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Influence of Prandtl number and slot height on free convection in a narrow slot
Influence of Prandtl Number and Slot Height on Free Convection in a Narrow Slot V. E. Nakoryakov M. P. Reznichenko V. M. Chupin Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
..An experimental investigation of free convection in a two-dimensional rectangular open cavity (the Hele-Shaw cell) was carried out using an IR imager. The effect of Prandtl number and of the height of the working cavity on heat transfer at Rayleigh numbers varying from 4 × 102 to 108 was studied. Water, alcohol, and transformer oil were used as working liquids. The experiments were conducted with a constant density of heat flux at one of the end walls of the cavity and under isothermal conditions at the other. An increase in the mean Nusselt number for alcohol and transformer oil is observed in a rectangular cell and for water in a shallow cavity with an aspect ratio of 0.5. Correlations are obtained for the local and mean Nusselt numbers for the three liquids and different heights of the working cavity. Also presented are the values of dimensionless temperature and distribution of the local Nusselt number on the heated wall. The findings presented supplement the data obtained during a study of heat exchange in a rectangular cavity filled with a porous medium. Keywords: Hele-Shaw cell, free convection, IR imager, open cavity,
narrow slot
INTRODUCTION Convective flow of liquid in a porous medium was theoretically and experimentally studied in Refs. 1-4 in relation to the investigation of the distribution of salts in sandy beds. Later, numerical methods were used to solve equations characterizing heat exchange in a porous medium filled with a gas and bounded by plane rectangular surfaces at different temperatures [5]. This model is related to the problem of radiant heat transfer from the walls of a fissile core of a nuclear reactor via a multirod structure containing the porous material. As a result of the investigations, correlations were derived for the heat transfer coefficients that involve a wide range of Rayleigh number values and aspect ratios of the cavity. Numerical solutions are presented in Ref. 6 for two-dimensional steady free convection in a rectangular cavity with a constant density of heat flux on one of the vertical walls. Visualization of the flow for a wide range of Rayleigh numbers and aspect ratios substantially differs from that observed with two isothermal walls. However, we note that no papers are available on experiments with mixed conditions on the vertical walls. The present paper presents experimental results of a study of free convection for liquids with different Prandtl numbers in a vertical narrow cavity (the Hele-Shaw cell) in which a uniformly distributed heat flux is supplied to one end wall and the temperature is maintained constant
on the other end wall while the side and bottom walls are thermally insulated and the top is open (Fig. 1). The flow of a viscous fluid in a narrow cavity formed by parallel plates is often used for simulating hydrodynamic characteristics of liquid percolation in a porous medium [7]. This model is based on similarity of differential equations describing percolation and equations of motion of a viscous liquid between parallel closely arranged plates. The Hele-Shaw cell, as a model object for studying convective heat exchange in a porous medium, was used by Vorontsov et al. [8] and Gorin et al. [9], who present theoretical and experimental data for water. It should be noted that using the Hele-Shaw cell as a model object for studying heat exchange in a porous medium essentially simplifies the methods of conducting the experiment and at the same time is of independent value for understanding heat transfer in cooling the elements of electronic equipment and in the design of modern heat exchangers.
EXPERIMENTAL SETUP A schematic representation of the experimental setup is depicted in Fig. 1. This setup is described in detail in Ref. 8. A brief description is given here to enable us to point out the source of errors and to understand the results of these experiments.
Address correspondence to Dr. V. M. Chupin, Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, 38-18 Tereshkova Street, Novosibirsk 630072, Russia.
Experimental Thermaland Fluid Science 1993; 7:103-110 © 1993 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010
0894-1777/93/$6.00 103
104
V.E. Nakoryakov ct al.
v-M 7
As a working liquid we used degassed water, ethyl alcohol, and transformer oil. The depth of the liquid varied, being 9, 18, or 36 cm. The time taken to attain the steady regime was 6 - 8 h. Visualization of current lines was carried out by injection of a luminophore dye into the working cavity while the temperature field was visualized using a TV-M IR imager. The technique is described in detail in Ref. 8. The measured thermal emf of the thermocouples was applied via a commutator to a digital voltmeter of the G-1212 type connected via a parallel interface to a computer that was used to record and process the experimental data. The local heat transfer coefficient was calculated using the formula % = q/(T,,-
T0,)]
T~) = W / [ F ( T ~ -
(1)
where F is the area of the heater, T0x is the temperature of the water in a cell core at level x, and W is the electric power supplied to the heater. Physical parameters of the fluid were determined for T.f (Tw + T0x)/2. The local Nusselt number was determined from the running height of the cell, =
Nu x = a x X / a
= qx/[ a(T w -
T,,x) ]
(2)
The mean Nusselt number Nu was calculated throughout the entire height of the cell (by the level of the liquid) and the mean temperature of the heater T, Nu = ~ H . / A Figure 1. Schematic representation of experimental setup. To avoid heat loss, the side walls were manufactured of foam plastic with a thickness 5 cm, with a heater 1 and a copper rectangular tube 2 of cross section 7 x 5 m m for pumping through cold water clamped between them. Sheets of glass textolite 1 m m thick were glued to the inside surfaces of the walls. The heater is made of nichrome foil 10-1 mm thick, 2 mm wide, and 3.6 x 10-1 m long. It is glued onto the end wall of the foam plastic plate, which is 2 m m thick. The working dimensions of the cavity are H = 3.6 x 10 -1 m, L = 1.8 x 10 -1 m, h = 2 mm. Copper wires to which a direct current voltage was alternately applied from the controlled source were soldered to the heater through the holes in the end wall plate at distances of 9, 18, and 36 cm from the cavity bottom. Thus, the aspect ratio of the working cavity, A = H / L , could vary from 0.5 to 2. The temperature of the heater was measured by 12 calibrated nichrome-constantan thermocouples 3, which were glued to the heater with an epoxy resin with a copper oxide filler. The thermocouples were arranged, 15, 35, 51, 67, 90, 125, 160, 190, 225, 260, 295, and 330 mm from the bottom of the working cavity. The heater was used as a plate for studying heat exchange. An opposite end wall of the cavity was cooled with running water at a given temperature T=. The temperature of the liquid in the working cavity was measured on a vertical axis of cell symmetry by 12 thermocouples 3. The distances between the thermocouples were the same as on the heater. All the thermocouples were made of a wire 10 -1 mm in diameter, and the working head was 1.6 x 10- 2 mm in diameter.
= WHJ[FA(T-
T~)]
(3)
where Hx is the running height of the cell equal to 9, 18, or 36 cm. The temperature of the end wall being cooled in the experiment varied within the range 12-18 ° C. Heat flux q on the wall being heated was set between 10 s and 9.2 x 103 W / m 2. All the experimental results presented were obtained at A = 0.5, 1, and 2 for the three working liquids with Prandtl numbers 7, 17, and 298 for water, ethyl alcohol, and transformer oil, respectively. The local Rayleigh number was determined as Ra* = g~Sqh2x 2 P r / 1 2 v 2
(4)
where h 2 / 1 2 is the permeability of the Hele-Shaw cell. For the running height of the cell H X, the mean Rayleigh number is Ra* = g f l q h 2 H 2 P r / 1 2 v 2
(5)
ERROR ESTIMATES Experimental errors appeared mainly during measurements of heat flux, heated plate temperature, and liquid temperature inside the working cavity. Losses of heat through the back side of the plate with the heater were calculated using the Fourier conduction law. If the conductivity and geometry of the side wall and its temperature at two points are known, then the losses can be determined according to AT
Q= aA--
Ay
where A T = T 1 - T 2 and A = thermal conductivity.
Free Convection in a Narrow Slot Here T 1, T 2 are temperatures measured at two points of the side wall with the distance Ay = 1.2 cm between them, and A is the cross-sectional area of the side wall. The maximal value of Q was 0.8% of the general value of the heat flux applied. Thermocouples were calibrated in a thermostat with a mercury thermometer with a scale reading to 0.05°C. Resulting tables were entered into a computer and were used during the experiments for actual measurement of temperature. The error of measuring emf is determined by the accuracy of the device G 1212 and is equal to 5 × 10 -4. Radiative heat transfer through a free liquid surface was neglected because working liquids are not transparent to infrared radiation. Heat losses through the front and back from plastic walls are negligible due to their low conductivity. The cold side wall was kept isothermal by pumping water at a fixed temperature through a copper tube. Temperature deviations along the surface were less than + 0.4° C. Temperature control was performed by thermocouples attached to the copper tube. Uncertainties of experimental data presented in Figs. 6 and 7 are connected with the uncertainty of determination of both Ra* and Nu x. Uncertainty of the temperature difference AT = (Tw - T~) measurements consists of errors in measurements of the hot wall temperature Tw(~ 0.2 ° C) and fluid temperature T0x( ~ 0.1° C). Corresponding to limits of the examined range of temperature difference, the uncertainty of values of Ra* changed from 12% at small Ra* to 8% at large Ra*. Uncertainty of Nusselt number, found during the experiments according to Eq. (2), consists of the uncertainty of values IV(~ 10%) and AT and is within the ranges from 8% at small Ra* to 6% large Ra*. The estimated errors in Nu and Ra* are 6 - 8 % and 7-9%, respectively. The errors in the thermophysical properties of matching fluid, oil and alcohol, a,/3, u, A, and ~, are about _+5% (based on information supplied by the manufacturer). This includes the small variations in properties with temperature not considered in these calculations.
1@5
RESULTS AND DISCUSSION The experiments were conducted within the range of Rayleigh numbers 4 X 10 2 < Ra* < 10 8 with A varying from 0.5 to 2. Figures 2 - 4 present visualizations of flow and temperature fields for water for some values of q and A. It should be noted that isotherms in cells of different height originate on either the heated wall or the cavity bottom and end on a free surface of the liquid. In the lower half of the working cavity at A = 1 and A = 2, isotherms are S-shaped. At A = 0.5, the isotherms are flatter, and S-shaped isotherms are not observed for any value of the heat flux. The thickness of the thermal boundary layer increases upward on a hot wall and downward on a cold wall, which follows from analyzing the temperature fields shown in Figs. 2-4. These temperature fields give rise to fluid flow in the working cavity upward at the heated surfaces and downward at the cooled surface. For a heat flux q < 2.3 x 103 W / m 2, the center of fluid rotation is near the cold wall and displaced toward the bottom. An increase in the heat flux causes a change of current lines. The center of rotation is displaced toward the hot wall and rises upward, and the current lines near the wall being heated are, for the most part, parallel to the surface of the heater at any height of the cavity. Note that for A = 0.5 all the liquid actively participates in the flow, whereas for A = 2, about a third of the liquid at the bottom is slow-moving. Figure 5 demonstrates the effect of Prandtl number on the stationary distribution of the temperature field for A = 1 and q = 6.4 x 103 W//m 2. For relatively high Pr, convection slightly affects the shape of the isotherms, which in this case are nearly parallel to the end walls. This is similar to the results obtained in the case of pure heat conduction. With decreasing Pr, isotherms in the central portion of the cavity tend to take a horizontal position. For relatively low Pr, the structure of the temperature field consists of two boundary layers, one ascending along the heated wall and the other descending along the cool wall. Nonetheless, isotherms in the central portion of the
Figure 2. Isotherms and current lines for q = 6.4 x 10 3 W / m E at A = 2, T~ = 17° C.
106
V . E . Nakoryakov et al.
F i g u r e 3 . I s o t h e r m s and current lines for q = 6.4 x l0 s W / m : at A = 1, T, = 17°(;.
Figure 4. I s o t h e r m s and current lines for q = 6.4 x 10 3 W / m 2 at A = 0.5, T~ = 17°C.
a
b
e
Figure 5. I s o t h e r m s for q = 6.4 x 103 W / m 2 for A = 1. (a) Water; (b) ethyl alcohol; (c) t r a n s f o r m e r oil.
Free Convection in a Narrow Slot working cavity are horizontal, and those in t h e lower portion are S-shaped, which indicates heat transfer by pure convection. The dependence of the local Nusselt number Nu~ on the heated wall on Ra* for water, ethyl alcohol, and transformer oil for A = 1 and A -- 2 is shown in Fig. 6, from which it is apparent that the curves of the function Nu~(Ra*) have different slopes for different working liquids. The rate of increase in heat exchange is higher in the lower half of the working cavity. This should have been expected because the liquid from a cool wall comes here and therefore the highest temperature gradient is observed here. As the fluid ascends along the heated wall, its temperature elevates and heat transfer intensity drops. The slope of the curve has a linear dependence in logarithmic coordinates. Hence, the Nusselt number can be expressed by the relation Nu x = c Ra *m
(6)
for any portion of the curve with a constant slope. The parameters of Eq. (6) are presented in Table 1. Note that the experimental data for transformer oil are above the water and ethyl alcohol curves. The experimental data on heat exchange in a low cell with A = 0.5 with different values of heat flux q are given for alcohol and transformer oil in Table 2 and for water in Fig. 7. An analysis of Fig. 7 and the data of Table 2 shows that the slope of the curves for low values of heat flux is steeper than for high q for all liquids. This is due to the fact that for a shallow cavity, heat exchange by pure heat conduction in the central portion is not low even for large Rayleigh numbers though heat transport is mainly accomplished by convection in the boundary layers on the hot and cool walls. This situation is referred to as a pseudoboundary layer [10]. Proceeding from this reasoning, we can account for the shape of the curves in Fig. 7 and the data in Table 2 for alcohol and transformer oil. With low values of heat flux, heat exchange in a cavity occurs under the pseudoboundary layer regime. Enhancement of heat flux makes the curve flatter due to a transition of the pseudoboundary layer to the boundary layer.
....
I
,
,,i..,,l
•
'
'''"'I
,
,,,,,,i
I
i
,,
.,,,,i
i
,
ii,,,,
10
Table 1. The Values of c and m in Eq. (6) c
m
Ra*
0.147 0.625 0.034 0.301 0.579 0.969
0.547 0.404 0.609 0.443 0.462 0.419
4 2 1 8 4 2
x × x × x ×
103-2 105-3 104-8 106-1 103-2 105-1
Working Liquid
× × × × × x
105 108 106 108 105 107
Water Water Alcohol Alcohol Oil Oil
The dependence of the mean Nusselt number Nu, determined by the mean temperature of the heated wall and the entire height of the working cavity for all three fluids and for A = 0.5, 1, and 2, on the Rayleigh number is shown in Fig. 8 and Table 3. An analysis of these data reveals an increase in Nu with growth in q for all three liquids and A values due to enhanced heat removal from the wall by convection. The maximum heat transfer, however, is observed at A = 0.5 though the rate of growth is higher for A = 1. Figures 9a, 10a, and l l a present distributions of the dimensionless temperature and local Nusselt number on the heated wall for A = 0.5, 1, and 2 and various Rayleigh numbers for water. The local Nusselt number Nu~ on the heated wall expressed in terms of local temperature difference Tx - T® is determined by Nu x = A X / 0 x
(7)
and Ra~. was calculated from the width L of the cavity. It is apparent that the temperature profile with high Rayleigh numbers is close to linear for a great portion of the wall, particularly for A = 0.5, except for the inflection near the upper open surface of the liquid and the bottom wall. This is due to a low velocity of flow in the top right corner of the cavity and near the bottom left corner (see Figs. 2a). Variation of Nu x versus the dimensionless height X is shown in Figs. 9b, 10b, and l l b for various Ra numbers for A = 0.5, 1, and 2. Note that at high Ra~., the local number Nu x first rises and then slightly drops near the free surface of the liquid. The distribution of Nu x indicates the variation in the dimensionless temperature on the heated wall. Nu x decreases at X = 1 due to decreasing convection near the free surface. It should be noted that variation of the dimensionless temperature and Nu x
Table 2. Values of the Coefficients for the Cavity with A = 0.5 fO ~.1.1
.
lO ~
i
I .....
I
. . . . .
I0 ~
I111
i
i
i l,,,J
lO s I0 ~ Ro*
i
i
,
i,
iiII
i
I0 ~
i
i iii
IJ
Figure 6. Local Nusselt number versus Rayleigh number. ( ~ ) Transformer oil, A = 1, A = 2; (O) water, A = 1; (zx) water, A = 2; (O) ethyl alcohol, A = 1; ( n ) ethyl alcohol, A=2.
107
c
m
0.001 0.046 0.155 0.303 0.524 0.634 0.709
0.973 0.644 0.532 0.474 0.557 0.489 0.462
Working Liquid
Ra*
2 × 104-1 4 x 104-2 5 x 104-3 5 × 104-1 103-105 4 x 103-2 5 × 103-8
× × x x
106 106 106 107
x 10 s X 105
Alcohol Alcohol Alcohol Alcohol Oil Oil Oil
V . E . Nakoryakov et al.
108
/0:5
i
l
/ 0 ') Nux
lll.ll
,
l,ll,,l
I
l
,,,,m
,
l
,lllll
I
I
'
I lll'~
l
I 'Ill
• Nux = 0.0808 Ra*°.625 O Nux = 0.194 Ra*°'526
q = 6.4x103 W/m2 q = 9.2 xl03 W/m2
i , ,ll,~
l
~
l Illllll
tO ~
¢0
L
I
l lllllnl
/0 ~
l
I,,,,,.I
I
,ItltlJ
fOy
tO ~
,
Table 3. Values of the Coefficients c and m for the
Cavity with A = 0.5, 1, and 2
q = 103 W / m 2 q = 2x103 W/m2
H=90 m
i
I
O Nux = 0.0273 Ra *0"773 • Nux = 0.0452 Ra*°'699
W~
¢0
I
c
m
Ra*
0.025 0.007 0.025 0.064 0.028 0.057
0.5 0.56 0.048 0.508 0.548 0.495
8 8 8 7 7 7
X × x × × ×
106-3 10~'-5 106-5 105-1 105-2 10s-2
X x × × × ×
109 109 109 108 108 108
A
Working Liquid
0.5 1 2 0.5 1 2
Alcohol Alcohol Alcohol Oil Oil Oil
I,.I,
tO 6
10 7
Ra* Figure 7. Local Nusselt number versus Rayleigh number for A = 0.5. is of a similar character on the heated wall for both ethyl alcohol and transformer oil. PRACTICAL SIGNIFICANCE This p a p e r demonstrates that the heat transfer rate for a saturated porous m e d i u m is a complex function of Prandtl n u m b e r and geometrical parameters. Several anomalous p h e n o m e n a were displayed in experimental data for alcohol and transformer oil (e.g., increase of the average Nusselt n u m b e r Nu at A = 1). Therefore, the correlation ratio [Eq. (6)] and the data in Tables 1-3 should be taken into consideration in the practical application and theoretical solutions of such problems. On the other hand, the numerically calculated temperature fields shown in Figs. 1 - 4 help to qualitatively estimate the thickness of the thermal b o u n d a r y layer.
constant density of heat flux on one of the end walls. Using an I R imager, t e m p e r a t u r e fields were obtained that allow a qualitative evaluation of the thickness of the thermal boundary layer in various liquids and for various aspect ratios. The experiments revealed a single-vortex circulatory flow inside the cavity filled with alcohol or transformer oil and a two-vortex flow that was unsteady with increasing heat flux for a cavity filled with water. U n d e r the conditions considered, the d e p e n d e n c e of the local Nusselt number on the Rayleigh n u m b e r for cavities with aspect ratio A = 1 or 2 has an inflection, while for the cavity with A = 0.5 there is observed a
t208
O Ra= 1.52x106 Ra=3.84x106 • Ra=6.90xl06 O Ra= 12.77x106
0.06 ~ae
Water 0.04
A=0.5
o.oe CONCLUSIONS The effect of Prandtl n u m b e r and the height of the working cavity on heat transfer by natural convection has been experimentally studied in the Hele-Shaw cell with a
o..oo
al*|a
2O
t l
i
Ii
p , . i .
0.?
1~
. l l l l
O.,~
i . . .
×
11
I,
. . . . . . . .
0. 6
|
, . . . . .
0. 8
i,
(.0
$ ,
/0 4
]0 Nu
i
,,,,1~
.i.ilu
I
,
,,HIll
I
i
i
g,l,,i
I
O
H=360 mm, N---u=O.058 Ra "--~0'45s
•
H=Ig0 mm, N--u=0.028 ~-~o..~
•
H=90 mm, N--u=0.005 ~--*o.6311
3
[0
i
i
,,,,,.|
,
i,..,,
,o I ............................ .1_. ......... .........1 ~ Ra--3'S4x10~
60
• Ra=6.90xl06 O Ra= 12.77x106
Nux //O ~
Water
Water A=0.5
~
0.2
0.~
1
.
IO
10
........
/03
I
........
[0 ~
I
0
. . . . . . . . . . . . . . . . . . . . . . . . . .
lOS
10G
/Or~ lOS
0.0
W ~ b
Ra*
Figure 8. Dependence of mean Nusselt number on Rayleigh number for water.
0.6
O8
1.0
X
Figure 9. (a) Distribution of temperature on heated wall, A = 0.5; (b) distribution of local Nusselt number on heated wall, A = 0.5.
Free Convection in a Narrow Slot
o.o,~
0.08 Ra= 1.4.9xt~ Ra=3.74x106 • Ra=7.55x106 o Ra= 13.47x106 Water A=I
0..oll v
0.02
V
-
. , , . , , u . ,
•
• •
0.06
,i
• •
0.06
,
~ , , , , ,
i , , , , , ,
,,,
i , , , , , , ,
,,i
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109
w . . . .
Raffi1.52x106 Ra=3.83x106 Ra=7.70xt06
O Ra= 13.77x106 Ra=23.44x106
0.0~
Water
w
o.02
(ZOO
, ........
0.0
'00~
N.x
t .........
0.~
,,,
OA
.....
×
,.I
.........
0.6
O.00 o.o
~ .........
0.8
• **|tit|.
f.O
I ....
*.l*.
O.2
It*|
* , , l , . | . * * . | t t
OA
O.6
0.8
/D
X
200 f
Ra=7.70xl" Ra= 13.77x106 Ra--23.44x106
Nux [
Wa~r
• Ra=7"55x10~ O Ra= 13.47x10~
Water
~
1
A=2 5O
IO0
~
0 0 b
02
OA
0.6
0.8
1.0
X
Figure 10. (a) Distribution of temperature on heated wall, A = 1; (b) distribution of local Nusselt number on heated wail, A = 1.
separation of the Nusselt number with respect to the density of heat flux. With a given heat flux the mean Nusselt number attains a maximum at A = 1 for alcohol or transformer oil and at A = 0.5 for water. Proceeding from the experimental data, the correlations are derived for the local and mean Nusselt numbers at A = 0.5, 1, and 2 for all three working liquids. The values for dimensionless temperature show that the maximum-to-minimum temperature ratio on the heated wall increases with increasing Rayleigh number and aspect ratio, the distribution of the local Nusselt number Nu x indicating a variation of the dimensionless temperature.
NOMENCLATURE Aspect ratio of the Hele-Shaw cell (= H/L), dimensionless a Thermal diffusivi~ of liquid, mE/s F Area of heater, m 2 g Free fall acceleration, m / s 2 H Height of working cavity, m h Horizontal thickness of cavity, m L Horizontal length of cavity, m Nu x Local Nusselt number (= axx/A), dimensionless
0.0 b
0.2
OA
O. 6
0.8
t.O
X
Figure 11. (a) Distribution of temperature on heated wall, A = 2; (b) distribution of local Nusselt number on heated wall, A = 2. Mean Nusselt number ( = aHx/A), dimensionless Prandtl number ( = v/a), dimensionless Density of heat flux at y = 0, W / m z Rayleigh number ( = g/3qhZx z P r / 1 2 v 2 ) , dimensionless Ra* Mean Rayleigh number (=g/3qh2H f Pr/12~,2), dimensionless Ra~ Rayleigh number over cavity length (=g/3qhZL 2 Pr/12 u2), dimensionless r~ Temperature of heated wall, ° C T~ Temperature of cooled wall, ° C W Heater power, W x, y, z Rectangular coordinates, m X Coordinate along x axis ( = x/H), dimensionless Greek Symbols a Heat transfer coefficient, W/m 2 • °C /3 Isobaric coefficient of thermal expansion of liquid, Nu Pr q Ra*
A
o c - I
0x
Temperature [ = (T - T~)/(qL/A)], dimensionless A Thermal conductivity of liquid, W / ( m . °C) u Kinematic viscosity of liquid, mE/s Subscripts w Parameter on heated wall x Local value X Local value on heated wall
110
V . E . Nakoryakov et al. REFERENCES
1. Horton, C. W., and Rogers, F. T., Jr., Convective Currents in a Porous Medium, J. Appl. Phys., 16, 347-351, 1945. 2. Morrison, H. L., Rogers, F. T., Jr., and Horton, C. W., Convective Currents in a Porous Medium. II. Observation of Condition at Onset of Convection, J. Appl. Phys., 20, 1027-1032, 1949. 3. Rogers, F. T., Jr., and Morrison, H. L., Convective Currents in a Porous Medium. III. Extended Theory of the Critical Gradient, J. Appl. Phys., 21, 1177-1180, 1950. 4. Rogers, F. T., Jr., Schilberg, L. E., and Morrison, H. L., Convective Current in a Porous Medium. IV. Remarks on the Theory, J. AppL Phys., 22, 1476-1479, 1951. 5. Newell, M. E., and Schmidt, P. W., Heat Transfer by Laminar Natural Convection Within Rectangular Enclosures, J. Heat Transfer, 92C(1), 106-109, 1970. 6. Prasad, V., and Kulacki, F. A., Natural Convection in a Reetan-
gular Porous Cavity with Constant Heat Flux on One Vertical Wall, J. Heat Transfer, 106(1), 142-150, 1984. 7. Bear, J., Zaslavsky, D., and Irmay, S., in Physical Principles of Water Percolation on Seepage, S. Irmay, Ed., UNESCO, 1968. 8. Vorontsov, S. S., et al., Natural Convection in Hele-Shaw Cell, Int. J. Heat Mass Transfer, 34(3), 703-709, 1991. 9. Gorin, A. V., Nakoryakov, V. E., and Chupin, V. M., Heat Transfer Under Natural Convection in Narrow Channel, Eurotherm Seminar, No. 16, Pisa, pp. 213-218, October 1990. 10. Prasad, V., and Kulacki, F. A., Convective Heat Transfer in a Rectangular Porous Cavity--Effect of Aspect Ratio on Flow Structure and Heat Transfer, J. Heat Transfer, 106(1), 149-151, 1984.
Received May 22, 1991; revised March 1, 1993
..An experimental investigation of free convection in a two-dimensional rectangular open cavity (the Hele-Shaw cell) was carried out using an IR imager. The effect of Prandtl number and of the height of the working cavity on heat transfer at Rayleigh numbers varying from 4 × 102 to 108 was studied. Water, alcohol, and transformer oil were used as working liquids. The experiments were conducted with a constant density of heat flux at one of the end walls of the cavity and under isothermal conditions at the other. An increase in the mean Nusselt number for alcohol and transformer oil is observed in a rectangular cell and for water in a shallow cavity with an aspect ratio of 0.5. Correlations are obtained for the local and mean Nusselt numbers for the three liquids and different heights of the working cavity. Also presented are the values of dimensionless temperature and distribution of the local Nusselt number on the heated wall. The findings presented supplement the data obtained during a study of heat exchange in a rectangular cavity filled with a porous medium. Keywords: Hele-Shaw cell, free convection, IR imager, open cavity,
narrow slot
INTRODUCTION Convective flow of liquid in a porous medium was theoretically and experimentally studied in Refs. 1-4 in relation to the investigation of the distribution of salts in sandy beds. Later, numerical methods were used to solve equations characterizing heat exchange in a porous medium filled with a gas and bounded by plane rectangular surfaces at different temperatures [5]. This model is related to the problem of radiant heat transfer from the walls of a fissile core of a nuclear reactor via a multirod structure containing the porous material. As a result of the investigations, correlations were derived for the heat transfer coefficients that involve a wide range of Rayleigh number values and aspect ratios of the cavity. Numerical solutions are presented in Ref. 6 for two-dimensional steady free convection in a rectangular cavity with a constant density of heat flux on one of the vertical walls. Visualization of the flow for a wide range of Rayleigh numbers and aspect ratios substantially differs from that observed with two isothermal walls. However, we note that no papers are available on experiments with mixed conditions on the vertical walls. The present paper presents experimental results of a study of free convection for liquids with different Prandtl numbers in a vertical narrow cavity (the Hele-Shaw cell) in which a uniformly distributed heat flux is supplied to one end wall and the temperature is maintained constant
on the other end wall while the side and bottom walls are thermally insulated and the top is open (Fig. 1). The flow of a viscous fluid in a narrow cavity formed by parallel plates is often used for simulating hydrodynamic characteristics of liquid percolation in a porous medium [7]. This model is based on similarity of differential equations describing percolation and equations of motion of a viscous liquid between parallel closely arranged plates. The Hele-Shaw cell, as a model object for studying convective heat exchange in a porous medium, was used by Vorontsov et al. [8] and Gorin et al. [9], who present theoretical and experimental data for water. It should be noted that using the Hele-Shaw cell as a model object for studying heat exchange in a porous medium essentially simplifies the methods of conducting the experiment and at the same time is of independent value for understanding heat transfer in cooling the elements of electronic equipment and in the design of modern heat exchangers.
EXPERIMENTAL SETUP A schematic representation of the experimental setup is depicted in Fig. 1. This setup is described in detail in Ref. 8. A brief description is given here to enable us to point out the source of errors and to understand the results of these experiments.
Address correspondence to Dr. V. M. Chupin, Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, 38-18 Tereshkova Street, Novosibirsk 630072, Russia.
Experimental Thermaland Fluid Science 1993; 7:103-110 © 1993 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010
0894-1777/93/$6.00 103
104
V.E. Nakoryakov ct al.
v-M 7
As a working liquid we used degassed water, ethyl alcohol, and transformer oil. The depth of the liquid varied, being 9, 18, or 36 cm. The time taken to attain the steady regime was 6 - 8 h. Visualization of current lines was carried out by injection of a luminophore dye into the working cavity while the temperature field was visualized using a TV-M IR imager. The technique is described in detail in Ref. 8. The measured thermal emf of the thermocouples was applied via a commutator to a digital voltmeter of the G-1212 type connected via a parallel interface to a computer that was used to record and process the experimental data. The local heat transfer coefficient was calculated using the formula % = q/(T,,-
T0,)]
T~) = W / [ F ( T ~ -
(1)
where F is the area of the heater, T0x is the temperature of the water in a cell core at level x, and W is the electric power supplied to the heater. Physical parameters of the fluid were determined for T.f (Tw + T0x)/2. The local Nusselt number was determined from the running height of the cell, =
Nu x = a x X / a
= qx/[ a(T w -
T,,x) ]
(2)
The mean Nusselt number Nu was calculated throughout the entire height of the cell (by the level of the liquid) and the mean temperature of the heater T, Nu = ~ H . / A Figure 1. Schematic representation of experimental setup. To avoid heat loss, the side walls were manufactured of foam plastic with a thickness 5 cm, with a heater 1 and a copper rectangular tube 2 of cross section 7 x 5 m m for pumping through cold water clamped between them. Sheets of glass textolite 1 m m thick were glued to the inside surfaces of the walls. The heater is made of nichrome foil 10-1 mm thick, 2 mm wide, and 3.6 x 10-1 m long. It is glued onto the end wall of the foam plastic plate, which is 2 m m thick. The working dimensions of the cavity are H = 3.6 x 10 -1 m, L = 1.8 x 10 -1 m, h = 2 mm. Copper wires to which a direct current voltage was alternately applied from the controlled source were soldered to the heater through the holes in the end wall plate at distances of 9, 18, and 36 cm from the cavity bottom. Thus, the aspect ratio of the working cavity, A = H / L , could vary from 0.5 to 2. The temperature of the heater was measured by 12 calibrated nichrome-constantan thermocouples 3, which were glued to the heater with an epoxy resin with a copper oxide filler. The thermocouples were arranged, 15, 35, 51, 67, 90, 125, 160, 190, 225, 260, 295, and 330 mm from the bottom of the working cavity. The heater was used as a plate for studying heat exchange. An opposite end wall of the cavity was cooled with running water at a given temperature T=. The temperature of the liquid in the working cavity was measured on a vertical axis of cell symmetry by 12 thermocouples 3. The distances between the thermocouples were the same as on the heater. All the thermocouples were made of a wire 10 -1 mm in diameter, and the working head was 1.6 x 10- 2 mm in diameter.
= WHJ[FA(T-
T~)]
(3)
where Hx is the running height of the cell equal to 9, 18, or 36 cm. The temperature of the end wall being cooled in the experiment varied within the range 12-18 ° C. Heat flux q on the wall being heated was set between 10 s and 9.2 x 103 W / m 2. All the experimental results presented were obtained at A = 0.5, 1, and 2 for the three working liquids with Prandtl numbers 7, 17, and 298 for water, ethyl alcohol, and transformer oil, respectively. The local Rayleigh number was determined as Ra* = g~Sqh2x 2 P r / 1 2 v 2
(4)
where h 2 / 1 2 is the permeability of the Hele-Shaw cell. For the running height of the cell H X, the mean Rayleigh number is Ra* = g f l q h 2 H 2 P r / 1 2 v 2
(5)
ERROR ESTIMATES Experimental errors appeared mainly during measurements of heat flux, heated plate temperature, and liquid temperature inside the working cavity. Losses of heat through the back side of the plate with the heater were calculated using the Fourier conduction law. If the conductivity and geometry of the side wall and its temperature at two points are known, then the losses can be determined according to AT
Q= aA--
Ay
where A T = T 1 - T 2 and A = thermal conductivity.
Free Convection in a Narrow Slot Here T 1, T 2 are temperatures measured at two points of the side wall with the distance Ay = 1.2 cm between them, and A is the cross-sectional area of the side wall. The maximal value of Q was 0.8% of the general value of the heat flux applied. Thermocouples were calibrated in a thermostat with a mercury thermometer with a scale reading to 0.05°C. Resulting tables were entered into a computer and were used during the experiments for actual measurement of temperature. The error of measuring emf is determined by the accuracy of the device G 1212 and is equal to 5 × 10 -4. Radiative heat transfer through a free liquid surface was neglected because working liquids are not transparent to infrared radiation. Heat losses through the front and back from plastic walls are negligible due to their low conductivity. The cold side wall was kept isothermal by pumping water at a fixed temperature through a copper tube. Temperature deviations along the surface were less than + 0.4° C. Temperature control was performed by thermocouples attached to the copper tube. Uncertainties of experimental data presented in Figs. 6 and 7 are connected with the uncertainty of determination of both Ra* and Nu x. Uncertainty of the temperature difference AT = (Tw - T~) measurements consists of errors in measurements of the hot wall temperature Tw(~ 0.2 ° C) and fluid temperature T0x( ~ 0.1° C). Corresponding to limits of the examined range of temperature difference, the uncertainty of values of Ra* changed from 12% at small Ra* to 8% at large Ra*. Uncertainty of Nusselt number, found during the experiments according to Eq. (2), consists of the uncertainty of values IV(~ 10%) and AT and is within the ranges from 8% at small Ra* to 6% large Ra*. The estimated errors in Nu and Ra* are 6 - 8 % and 7-9%, respectively. The errors in the thermophysical properties of matching fluid, oil and alcohol, a,/3, u, A, and ~, are about _+5% (based on information supplied by the manufacturer). This includes the small variations in properties with temperature not considered in these calculations.
1@5
RESULTS AND DISCUSSION The experiments were conducted within the range of Rayleigh numbers 4 X 10 2 < Ra* < 10 8 with A varying from 0.5 to 2. Figures 2 - 4 present visualizations of flow and temperature fields for water for some values of q and A. It should be noted that isotherms in cells of different height originate on either the heated wall or the cavity bottom and end on a free surface of the liquid. In the lower half of the working cavity at A = 1 and A = 2, isotherms are S-shaped. At A = 0.5, the isotherms are flatter, and S-shaped isotherms are not observed for any value of the heat flux. The thickness of the thermal boundary layer increases upward on a hot wall and downward on a cold wall, which follows from analyzing the temperature fields shown in Figs. 2-4. These temperature fields give rise to fluid flow in the working cavity upward at the heated surfaces and downward at the cooled surface. For a heat flux q < 2.3 x 103 W / m 2, the center of fluid rotation is near the cold wall and displaced toward the bottom. An increase in the heat flux causes a change of current lines. The center of rotation is displaced toward the hot wall and rises upward, and the current lines near the wall being heated are, for the most part, parallel to the surface of the heater at any height of the cavity. Note that for A = 0.5 all the liquid actively participates in the flow, whereas for A = 2, about a third of the liquid at the bottom is slow-moving. Figure 5 demonstrates the effect of Prandtl number on the stationary distribution of the temperature field for A = 1 and q = 6.4 x 103 W//m 2. For relatively high Pr, convection slightly affects the shape of the isotherms, which in this case are nearly parallel to the end walls. This is similar to the results obtained in the case of pure heat conduction. With decreasing Pr, isotherms in the central portion of the cavity tend to take a horizontal position. For relatively low Pr, the structure of the temperature field consists of two boundary layers, one ascending along the heated wall and the other descending along the cool wall. Nonetheless, isotherms in the central portion of the
Figure 2. Isotherms and current lines for q = 6.4 x 10 3 W / m E at A = 2, T~ = 17° C.
106
V . E . Nakoryakov et al.
F i g u r e 3 . I s o t h e r m s and current lines for q = 6.4 x l0 s W / m : at A = 1, T, = 17°(;.
Figure 4. I s o t h e r m s and current lines for q = 6.4 x 10 3 W / m 2 at A = 0.5, T~ = 17°C.
a
b
e
Figure 5. I s o t h e r m s for q = 6.4 x 103 W / m 2 for A = 1. (a) Water; (b) ethyl alcohol; (c) t r a n s f o r m e r oil.
Free Convection in a Narrow Slot working cavity are horizontal, and those in t h e lower portion are S-shaped, which indicates heat transfer by pure convection. The dependence of the local Nusselt number Nu~ on the heated wall on Ra* for water, ethyl alcohol, and transformer oil for A = 1 and A -- 2 is shown in Fig. 6, from which it is apparent that the curves of the function Nu~(Ra*) have different slopes for different working liquids. The rate of increase in heat exchange is higher in the lower half of the working cavity. This should have been expected because the liquid from a cool wall comes here and therefore the highest temperature gradient is observed here. As the fluid ascends along the heated wall, its temperature elevates and heat transfer intensity drops. The slope of the curve has a linear dependence in logarithmic coordinates. Hence, the Nusselt number can be expressed by the relation Nu x = c Ra *m
(6)
for any portion of the curve with a constant slope. The parameters of Eq. (6) are presented in Table 1. Note that the experimental data for transformer oil are above the water and ethyl alcohol curves. The experimental data on heat exchange in a low cell with A = 0.5 with different values of heat flux q are given for alcohol and transformer oil in Table 2 and for water in Fig. 7. An analysis of Fig. 7 and the data of Table 2 shows that the slope of the curves for low values of heat flux is steeper than for high q for all liquids. This is due to the fact that for a shallow cavity, heat exchange by pure heat conduction in the central portion is not low even for large Rayleigh numbers though heat transport is mainly accomplished by convection in the boundary layers on the hot and cool walls. This situation is referred to as a pseudoboundary layer [10]. Proceeding from this reasoning, we can account for the shape of the curves in Fig. 7 and the data in Table 2 for alcohol and transformer oil. With low values of heat flux, heat exchange in a cavity occurs under the pseudoboundary layer regime. Enhancement of heat flux makes the curve flatter due to a transition of the pseudoboundary layer to the boundary layer.
....
I
,
,,i..,,l
•
'
'''"'I
,
,,,,,,i
I
i
,,
.,,,,i
i
,
ii,,,,
10
Table 1. The Values of c and m in Eq. (6) c
m
Ra*
0.147 0.625 0.034 0.301 0.579 0.969
0.547 0.404 0.609 0.443 0.462 0.419
4 2 1 8 4 2
x × x × x ×
103-2 105-3 104-8 106-1 103-2 105-1
Working Liquid
× × × × × x
105 108 106 108 105 107
Water Water Alcohol Alcohol Oil Oil
The dependence of the mean Nusselt number Nu, determined by the mean temperature of the heated wall and the entire height of the working cavity for all three fluids and for A = 0.5, 1, and 2, on the Rayleigh number is shown in Fig. 8 and Table 3. An analysis of these data reveals an increase in Nu with growth in q for all three liquids and A values due to enhanced heat removal from the wall by convection. The maximum heat transfer, however, is observed at A = 0.5 though the rate of growth is higher for A = 1. Figures 9a, 10a, and l l a present distributions of the dimensionless temperature and local Nusselt number on the heated wall for A = 0.5, 1, and 2 and various Rayleigh numbers for water. The local Nusselt number Nu~ on the heated wall expressed in terms of local temperature difference Tx - T® is determined by Nu x = A X / 0 x
(7)
and Ra~. was calculated from the width L of the cavity. It is apparent that the temperature profile with high Rayleigh numbers is close to linear for a great portion of the wall, particularly for A = 0.5, except for the inflection near the upper open surface of the liquid and the bottom wall. This is due to a low velocity of flow in the top right corner of the cavity and near the bottom left corner (see Figs. 2a). Variation of Nu x versus the dimensionless height X is shown in Figs. 9b, 10b, and l l b for various Ra numbers for A = 0.5, 1, and 2. Note that at high Ra~., the local number Nu x first rises and then slightly drops near the free surface of the liquid. The distribution of Nu x indicates the variation in the dimensionless temperature on the heated wall. Nu x decreases at X = 1 due to decreasing convection near the free surface. It should be noted that variation of the dimensionless temperature and Nu x
Table 2. Values of the Coefficients for the Cavity with A = 0.5 fO ~.1.1
.
lO ~
i
I .....
I
. . . . .
I0 ~
I111
i
i
i l,,,J
lO s I0 ~ Ro*
i
i
,
i,
iiII
i
I0 ~
i
i iii
IJ
Figure 6. Local Nusselt number versus Rayleigh number. ( ~ ) Transformer oil, A = 1, A = 2; (O) water, A = 1; (zx) water, A = 2; (O) ethyl alcohol, A = 1; ( n ) ethyl alcohol, A=2.
107
c
m
0.001 0.046 0.155 0.303 0.524 0.634 0.709
0.973 0.644 0.532 0.474 0.557 0.489 0.462
Working Liquid
Ra*
2 × 104-1 4 x 104-2 5 x 104-3 5 × 104-1 103-105 4 x 103-2 5 × 103-8
× × x x
106 106 106 107
x 10 s X 105
Alcohol Alcohol Alcohol Alcohol Oil Oil Oil
V . E . Nakoryakov et al.
108
/0:5
i
l
/ 0 ') Nux
lll.ll
,
l,ll,,l
I
l
,,,,m
,
l
,lllll
I
I
'
I lll'~
l
I 'Ill
• Nux = 0.0808 Ra*°.625 O Nux = 0.194 Ra*°'526
q = 6.4x103 W/m2 q = 9.2 xl03 W/m2
i , ,ll,~
l
~
l Illllll
tO ~
¢0
L
I
l lllllnl
/0 ~
l
I,,,,,.I
I
,ItltlJ
fOy
tO ~
,
Table 3. Values of the Coefficients c and m for the
Cavity with A = 0.5, 1, and 2
q = 103 W / m 2 q = 2x103 W/m2
H=90 m
i
I
O Nux = 0.0273 Ra *0"773 • Nux = 0.0452 Ra*°'699
W~
¢0
I
c
m
Ra*
0.025 0.007 0.025 0.064 0.028 0.057
0.5 0.56 0.048 0.508 0.548 0.495
8 8 8 7 7 7
X × x × × ×
106-3 10~'-5 106-5 105-1 105-2 10s-2
X x × × × ×
109 109 109 108 108 108
A
Working Liquid
0.5 1 2 0.5 1 2
Alcohol Alcohol Alcohol Oil Oil Oil
I,.I,
tO 6
10 7
Ra* Figure 7. Local Nusselt number versus Rayleigh number for A = 0.5. is of a similar character on the heated wall for both ethyl alcohol and transformer oil. PRACTICAL SIGNIFICANCE This p a p e r demonstrates that the heat transfer rate for a saturated porous m e d i u m is a complex function of Prandtl n u m b e r and geometrical parameters. Several anomalous p h e n o m e n a were displayed in experimental data for alcohol and transformer oil (e.g., increase of the average Nusselt n u m b e r Nu at A = 1). Therefore, the correlation ratio [Eq. (6)] and the data in Tables 1-3 should be taken into consideration in the practical application and theoretical solutions of such problems. On the other hand, the numerically calculated temperature fields shown in Figs. 1 - 4 help to qualitatively estimate the thickness of the thermal b o u n d a r y layer.
constant density of heat flux on one of the end walls. Using an I R imager, t e m p e r a t u r e fields were obtained that allow a qualitative evaluation of the thickness of the thermal boundary layer in various liquids and for various aspect ratios. The experiments revealed a single-vortex circulatory flow inside the cavity filled with alcohol or transformer oil and a two-vortex flow that was unsteady with increasing heat flux for a cavity filled with water. U n d e r the conditions considered, the d e p e n d e n c e of the local Nusselt number on the Rayleigh n u m b e r for cavities with aspect ratio A = 1 or 2 has an inflection, while for the cavity with A = 0.5 there is observed a
t208
O Ra= 1.52x106 Ra=3.84x106 • Ra=6.90xl06 O Ra= 12.77x106
0.06 ~ae
Water 0.04
A=0.5
o.oe CONCLUSIONS The effect of Prandtl n u m b e r and the height of the working cavity on heat transfer by natural convection has been experimentally studied in the Hele-Shaw cell with a
o..oo
al*|a
2O
t l
i
Ii
p , . i .
0.?
1~
. l l l l
O.,~
i . . .
×
11
I,
. . . . . . . .
0. 6
|
, . . . . .
0. 8
i,
(.0
$ ,
/0 4
]0 Nu
i
,,,,1~
.i.ilu
I
,
,,HIll
I
i
i
g,l,,i
I
O
H=360 mm, N---u=O.058 Ra "--~0'45s
•
H=Ig0 mm, N--u=0.028 ~-~o..~
•
H=90 mm, N--u=0.005 ~--*o.6311
3
[0
i
i
,,,,,.|
,
i,..,,
,o I ............................ .1_. ......... .........1 ~ Ra--3'S4x10~
60
• Ra=6.90xl06 O Ra= 12.77x106
Nux //O ~
Water
Water A=0.5
~
0.2
0.~
1
.
IO
10
........
/03
I
........
[0 ~
I
0
. . . . . . . . . . . . . . . . . . . . . . . . . .
lOS
10G
/Or~ lOS
0.0
W ~ b
Ra*
Figure 8. Dependence of mean Nusselt number on Rayleigh number for water.
0.6
O8
1.0
X
Figure 9. (a) Distribution of temperature on heated wall, A = 0.5; (b) distribution of local Nusselt number on heated wall, A = 0.5.
Free Convection in a Narrow Slot
o.o,~
0.08 Ra= 1.4.9xt~ Ra=3.74x106 • Ra=7.55x106 o Ra= 13.47x106 Water A=I
0..oll v
0.02
V
-
. , , . , , u . ,
•
• •
0.06
,i
• •
0.06
,
~ , , , , ,
i , , , , , ,
,,,
i , , , , , , ,
,,i
,,,
* nl
|lt*l.tt,
109
w . . . .
Raffi1.52x106 Ra=3.83x106 Ra=7.70xt06
O Ra= 13.77x106 Ra=23.44x106
0.0~
Water
w
o.02
(ZOO
, ........
0.0
'00~
N.x
t .........
0.~
,,,
OA
.....
×
,.I
.........
0.6
O.00 o.o
~ .........
0.8
• **|tit|.
f.O
I ....
*.l*.
O.2
It*|
* , , l , . | . * * . | t t
OA
O.6
0.8
/D
X
200 f
Ra=7.70xl" Ra= 13.77x106 Ra--23.44x106
Nux [
Wa~r
• Ra=7"55x10~ O Ra= 13.47x10~
Water
~
1
A=2 5O
IO0
~
0 0 b
02
OA
0.6
0.8
1.0
X
Figure 10. (a) Distribution of temperature on heated wall, A = 1; (b) distribution of local Nusselt number on heated wail, A = 1.
separation of the Nusselt number with respect to the density of heat flux. With a given heat flux the mean Nusselt number attains a maximum at A = 1 for alcohol or transformer oil and at A = 0.5 for water. Proceeding from the experimental data, the correlations are derived for the local and mean Nusselt numbers at A = 0.5, 1, and 2 for all three working liquids. The values for dimensionless temperature show that the maximum-to-minimum temperature ratio on the heated wall increases with increasing Rayleigh number and aspect ratio, the distribution of the local Nusselt number Nu x indicating a variation of the dimensionless temperature.
NOMENCLATURE Aspect ratio of the Hele-Shaw cell (= H/L), dimensionless a Thermal diffusivi~ of liquid, mE/s F Area of heater, m 2 g Free fall acceleration, m / s 2 H Height of working cavity, m h Horizontal thickness of cavity, m L Horizontal length of cavity, m Nu x Local Nusselt number (= axx/A), dimensionless
0.0 b
0.2
OA
O. 6
0.8
t.O
X
Figure 11. (a) Distribution of temperature on heated wall, A = 2; (b) distribution of local Nusselt number on heated wall, A = 2. Mean Nusselt number ( = aHx/A), dimensionless Prandtl number ( = v/a), dimensionless Density of heat flux at y = 0, W / m z Rayleigh number ( = g/3qhZx z P r / 1 2 v 2 ) , dimensionless Ra* Mean Rayleigh number (=g/3qh2H f Pr/12~,2), dimensionless Ra~ Rayleigh number over cavity length (=g/3qhZL 2 Pr/12 u2), dimensionless r~ Temperature of heated wall, ° C T~ Temperature of cooled wall, ° C W Heater power, W x, y, z Rectangular coordinates, m X Coordinate along x axis ( = x/H), dimensionless Greek Symbols a Heat transfer coefficient, W/m 2 • °C /3 Isobaric coefficient of thermal expansion of liquid, Nu Pr q Ra*
A
o c - I
0x
Temperature [ = (T - T~)/(qL/A)], dimensionless A Thermal conductivity of liquid, W / ( m . °C) u Kinematic viscosity of liquid, mE/s Subscripts w Parameter on heated wall x Local value X Local value on heated wall
110
V . E . Nakoryakov et al. REFERENCES
1. Horton, C. W., and Rogers, F. T., Jr., Convective Currents in a Porous Medium, J. Appl. Phys., 16, 347-351, 1945. 2. Morrison, H. L., Rogers, F. T., Jr., and Horton, C. W., Convective Currents in a Porous Medium. II. Observation of Condition at Onset of Convection, J. Appl. Phys., 20, 1027-1032, 1949. 3. Rogers, F. T., Jr., and Morrison, H. L., Convective Currents in a Porous Medium. III. Extended Theory of the Critical Gradient, J. Appl. Phys., 21, 1177-1180, 1950. 4. Rogers, F. T., Jr., Schilberg, L. E., and Morrison, H. L., Convective Current in a Porous Medium. IV. Remarks on the Theory, J. AppL Phys., 22, 1476-1479, 1951. 5. Newell, M. E., and Schmidt, P. W., Heat Transfer by Laminar Natural Convection Within Rectangular Enclosures, J. Heat Transfer, 92C(1), 106-109, 1970. 6. Prasad, V., and Kulacki, F. A., Natural Convection in a Reetan-
gular Porous Cavity with Constant Heat Flux on One Vertical Wall, J. Heat Transfer, 106(1), 142-150, 1984. 7. Bear, J., Zaslavsky, D., and Irmay, S., in Physical Principles of Water Percolation on Seepage, S. Irmay, Ed., UNESCO, 1968. 8. Vorontsov, S. S., et al., Natural Convection in Hele-Shaw Cell, Int. J. Heat Mass Transfer, 34(3), 703-709, 1991. 9. Gorin, A. V., Nakoryakov, V. E., and Chupin, V. M., Heat Transfer Under Natural Convection in Narrow Channel, Eurotherm Seminar, No. 16, Pisa, pp. 213-218, October 1990. 10. Prasad, V., and Kulacki, F. A., Convective Heat Transfer in a Rectangular Porous Cavity--Effect of Aspect Ratio on Flow Structure and Heat Transfer, J. Heat Transfer, 106(1), 149-151, 1984.
Received May 22, 1991; revised March 1, 1993