature of 29.78°C. There were several reasons for selecting gallium. First, the thermophysical properties are reasonably well established [9]. Second, it has a fusion temperature close to the ambient, which is conducive to experimentation.

Third, the metal is important technologically as it is usually combined with other pure elements to form electronic and industrial materials such as semiconductors, laser diodes, solar cells, and magnetic bubble devices. The main disadvantages are that it is very expensive (about one dollar per gram, depending on purity) and is anisotropic. Pure solid gallium appears in polycrystalline form and the crystals have an orthorhombic structure. The thermal conductivity of the solid crystal is highly anisotropic and is 40.82 W/mK along the a axis of the crystal, 88.47 W/mK along the b axis, and 15.99 W/mK along 7777777 the c axis [9]. The thermal conductivity of the liquid is still anisotropic but not much different along each of the crystal axes. Fig. 1 Schematic diagram of the test cell, top view (a) and front view (b): (1) heat source and/or sink, (2) Plexiglass wall, (3) air gap, (4) phaseBefore each experiment the metal was melted and poured change material, (5) thermocouple rack, (6) thermocouples along the into the preheated test cell through the sprue on the left side walls, (7) small-diameter thermocouples, (8) hole for filling material, and (Fig. 1). Provision was made to avoid air bubble entrapment (9) constant temperature baths in the test cell. This was accomplished during the filling procedure by slightly lifting one of the sidewalls where a small thermocouple wires was sheathed in a 0.584-mm-dia stainless hole had been drilled near the top plate. This hole was sealed steel tube and was then attached perpendicularly to a 2.38- after the test cell was filled. mm-dia stainless steel tube.

The sheathed thermocouples were The initial temperature inside the test cell was kept uniform arranged in a horizontal plane and in a direction per- and only a single phase was allowed to exist. This was done by pendicular to the front and the back walls where the tem- circulating a constant temperature fluid in both heat experature gradient in the liquid was the smallest. The changers at a temperature slightly above or below the fusion arrangement minimized heat conduction along the small point for a sufficient period of time. Melting or solidification stainless steel tubes and therefore reduced the measurement was initiated by switching one of the heat exchangers to error. another constant temperature bath preset at a different Two additional thermocouples were inserted through small temperature.

A uniform solid temperature was reached, holes drilled in the bottom wall to measure the temperature usually from 1 °C to 2°C below the fusion temperature of the fluctuations. One was located in the central region away from material, before melting was initiated. This was done in an the heated vertical wall; the other was located 24 mm away attempt to minimize the subcooling as a parameter of the from the wall. The thermocouple junctions were coated with a problem. thin layer of high-thermal-conductivity epoxy for protection. The pour-out method and the probing method were The d-c part of the signal of the temperature fluctuations was adopted to examine and/or measure the shape of the solidsuppressed and the small a-c component could be amplified liquid interface [7]. At predetermined times the phase-change and recorded. process was terminated, and the liquid (melt) was rapidly The test cell was placed in a transparent box made of Lexan poured out. In the process of draining the liquid, some forced

Nomenclature
A Aw c Fo H Ahf L Nu k Ra = = = = = = = = = = aspect ratio = H/L heat transfer area specific heat Fourier number = at/L2 height of the cavity latent heat of fusion length of cavity Nusselt number = hH/k/ thermal conductivity Rayleigh number = g/3(Tw —

s T t V V0 W x

TrfHP/av
Ste = Stefan number = cj(Twb — Tf) /Ahf for melting and cs (7} — Tw,/Ahf for solidification Journal of Heat Transfer

= solid-liquid interface position = temperature — time = volume = volume of test cell — cavity width = coordinate along the horizontal length of the test cell a = thermal diffusivity /3 = thermal expansion coefficient 5 = dimensionless interface position = s/H

v = kinematic viscosity r = dimensionless time; 77 = Fo/Ste/ for melting and TS = FOjSte^. for solidification Subscripts c = characteristic length or cooled h = heated wall / = fusion point of material / = liquid phase s = solid phase w = wall FEBRUARY 1986, Vol. 108/175

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Time (min)

Time(min)
I I I

6 8
I I

10 12.5 15 17 19
I I

2
I

5 6 | 8 10 12 14
I I I I I I

17
I

Fig. 2 Interface shape traced at preselected times during melting from a side using pour-out method: (a) Twh = 38.0C, A = 0.714, (fa) Twh = 38.3°C,A = 0.5,and(c)T w h = 38.7°C,A = 0.286 0.8

I I Interface Location Data V Top Wall D Center A Bottom Wall

I

I

I

I
V

I V

|

I

D
V

0.6

D

0.4 V

a
V

Neumann Model

a
D


A
A A A

0.2

v

V

n
A

A

-‘
/ & •

Q ^ * ^

A A

I

I

I

I

I

I

I

I

,
10

TjXIO* Fig. 3 Comparison of the local melting rate with the Neumann solution for Ste s = 0.0408,4 = 0.714

convection effect was introduced, but since the interface was bared very rapidly the effect on the interface is considered to be minimal. The solid-liquid interface position was traced and/or photographed. Melting From a Vertical Wall Interface Shape. The interface shape observed during the melting of gallium from the side was smooth and flat in the direction perpendicular to the front and back walls of the test cell. This suggests that fluid motion and heat transfer in that direction was small, and the effect of natural convection recirculation on the interface motion can justifiably be neglected. The interface shape traced at preselected times is shown in Fig. 2. At very early times, before the buoyancydriven flow was initiated or when fluid motion was still very weak, the interface shape was flat and parallel to the heated wall of the test cell. Heat transfer was dominated by conduction. As the heating progressed the buoyancy-driven convection 176/Vol. 108, FEBRUARY 1986

in the melt started to develop and continued to intensify. This is evidenced by the appearance of a nonuniform melt layer (Fig. 2a, t = 3 min) receding from the top to the bottom of the test cell. This is particularly evident in Fig. 2(a) for the larger aspect ratio test cell (A = 0.714). A flow structure typical of natural convection-driven phase-change systems is observed.

Fluid is heated as it rises along the hot wall. The flow is deflected and descends along the solid-liquid interface with higher melting occurring at the top where the fluid is at higher temperatures. Fluid motion associated with phase change at the interface was negligible since there is very little difference in the density of the solid and liquid phases. Melting at the bottom is nearly terminated at later times (for times from 12.5 to 17 min), and the interface shape becomes almost flat. This is in part due to the very high thermal conductivity of liquid gallium, which has been cooled to nearly fusion temperature in the vicinity of the interface in the lower part of the test cell. Experiments performed with higher sidewall temperatures yielded similar results. Significant differences in the local melting rate that alter the Transactions of the ASME

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0.8

0.6

v/v 0
0.4

0.2

0.05

0.10
T 0.843

0.15

0.20

0.25

0.30

0.35

RQ0.0504A-0.I4

Fig. 4

Correlation of the melted volume data

shape of the interface are evident from the interface contours shown in Fig. 2. Physically, heat transfer near the bottom of the melt-filled cavity is lower because of the recirculation pattern established in the liquid. For a higher aspect ratio cavity the recirculation is stronger and produces a lower melting rate in this region (compare Figs, la and 2c at t= 17 min). A comparison of the measured interface positions near the bottom wall with the predictions based on the Neumann model [10] showed that the interface moved more slowly (Fig. 3). The results suggest that the recirculation which was established in the liquid became more intense as time progressed and decreased the melting rate near the bottom of the cell where the cooled liquid flowed from the interface toward the heated wall.

Comparison of the results given in Fig. 2 with those reported in the literature for melting of n-octadecane [11, 12] shows that there is much more melting near the bottom of the test cell with gallium. The shapes of the interface contours are also significantly different. This is attributed, in part, to large differences in the thermophysical properties between the paraffin and the metal and due to the difference in the flow structure in the two phase-change materials. Melting Rate. The melted volume fractions at each time for different aspect ratio cavities and Rayleigh numbers (or Stefan numbers) were measured and are presented in Fig. 4. Attempts were made to correlate V/V0 with the group T/Ra1/4A4 as a parameter which is similar to the one suggested in the literature [12], except that A was included here to take into account the different length scales defined in jt and Ra; however, the data could not be collapsed onto a single line. Therefore, each of the variables in the group rRa^4 was treated as a separate parameter in the regression analysis. A least-squares fit of the data was found to yield the following equation = 2.708r?- 843 Ra 0 – 0504 /r (1)

39

36 T(°C) 33 \

fi

8 ggg g •
A

Data t(min) u 0 4 • A 8 A 12 O 16 20 a
D

30
il^Bl

27 0

•, •
0.2

mm*

m1 j 1 j • | | j |
0.6 x/L 0.8 1.0

0.4

Fig. 5 Temperature distribution along the centeriine during melting from a side: Twh = 38.3°C,A = 0.5

there should be no dependence on the Rayleigh number during this period. The effect of the aspect ratio on the melting rate could not be conclusively deduced from equation (1), because the cell height which is used in the Rayleigh number may not be meaningful for the problem considered. For natural convection without phase change in a cavity heated from a side the heat transfer rate increases with the aspect ratio [15]. Therefore, an increase in the melting rate with aspect ratio should have been expected. In a correlation where the Stefan number instead of the Rayleigh number was chosen as a correlation parameter, a similar correlation equation to the one found earlier [14] in a different system could be obtained. The melted volume fraction was found to be ,0.0393 V/V0= 6.278 T?’842Ste: A 0.0137

Vc

(2)

When heat transfer in the melt is by conduction (only during about the first 2 min of the experiments, see Fig. 2), the melted volume V/V0 is proportional to rf5 In the presence of natural convection, the melted volume VIV0 is proportional to T1M3 . Since the empirical correlation was obtained by a least-squares fit of the experimental data the short times (about 2 min time period compared to the total melting time of about 20 min) when heat transfer is by conduction only are not reflected in the correlation, because Journal of Heat Transfer

This correlation also indicates a very weak dependence of the melted volume fraction on the aspect ratio. The melting rate measured has been compared with the Neumann solution [10]. Near the end of the process, the melted volume is about 1.75 times that which would be expected in the absence of natural convection in the melt. Temperature Distribution. During the early melting stages the horizontal temperature distributions along the top wall, FEBRUARY 1986, Vol. 108/177

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Ste„ 0.4

• ©

• A
fi
A

0.3 Nu Ra^4 0.2

A A O A

0.041 0.042 0.044 0.009 0.083
A

0.714 0.500 0.286 0.500 0.714

°
A

m

m
® ” 4 « -&»

*

8 CM
A
i i

A
i

A 1

0

0.2

0.4

0.6
TJJ

0.8
Z

1.0

1.2

X I0

Fig. 6 Timewise variation of the heat transfer parameter
ste

10′

r
m ® A A o

i

Nu r

0.041 0.042 0.044 0.009 0.083

A 0.714 0.500 0.286 0.500 0.714
B

o%x

a
10°
i i

fgF*^
\—Eq.(6)

i

i

i

I0 3

10″

I0 5 Rac/s\eJt

I0 6

I0 7

I0 8

Fig. 7 Correlation of the average Nusselt number in terms of the characteristic Bayleigh number Rac

center, and the bottom wall were linear and were nearly the same. Conduction was the dominant mode of heat transfer. At later times, the fluid motion produced by buoyancy resulted in nonlinear temperature distributions which were different at each height [13]. Due to the expected clockwise circulation of the melt driven by buoyancy, steep horizontal temperature gradients were observed at the two boundaries of the liquid-filled cavity (Fig. 5). Along the vertical heated wall the largest temperature gradient occurred near the bottom region and the smallest near the upper region. Along the interface the trends were opposite.

The horizontal temperature gradients at each height near the heated wall and the interface changed little after natural convection was initiated and established. The temperature gradients at the three different heights in the region away from the heated wall and interface gradually decreased as the melting proceeded. The reduction of the temperature gradient is attributed to the growth of the main circulation flow and of the melt layer. The temperature gradient along the center line (Fig. 5) decreased the most and approached zero at later stages of the melting process (? = 16 min). No reversal of temperature gradient was ever observed. The gradual change of temperatures suggests that no significant change of flow structure can be expected. The flow was laminar and remained so for the entire duration of the melting process. Temperature fluctuations were measured near (4.74 mm) 178/Vol. 108, FEBRUARY 1986

SM
•I i ! )
/”:•/ •’/

//If
A]

if/f y /
(b)

Interface Position Time(min.)

5.0 7,5 12.5 15.0 20.0 25.0
Fig. 8 Traces of interface shape along (a) the front wall and (b) the back
wall of the test cell during solidification from a side: Twc = 10.2°C, Twh = 33.2°C,4 = 0.714

and farther away (24 mm) from the heated wall. The traces of the thermocouple output indicated a sharp increase and small fluctuations shortly after the interface passed over the sensor [13]. However, the frequency and the mean temperature showed gradual increases and became steady at later times as the fluid circulated in a much more stable manner. Transactions of the ASME

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0.8

1

1

1 Ste s 0.0909

1 0wh \.\/4f

1

1

1

I

1

1

Analysis Data J o

o

0.6

°
v/v0
0.4

° —— –

o

0

-Z^^

0.2

-/
1 1 1 1
1 12 T, XIO2 Fig. 9 Comparison between experimental data and predictions for the melted volume fraction 1 1 1 1 20 1 24

16

Average Heat Transfer Coefficient. With the knowledge of the melted volume at a specific time the total heat transferred to the phase change material can be estimated and the timeaverage (spatial) heat transfer coefficient can be calculated [14]. The time-average heat transfer coefficient was defined by

35

Bag

• @@B §§§

m • • • m m BHBB
1 m. • m m A
A

30

A i

ft
A„(T„-Tf)t

ttl

qdA dt
(3)

25 T(°C)

a o A aa o a a

A

m
@

0 A A

A„{T„-Tf)t

20 _Data t(min)

a o a

where the total heat input Q, or heat stored in the phasechange material is given by


m
15 A A O

a=

Lo>-

•cAT/-T0(p))dVsip)+p,V,Ui/

+\

Plc,(Tl(p)-Tf)dVl

(4)

10 0

a

0 2 6 10 14 26
1 l

A A O A a O

m
A

0

a

AA d &

i

I

i

J vtU)

0.2

0.4 x/L

0.6

0.8

1.0

The time-averaged heat transfer coefficient can be written in dimensionless form as Nu = — = ( VI K„)(l + Ste/2)A4r, ki
(5)

Fig. 10 Temperature distribution along the center line during solidification from a side: Twc = 12.5°C, T w /, = 34.0°C, and A = 0.5

The heat transfer coefficients were correlated in terms of standard heat transfer parameter Nu/Ra 1 / 4 , where the height H of the test cell was used as characteristic length. The timewise variation of the Nusselt number is presented in Fig. 6. At early times, the parameter Nu/Ra 1 / 4 is large, because the melt layer is relatively thin and heat transfer is dominated by conduction. Growth of the melt layer increases the thermal resistance and decreases the heat transfer coefficient. The decrease in the parameter Nu/Ra 1 / 4 levels off after natural convection is initiated. At later times in the process, the heat transfer parameter Nu/Ra 1 / 4 indicates that convectiondominated quasi-steady melting has been reached for the largest but not for the smallest aspect ratio test cells. As shown in Fig. 6, the parameter Nu/Ra 1 / 4 is a function of the Stefan number and the aspect ratio. For smaller Stefan numbers the dimensionless heat transfer parameter shifts up and to the right. This was also found in a different melting Journal of Heat Transfer

system [14] and was attributed to the use of improper characteristic length for defining the Rayleigh number. The most probable reason for the heat transfer parameter being inversely proportional to the Stefan number may be that the parameter itself could not completely characterize the natural convection flow and heat transfer process in the liquid during melting when the size and shape of the liquid-filled cavity changes with time. For quasi-steady melting the average Nusselt number was well correlated with the Rayleigh number [11, 16]. However, since convection-dominated quasi-steady melting does not appear to have been fully reached in the present experiments no attempt was made to correlate the data in a form of a standard natural convection correlation. Even though the entire melting process is transient in nature, and one of the boundaries of the heat sink (solidliquid interface) is not planar, an average melt layer thickness FEBRUARY 1986, Vol. 108/179

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has sometimes been used as a characteristic length lc in correlating the heat transfer data. A simple relation between Nu c and Ra c as well as Ste/ has been obtained for a different system [14]. The least-squares fit of the data was found to yield the following empirical equation hlc / Ra c \ °-274 Nu c = ^ = 0 . 0 6 3 l ( ^ ) (6)

solid layer. During the final stages, the rate is primarily retarded by natural convection in the melt. Temperature Distribution. Despite the irregular interface shape and the flow field in its vicinity, which were not reproducible for each experiment, the temperature distributions exhibited trends which were somewhat similar to those during the melting when natural convection was well established. The highest temperature gradient along the heated vertical plate was found near the bottom region and the smallest was found near the top. An opposite trend for the temperature gradient was observed along the solid-liquid interface. Along the center line, the core was nearly isothermal. An increase in temperature from the bottom to the top region was observed [13]. Therefore, a clockwise circulation of flow can be inferred. The temperatures along the center and the top wall in the melt remained nearly the same shortly after the solidification process was initiated and remained so until just before the solid-liquid interface approached the thermocouple junctions, whereas the temperature along the bottom wall decreased slowly.

This suggests that the flow in the upper region of the melt is more readily established than near the bottom region. The temperature distributions along the center, the top, and the bottom walls in the solid are linear and nearly the same [11]. The slightly higher temperature along the top and the bottom walls may be due to the smaller heat conduction along or across the Plexiglass wall from either the melt or the environment. The temperature gradient in the solid is large initially as compared to the temperature gradient near the interface in the melt. Natural convection begins to play an important role in controlling the solidification process during the later stages when the temperature gradient in the solid decreases with the increase of the solid thickness. Conclusions In spite of the large thermal conductivity, the experimental results have clearly established the important role played by natural convection during melting of a pure metal from a vertical sidewall.

The phase-change boundary, the melting rate, and heat transfer were greatly affected by buoyancydriven natural convection. Qualitative flow structure was deduced indirectly from the interface shape and motion and reaffirmed by temperature measurements. Laminar flow persisted throughout the melting process, and nearly quasisteady melting was attained at later stages. The melted volume fraction and heat transfer were well correlated in terms of relevant dimensionless parameters. During solidification of gallium from a sidewall, the interface shape was controlled not only by the natural convection circulation in the melt but also by the combined crystallographic effects and anisotropy of the gallium crystal. The interface morphology was irregular, and the shape was not reproducible.

The buoyancy-induced flow and its structure inferred from the observations of phase-change boundary shape and motion and the temperature distribution in the melt were found to be similar to those during melting. The solidification rate was higher near the bottom than near the top. The fraction of the total solidified volume measured at different times agreed reasonably well with the predictions based on a one-dimensional model. Acknowledgments The work described in this paper was supported by the National Science Foundation Heat Transfer Program under grant No. MEA-8313573. References 1 Flemings, M., Solidification Processing, McGraw-Hill, New York, 1974.

A comparison of the data with the equation is presented in Fig. 7. All the data points collapse nicely onto a single line. The correlation is valid in the conduction, transition, and natural convection regimes.

Solidification on a Vertical Wall

Interface Shape. The exposed interface shape at different times was examined and traced along the front and back walls by pouring out the liquid at specific times during each run (Fig. 8). The interface shape was very irregular. The anisotropy in the thermal conductivity and the crystallographic effects [17, 18] played a dominant role in controlling the interface shape, not only during the early time when conduction in the solid predominated over convection in the melt but also at later time when natural convection played an important role in controlling the interface motion. The irregular interface shape has prohibited inferring the flow structure in the melt filled cavity.

However, because of the arrangement of the heat source and the cavity geometry, a clockwise circulation of liquid which rose near the isothermal side (left) and fell near the cool interface could be expected. The local flow structure and heat transfer near the interface would be influenced by the irregular interface morphology and shape and vice versa. Therefore, especially at later times the interaction of natural convection in the melt with anisotropic conduction in the solid and the crystallographic effects made the analysis of the growth morphology and of the flow structure along the interface more complicated. Experiments were repeated, but the interface shape and the flow structure in its vicinity were not reproducible. The interface and the fluid motion near the bottom of the test cell sensed the upstream effects, i.e., the irregular interface shape and its interaction with the flow. Solidification Rate. At early times, the local solidification rate along the interface is controlled primarily by conduction in the solid and crystallographic effects at the interface.

The effect of natural convection heat transfer on the interface shape and motion is relatively small. At later times when natural convection plays an important role in controlling the interface motion, higher solidification rates in the bottom region can be observed (Fig. 8). The total frozen volume fractions at different times were measured and are compared in Fig. 9 with the predictions based on planar interface motion [7]. The heat transfer coefficient at the interface was obtained from the literature [15] for steady-state natural convection in a cavity heating from the side. The coefficient has been multiplied by a factor of 0.45 as suggested because the ratio W/H is close to unity. The dashed line in Fig. 9 shows the predictions based on the heat transfer coefficient being reduced by 20 percent.

The predicted solidification rates are lower than the measured ones. The higher rate measured at later times is attributed to the anisotropy in the thermal conductivity of gallium crystal and possibly to the increase in the surface area afforded by the irregular crystalline structure relative to a plane surface. In addition, the assumption of a planar interface shape and the use of steady-state heat transfer coefficient (in the absence of phase change) in the model may contribute to the discrepancy. Initially, the solidification rate is relatively high due to the sharp temperature gradient in the solid but gradually decreases due to the increase in the thermal resistance of the 180/Vol. 108, FEBRUARY 1986

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