Effect of density extremum on the solidification of water on a vertical wall of a rectangular cavity

Effect of Density Extremum on the Solidification of Water on a Vertical Wall of a Rectangular Cavity S. L. Braga Department of Mechanical Engineering, Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil R. Viskanta School of Mechanical Engineering, Purdue University, West Lafayette, Indiana

• Experiments were performed in a two-dimensional rectangular cavity to study the transient flow in an initially isothermal and motionless fluid due to a step decrease in temperature on one of the two vertical end walls. In the experiments water was used as the phase-change medium, with the cold-wall temperature maintained below the freezing temperature. The opposite vertical wall was kept at the initial temperature, greater than the temperature where the density extremum occurs. The growth of ice and the transient flow in the cavity were visualized with the aid of a tracer technique to examine the effect of density inversion. The temperature field was continuously recorded by an array of thermocouples. It was found that the density inversion of water strongly influences both the growth of ice and the convective flow in the liquid region of the test cavity. Keywords: density extremum, solidification of water, natural convection

INTRODUCTION Natural convection in cavities is a problem of interest because of numerous applications in many natural and engineering processes. Solidification is relevant to the growth of crystals from melts and solutions for the semiconductor industry, metallurgy, convection in magma chambers, oceanography, nuclear reactor safety, and elsewhere. Natural convection in enclosures has been studied both numerically and experimentally by numerous investigators. Most earlier work on natural convection in cavities, however, was concerned with steady-state situations, and the density-temperature relationship of the fluid in question is usually based on the Boussinesq approximation. Several fluids, including tellurium, gallium, antimony, and bismuth, exhibit an extremum in their densities at specific temperatures. Water, however, is the most important fluid where this kind of maximum in density occurs. The effect of density inversion is known to be important with respect to convective heat transfer in the vicinity of the temperature where this maximum density occurs. During solidification of this type of fluid, the liquid motion driven by buoyancy develops in a complicated manner and controls the growth of the solid phase. For the specific case of water, the maximum density is attained at a temperature of Tm = 3.98°C, and density thereafter decreases in a nonlinear manner as the temperature passes through the critical value [1]. Numerical studies concerned with convective flow in cavities with different types of boundary conditions in the presence of the density inversion have been reported in

the literature. Some authors have simulated the behavior of the water under steady-state [2-7] or transient [8-12] natural convection conditions. All investigators have found that flow structure and heat transfer are strongly influenced by the density inversion. The freezing process, however, was not considered in these works. A small number of experimental studies have been identified that are concerned with steady-state natural convective flow of water in a range of temperature about the density inversion. Inaba and Fukuda [13] investigated the flow in an inclined rectangular cavity where the temperature of one wall was maintained at 0°C, while the temperature of the opposite wall was varied from 2 to 20°C. Different inclinations of the test cell were examined. Lankford and Bejan [14] conducted experiments in a vertical enclosure where a constant heat flux boundary condition was imposed on the hot vertical wall while the opposite wall was cooled. Seki et al [15] reported on an experimental and analytical investigation of two-dimensional natural convection in a rectangular cavity where the cold vertical wall was kept at 0°C while the opposite hot wall was maintained at a constant temperature between 1 and 12°C. Only three papers dealing with transient natural convection in rectangular enclosures filled with water were identified during the literature search. Yewell et al [16] investigated transient natural convection in low-aspectratio (A = 0.0625 and A = 0.112) enclosures at high Rayleigh numbers. They used water as a working fluid at temperatures between 15 and 35°C, where the density inversion does not occur. Ivey [17] conducted experiments

Address correspondence to Professor S. L. Braga, Department of Mechanical Engineering, Catholic University of Rio de Janeiro, Rio de Janeiro, RJ-22453, Brazil.

Experimental Thermal and Fluid Science 1992; 5:703-713 © 1992by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010

0894-1777/92/$5.00

703

704 S.L. Braga and R. Viskanta in a square enclosure to study an initially isothermal cavity at temperature T0, following a rapid change of the two vertical end walls to temperatures of To _+ AT, respectively. He used water and a mixture of glycerol and water as the fluids. Once again, the density extremum in the fluid was not present. Braga and Viskanta [18] studied, both numerically and experimentally, the effects of the density inversion on transient natural convection of water in a rectangular cavity. The initially isothermal (8°C < T h < 20°C) and motionless fluid was suddenly subjected to a cold temperature (Tc = 0°C) imposed at a vertical wall while the opposite wall was maintained at the initial hot-wall temperature. Few papers were found in which the freezing of water was studied. Cheng et al [19] investigated the effects of steady-state natural convection on ice formation around an isothermal cooled horizontal cylinder. Chellaiah and Viskanta [20] studied both experimentally and numerically the transient freezing of superheated water-porous media contained in a rectangular cavity. This work studied the effect of an imposed horizontal temperature difference on the two vertical walls of a cavity that was initially filled with an isothermal and motionless fluid. Also, working with a water-filled porous layer, Sugawara et al [21] conducted a transient experiment where the top, cooled wall was kept at -20°C, while the lower, heated surface was maintained at a temperature between 0 and 30°C. Once again, it was found in these experiments that the density inversion of water influences the convective motion as well as the solidification process. The present work was motivated by the need to understand the development of the flow structure in the presence of the density inversion in water because it affects the local rate of solidification. Experiments were conducted in a water-filled rectangular cavity where it was possible to control the temperature at two vertical opposite walls. Five series of experiments were carried out with initial water temperatures between 6.5 and 20.2°C. Two different cold-wall temperatures were considered, T, = - 10 and - 20°C. EXPERIMENTAL METHODS The experiments were performed in a rectangular test cell having inside dimensions of 150 mm in height, 300 mm in width, and 75 mm in depth. Figure 1 is a schematic diagram of the experimental apparatus. The right and left vertical walls E in Fig. la consisted of multipass heat exchangers made of polycarbonate in which channels were machined to allow the circulation of a refrigerant. These channels were covered with a heat exchanger wall made of copper. At first, copper heat exchangers were used, but owing to their large thermal inertia they were replaced to enable the production of rapid changes in the temperatures of the heat transfer surfaces. The copper sheets were plated with nickel and chrome to avoid corrosion. As shown in Fig. la, the heat exchangers were connected through a valve system to two constant-temperature baths. Each exchanger contained three separate sections with passages through which the flow rate could be controlled independently. Five copper-constantan thermocouples (type T) were placed very close to each heat transfer surface for continuous monitoring of its temperature. By adjusting the flow rate through the passages, it was possi-

ble to maintain the temperature of each heat exchanger uniform to within +0.1°C of the desired value. With an appropriate valve setting, the vertical end walls could be maintained at either the same or different temperatures. Ethyl alcohol was used as working fluid in the constanttemperature baths. The horizontal bottom and top connecting walls as well as the front and back vertical walls were made of acrylic to allow for flow visualization of the fluid motion inside the test cell. To minimize the heat gain from or loss to the ambient, the front and back walls were constructed from a pair of acrylic sheets with an air gap between them. The entire test section was covered with 50 mm thick styrofoam insulation. Two removable windows (shown in Fig. lb) in the insulation (top and front walls) allowed for illumination and visual observations. Measurements of the temperature distributions inside the test cell were also made with (type T) thermocouples. They were placed in a stainless steel tube with an outside diameter of 0.9 mm. These probes were inserted through both horizontal walls where 42 holes (21 on each side) allowed for installation of probes in several positions along the test cell. Typically, five probes were used in each wall (top and bottom), and the temperatures were monitored in two different planes (see Fig. 1). A thermocouple rack was used to measure the temperature at the center plane of the test section. Figures la and lb show the configuration and the positions of the thermocouples. The thermocouple rack was made to minimize heat conduction along the probes. As the flow is two-dimensional, it is assumed that the temperature does not vary in the third direction. In this case the thermocouple probe ends are exposed to a uniform temperature, and only the vertical tube (see Fig. 1) experiences variations in the temperature field, reducing the errors in the thermocouple readings. A probe in a vertical position conducts heat and disturbs the temperature of the fluid to be measured. Therefore, to reduce the error in the temperature readings, the measurements in the lower and upper planes were made close enough to the horizontal walls to minimize the variations in the temperature field in the regions where the thermocouple probes were inserted. For reference, each thermocouple position will be associated with the cardinal points, as indicated in Fig. la. The outputs of all thermocouples were recorded by a data logger system at preselected time intervals between consecutive measurements. Distilled water was used in all experiments, and the water was carefully syphoned into the test cell to avoid introduction of air. The test cell was not filled completely full of water, and a small ( ~ 3 mm) gap was left at the top in order to have a free surface (see Fig. 1) and to allow water to expand a n d / o r contract. Before starting an experiment, both heat exchangers were connected to the same bath, and their surfaces were maintained at the desired temperature of the hot wall (Th). Enough time was allowed for all the fluid in the test section to reach thermal equilibrium. In all cases, this uniform initial temperature was obtained within +0.1°C of the desired T h value. After this was attained, the second constant-temperature bath, with a preselected initial temperature (below (0°C), was connected to one of the heat exchangers (left wall). Typically, the desired temperature at the cold wall (Tc < 0°C) was reached within 1--4 min, depending on both the initial temperature of water in

Effect of Density Extremum on Solidification

705

(~) Heal Exchanger (a) Gloss

Cylinder

Z

Dt

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Loser

Data Logger

a Logger

~) Removable Window 0a)

Figure 1. Schematic diagram of the test cell. (a) Front view; and (b) side view.

7tl6

S.L. Braga and R. Viskanta

the test cell and the initial bath temperature. To attain the cold-wall temperature within a short time interval, the second bath temperature was set at the level of - 5 0 or - 6 0 ° C . After the desired temperature was reached at the cold left wall, both heat exchanger walls were kept at uniform temperatures by individually controlling the flows through each of the heat circulating loops. Owing to the transient nature of the experiments, the temperature settings of the constant-temperature baths a~well as the flow rates in the heat exchangers had to be frequently readjusted. The thermocouple probes were calibrated with an accuracy of + 0.1°C before installation and were checked before each new series of experiments. The thermocouples were connected to a Hewlett-Packard data acquisition system and computer with which the temperature readings could be stored at preselected time intervals. To visualize the flow patterns, the water was seeded with a small amount of neutrally buoyant particles (Pliolite). A helium-neon laser was used as the illumination source. The laser beam was passed through a cylindrical glass rod to produce a sheet of laser light before passing through the test cell wall. Figure l b shows schematically the illumination system as well as the positions of the windows in the insulation. During the experiments the removable insulation (shown in Fig. lb) was kept in place; it was removed only for flow visualization. By controlling the quantity of particles and using a photographic camera it was possible to obtain qualitative information about the seeded flow. Photographs of the flow patterns were taken using ASA 400 (T-MAX) film. The exposure time was about 100 s. RESULTS AND DISCUSSION

Because of the density inversion of water, for the region where T < Tm the fluid descends when it is heated and flows upward when it is cooled. Because of this phenomenon, a small clockwise circulation grows with time in the lower left-hand corner of the cavity where the water attains its minimum temperature. Figure 2 shows schematically the two circulations and the stagnant region inside the test section. In the large rotating cell (T > Tin), the water is cooled by the solid surface (T = T f ) from point 1 to point 2 and by the small and colder circulation (T < T,,) while it flows to 3. Between points 4 and 5 the fluid rises because it is heated by the hot wall. The opposite occurs at the lower left corner, where the liquid is cooled while it flows upward in the vicinity of the solid surface (6 to 7), and it descends from 7 to 8 while it is heated by the larger circulation. In Ref. 18 it is shown that the isotherm T = Tm is located between the i'otating cells, as is evident from Fig. 2. Conduction through the bottom affects the location of this isotherm, moving it to the right, in the lower region of the cavity. In Ref. 18, a comparison between experimental and predicted (with insulated horizontal boundaries) flow patterns shows that the calculated cold circulation appears smaller in the x direction, owing to conduction effects. In the upper boundary the air layer reduces this difference and the phenomenon can be neglected. Experimental Conditions Five series of experiments were carried out with different initial and cold-wall temperatures in order to check their influences in the solidification and convection processes. Table 1 summarizes the test conditions where the Rayleigh number, the density distribution parameter, and the Prandtl number are, respectively,

Description of the Phenomena Before presentation of the results, it is desirable to describe the transient process that occurs inside the test cavity. Braga and Viskanta [18] have shown that in the early stages of the cooling process, heat transfer is by conduction. Because of the similar initial conditions in the present study, the same behavior is evident. Observations of the fluid motion over time reveal that initially the flow is dominated by the vertical downward flow over the cold (left) wall, and this motion is gradually extended throughout the cavity. A large counterclockwise rotating cell embraces almost the entire test section. The hot fluid descends along the cold wall while it is being cooled in the left side of the circulation, and it flows upward while it is heated by the right wall.

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(1)

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(2)

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Table 1. Experimental Conditions

T >Tin m

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Ra =

Expt.

Th

Tc

No.

(°C)

(°C)

1 2 3 4 5

10.2 10.2 15.1 20.2 6.5

-

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Ra (× 10 - 8)

R

Pr

1.07 1.07 2.40 4.46 0.43

0.39 0.39 0.26 0.20 0.61

11.4 11.4 10.5 9.6 12.1

Effect of Density Extremum on Solidification

707

could be neglected.

sent. No references were found where this matter is treated specifically. The density distribution parameter R shows the occurrence of the density inversion in the experiment. If T. < Tm< Th (or 0 < R < 1), the temperature where the Yensity maximum occurs (Tm) will be present inside the test section. For example, at steady state when R = 0.5, two counterrotating cells of about the same size are expected to appear in the liquid region. Some experiments were performed twice to check whether the removal of a part of the insulation affected the temperature field when the flow was visualized and photographed. Experiment 4 was also repeated in the absence of the particles with this objective in mind. In both cases it was found that this effect on the temperature

Experiment 1 Figures 3 a - 3 d show the flow patterns for Experiment 1. It is evident that in the beginning of the test the solid-liquid interface is almost planar because conduction is the dominant heat transfer mode in early times. In Fig. 3a (t = 15 min) it is possible to see the large counterclockwise circulation that embraces almost the entire test cavity, while a small clockwise cell starts to develop in the lower left-hand corner. It is interesting to note that at this location, due to its low temperature, the ice layer is a little thicker than in the other positions. As time passes, this smaller cell grows in both the x and y directions. Figures 3b-3d show this

(a)

(b)

(c)

(d)

(e) Figure 3. Photographs of flow patterns for Experiment 1 with Th = 10.2°C and Tc = - 10°C. (a) t = 15 min; (b) t = 60 min; (c) t = 120 min; (d) t = 180 min; (e) t = 300 min.

708 S.L. Braga and R. Viskanta evolution; the importance of the density inversion in the convective motion is clearly evident from the appearance of the two eddies. The solidification process is also greatly influenced by the maximum in the density of water. The average thickness of the solid region close to the colder cell is much greater than the thickness of the ice in contact with the hot eddy. It is easy to understand that the lowest temperature in the circulating fluid occurs in the neighborhood of point 7; this is depicted schematically in Fig. 2. In this region, the fluid inside the colder cell attains its lowest temperature. Because of this, the ice close to this position experiences the maximum growth rate. Observations of the water solidification process show that for large growth rates the forming solid is not transparent as it is in the cases in which small solidification rates are experienced. Figure 4 presents the solid-liquid interface positions for t = 15 min up to t = 300 min for Experiment 1. The shadowed parts in the left-hand side (solid region) were obtained from Fig. 3e and denote the regions with the greatest solidification rates. Close to the cold wall (x = 0), the high rate of freezing is explained by the large temperature gradients at the beginning of the experiment. It is evident in this figure that the location of the maximum growth rate moves to higher positions accompanying the movement of the coldest point of the cold cell. Figure 5 shows the timewise temperature variation for six of the 15 different positions where thermocouples were installed. Typically, the five probes positioned in the upper plane of the cavity (see Fig. 1) revealed the same trend and suggest about the same stratified temperature during the test period (5 h). Because of this, just one record from the upper thermocouples is shown. In the center plane the situation is exactly the same from t = 0 min until t = 200 min, when the temperature measured by the thermocouple near the cold wall decreases. This happens because at this time the cold circulation cell is close to enveloping the thermocouple tip, which occurs at about t = 235 min. Once its end is situated inside the cold eddy, the probe shows T < T m. In the lower part of the test cell, the temperatures reveal larger temperature gradients due to the downward motion of the fluid over the cold wall, and then the liquid flows horizontally along the bottom. This causes stratification inside the large circulation cell. The records from three of the five thermocouple probes in the lower plane show the same behavior as in the center plane. Here, the thermocouple on the left-hand side (SW) is inside the cold eddy for t > 25 min (see Figs. 3a and 3b). The vertical dashed lines in Fig. 5 represent the times

t : ~5 min

=60rnin -'32C~0mmi~n

Figure 4. Interface position versus time for Experiment 1.

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Experiment 2 The second experiment was very similar to the first. Here, the temperature imposed at the cold wall was T,. = - 20°C, and consequently the ice grew at a greater rate. The convective motion, however, shows the same behavior once the liquid is subjected to about the same boundary conditions. It is limited by the same hot and fusion temperatures ( T h and Tf); the only difference is in the local interface velocity. For short times this velocity does not significantly modify the flow patterns, and this can be seen in Figs. 6 a - 6 e . Figures 3b and 3c and 6b and 6c present the flows for t = 6 0 min and t = 120 min for Experiments 1 and 2, respectively. It is evident that the flow structures are very similar for the corresponding cases and that the sizes of the circulations are basically the same, except for a difference in the ice layer thickness. The thermocouples used to record the temperature evolution in Experiment 1 were also employed here. Figure 7 shows that the trend for this experiment differs little from the one observed in Experiment 1. It is interesting to compare Figs. 6 and 7 and note the temperature at SW and the relative position with respect to the cold cell for t = 20 min and t = 60 min. Observations of the W thermocouple records are also revealing. For t = 120 min (Fig. 6c) the thermocouple tip is inside the hot eddy and T(W) > 6°C. Fifteen minutes later (Fig. 6d) the probe is between the two circulation cells and T(W)--4°C. Finally, when t = 150 min T(W) < 2°C, and the thermocoupie head is in the upper part of the cold cell. In general, for the left-hand thermocouples (NW, W, and SW), the temperature drop due to the presence of the cold circulation cell occurs early in this experiment compared to the first. It is evident that the reason for that is the greater rate of ice growth in Experiment 2. This growth, however, does no significantly affect the central and right-hand-side temperatures.

Effect of Density Extremum on Solidification

(a)

(b)

(c)

(d)

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Figure 6. Photographs of flow patterns for Experiment 2 with t = 120 min; (d) t = 135 min; (e) t = 150 min.

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Experiment 3

Experiment 4

The growth of both the solid region and the lower left side circulation is greatly influenced by the increase of the hot initial and right-wall temperatures. These processes were observed in the third and fourth experiments. Figures 8a and 8b show the seeded flow for Th = 15°C (Experiment 3) at t = 30 and 90 min, respectively. Comparisons with previous cases demonstrate that even for earlier times the cold circulation cell appears bigger when T h is smaller (compare Figs. 6b and 8b). Figure 9 presents the temperature-time records, and it can be confirmed that during the test period only the SW temperature probe registered a temperature below the T,n expected inside the cold cell.

The highest initial temperature for all tests was T h = 20.2°C in Experiment 4. The first picture in this case (Fig. 10a) was taken at t = 17 min, and it is not possible to see a clockwise-rotating cell. Only the large eddy is evident, and it embraces the entire liquid region. As time progresses, the cold circulation appears weaker than in the other experiments. The temperature history presented in Fig. 11 shows the expected trends. Experiment 5 In Experiment 5,

Th

= 6.5°C, Tc = - 2 0 ° C , and the den-

710

S . L . Braga and R. Viskanta

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(b) 0a) Figure 8. Photographs of flow patterns for Experiment 3 with T h = 15.1°Cand Tc = - 2 0 ° C . ( a ) t = 30 rain; (b) t = 90 rain.

Figure 10. Photographs of flow patterns for Experiment 4 with T h = 20.2°C and T,, = - 2 0 ° C . (a) t = 17 min; (b) t = 90 min.

Effect of Density Extremum on Solidification sity distribution parameter R = 0.61 has the maximum value. Because of this, at steady state the cold circulation area is expected to be bigger than the hot one. The Rayleigh number is smallest, and convection is weak. Analysis of the results reveals that the convective flow is weaker than it appears in the other experiments. Figure 12a shows a flat and vertical solid-liquid interface due to the predominance of the heat transfer by conduction. At time t = 17 min the cold circulation starts to grow. One hour after the beginning of the test, the ice is thicker close to the upper part of the cold eddy, while above the eddy the thickness decreases owing to contact with the hot cell (see Fig. 12b). This behavior is exactly the same as that which occurs in the other experiments. In this case, however, the cold circulation cell is relatively thin and high, and within a 2 h period it reaches the top of the cavity as revealed in Fig. 12c. After this (Figs. 12d and 12e), both the ice thickness and the width of the cold cell increase with time in the upper part of the cavity. This happens because the water density decreases with the decrease in temperature if T < Tin. In this case, after the cold eddy has reached the free surface, its tendency is to enlarge by the accumulation of the colder and lighter fluid in this region. After this condition has been reached, the maximum ice growth rate will occur in this region of the test cell. Chao and Schoenhals [23] conducted an experimental study of a closed two-phase thermosyphon for ice production. For the results reported, the water was precooled to near freezing before the experiments were begun. It is very interesting to note there that, for all presented cases, the thickness of the ice increased from the bottom to the top of the tank. This reveals the same behavior as was obtained in our last experiment after the cold cell had reached the free surface. In Ref. 23, as the initial water temperatures were certainly below Tm, the moving fluid arose along the solid-liquid interface while it was becoming colder and lighter. The evolution with time of the temperatures inside the test section for Experiment 5 is presented in Fig. 13. In this experiment, the records from the thermocouples situated in the left (NW, W, SW) and center vertical (N, C, S) planes are shown. It can readily be seen when the NW

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and W thermocouples were enveloped by the cold cell and later by the ice. PRACTICAL S I G N I F I C A N C E It is well known that convective motion influences strongly the solidification process. The main objectives of this study are to identify how the natural convection motion develops in a fluid that has a maximum density and how this extremum influences the growth of the solid region. Knowledge of this phenomenon can be used to optimize the phase-change process when the solidifying melt exhibits this behavior. By means of a flow visualization technique it was observed that there is a region where the maximum rate of freezing occurs with time. This would not occur with Boussinesq fluids and explains why, in some cases, water freezes first at the free surface and then later the solidification front propagates downward. CONCLUSIONS An experimental study was performed in order to better understand the transient thermally driven natural convection in the presence of the density inversion in water and its effect on the solidification process in a partially filled (with a free surface) rectangular test cavity. Five experiments were carried out in which the initial temperature was varied from 6.5°C to 20.2°C in order to examine its influence on the flow structure and the temperature distribution. For one initial temperature ( T h = 10.2°C), two cold-wall temperatures were tested ( T c = - 1 0 ° C and -20°C). The maximum modified Rayleigh number investigated here was around 4.5 x 108, and no disturbances were observed in the laminar flow. The results obtained in the present study can be summarized as follows: • The density inversion of water has an important bearing on natural convective flow and heat transfer, which strongly affects the local freezing rate. • Two counterrotating eddies arose in all experiments because of the density inversion. • The maximum rate of solidification occurs near the top of the cold eddy where the lowest temperature region in the moving fluid is encountered. This work was done when one of the authors (S. L. B.) held a visiting appointment at Purdue University. He would like to express his appreciation to the Brazilian government for supporting his sabbatical leave during the 1988-1989 academic years. He would like also to acknowledge the support of the Secretaria de Ci~ncia e Tecnologia.

25

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NOMENCLATURE

T (*C)

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1 60 t (min)

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120

A c g H k L Pr R

aspect ratio ( = H / L ) , dimensionless specific heat, J / ( k g - ° C ) gravitational acceleration, m / s 2 height of liquid level, m thermal conductivity, W / ( m . °C) length of cavity, m Prandtl number, Eq. (3), dimensionless density distribution parameter, Eq. (2), dimensionless

712

S.L. Braga and R. Viskanta

Ra T t x y

Rayleigh n u m b e r , Eq. (1), d i m e n s i o n l e s s t e m p e r a t u r e , °C time, s horizontal distance from cold wall, m vertical coordinate, m

d y n a m i c viscosity, N . s / m 2 k i n e m a t i c viscosity, m 2 / s

c f h m r

cold fusion hot maximum reference

Subscripts

Greek Symbols a p

/~ v

t h e r m a l diffusivity, m 2 / s density, k g / m 3

(a)

(b)

(c)

(d)

(e) Figure 12. Photographs of flow patterns for Experiment 5 with T h = 6.5°C and 7",. = - 20°C. (a) t = 17 min; (b) t = 60 min; (c) t = 120 min; (d) t = 180 rain; (e) t = 255 ram.

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Figure 13. T e m p e r a t u r e versus time for E x p e r i m e n t 5.

REFERENCES 1. Gebhart, B., and Mollendorf, J., A New Density Relation for Pure and Saline Water, Deep Sea Res., 24, 831-848, 1977. 2. Lin, D. S., and Nansteel, M. W., Natural Convection in a Vertical Annulus Containing Water Near the Density Maximum, J. Heat Transfer, 109, 899-905, 1987. 3. Ho, C. J., and Lin, Y. H., Natural Convection Heat Transfer of Cold Water Within an Eccentric Horizontal Cylindrical Annulus, J. Heat Transfer, 110, 894-900, 1988. 4. Gebhart, B., Bendel, M. S., and Shaukatullah, H., Buoyancy Induced Flows Adjacent to Horizontal Surfaces in Water Near Its Density Extremum, Int. J. Heat Mass Transfer, 22, 137-149, 1979. 5. Mollendorf, J. C., and Jahn, K. H., Onset of Convection in a Horizontal Layer of Cold Water, J. Heat Transfer, 105, 460-464, 1983. 6. Watson, A., The Effect of the Inversion Temperature on the Convection of Water in an Enclosed Rectangular Cavity, Q. J. Mech. AppL Math., 25, 423-446, 1972. 7. Lin, D. S., and Nansteel, M. W., Natural Convection Heat Transfer in a Square Enclosure Containing Water Near Its Density Maximum, Int. J. Heat Mass Transfer, 30, 2319-2328, 1987. 8. Forbes, R. E., and Copper, J. W., Natural Convection in a Horizontal Layer of Water Cooled from Above to Near Freezing, J. Heat Transfer, 97, 47-53, 1975.

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9. Robillard, L., and Vasseur, P., Convective Response of a Mass of Water Near 4°C to a' Constant Cooling Rate Applied on Its Boundaries, J. Fluid Mech., 118, 123-141, 1982. 10. Robillard, L, and Vasseur, P., Transient Natural Convection Heat Transfer of Water with Maximum Density Effect and Supercooling, J. Heat Transfer, 103, 528-534, 1981. 11. Robillard, L., and Vasseur, P., Effet du maximum de densit6 sur la convection libre de l'eau dans une cavit6 ferm6e, Can. J. Civil Eng., 6(4), 481-493, 1979. 12. Vasseur, P., and Robillar, L., Transient Natural Convection Heat Transfer in a Mass of Water Cooled Through 4°C, Int. J. Heat Mass Transfer, 23, 1195-1205, 1980. 13. Inaba, H., and Fukuda, T., An Experimental Study of Natural Convection in an Inclined Rectangular Cavity Filled with Water at Its Density Extremum, J. Heat Transfer, 106, 109-115, 1984. 14. Lankford, K. E., and Bejan, A., Natural Convection in a Vertical Enclosure Filled with Water Near 4°C, 3. Heat Transfer, 108, 755-763, 1986. 15. Seki, N., Fukusako, S., and Inaba, H., Free Convective Heat Transfer with Density Inversion in a Confined Rectangular Vessel, Wiirrne Stoffiibertr., 11, 145-156, 1978. 16. Yewell, R., Poulikakos, D., and Bejan, A., Transient Natural Convection Experiments in Shallow Enclosures, J. Heat Transfer, 104, 533 538, 1982. 17. Ivey, G. N., Experiments on Transient Natural Convection in a Cavity, J. Fluid Mech., 144, 389-410, 1984. 18. Braga, S. L., and Viskanta, R., Transient Natural Convection of Water in a Rectangular Cavity Near Its Density Extremum, Int. J. Heat Mass Transfer, 35, 861-875, 1992. 19. Cheng, K. C., Inaba, H., and Gilpin, R. R., Effect of Natural Convection on Ice Formation Around an Isothermally Cooled Horizontal Cylinder, J. Heat Transfer, 110, 931-937, 1988. 20. Chellaiah, S., and Viskanta, R., Freezing of Water Saturated Porous Media in the Presence of Natural Convection: Experiments and Analysis, J. Heat Transfer, 111, 425-432, 1989. 21. Sugawara, M., Inaba, H., and Seki, N., Effect of Maximum Density of Water on Freezing of a Water-Saturated Horizontal Porous Layer, J. Heat Transfer, 110, 155-159, 1988. 22. Gebhart, B., Jaluria, Y., Mahajan, R. L., and Sammakia, B., Buoyancy-lnduced Flows and Transport, Hemisphere, New York, 1988. 23. Chao, S. L., and Schoenhals, R. J., An Experimental Study of a Closed Two-Phase Thermosyphon for Ice Production, 20th Joint A S M E / A I C h E National Heat Transfer Conf., Paper 81-HT-16, 1981.

Received July 3, 1991; revised February 5, 1992