Natural convection experiments in a triangular enclosure

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Natural Convection Experiments in a Triangular Enclosure D. Poulikakosr and A. Bejan2 Nomenclature g f{ k L Nu Q

= = : : =

gravitationalacceleration,ms -2 maximum height of the enclosure,m thermal conductivity, Wm rK

length of the enclosure,m Nusseltnumber, equation (3) net heat transfer rate, W, equation (3) Ra : Rayleigh number basedon maximum height, equation(4) Re : Reynoldsnumber, equation (2) St : Strouhal number, equation(l) TB : bursting period, s, equation(l) T, : cold (top) wall temperature,K TH = warm (bottom) wall temperature, K 7,, : mean temperature, K, 7,, = (Tu + T'\ /2 AT = temperature difference between warm and cold walls, K At = time interval, s V - s t r e a kv e l o c i t yt o t h e r i g h t , m s - r W - width of the enclosure,m x - longitudinal position, m

phenomenon of natural circulation in air-filled triangular e n c l o s u r e si ,n t h e r a n g e7 . 5 x 1 0 4 < R a ( 1 0 6 .T h i s r a n g e o f Rayleigh numbers characterizes solar collector spacesof triangular shape;however,it is much too low to be relevant to the study of natural convection in building-sizedspaces. For example, in an attic the Rayleigh number reachesas highas 108-10e. The experiment describedin this note was designedto shed light on the unknown phenomenon which might occurin a large-scalesystem,for example, in a life-sizeattic spaceor in ocean waters near shores with sloping bottoms. For this reason, the experimental study focused on: (i) a high Rayleigh number range not studied before, 106-l0e; (ii) the flow regime and, especially, those flow features which may prove essential in the future effort of modeling the phenomenon analytically. These objectives were achievedusinga relatively large apparatus filled with air or water. The experiment simulated the "night" mode of operation of an attic space, or the "day" mode of a shallow-sloped coastal body of water heated from below by solar radiation. This simulation consisted of cooling the upper wall and heatingthe bottom wall of the experimental enclosure. The oppositeoperating regime (warm top, cold bottom) was shown earlierto h dominatedby pure conduction [4]. Experimental Apparatus

The main feature of the apparatus is a triangularcavityof length L : 737 mm, heightH : 152mm, and width ll = 559 mm. The height to length ratio is therefore,H/L = 0.207. The enclosureis shown drawn to scalein Fig. l. Thetopand Greek Symbols bottom walls were constructed out of massivepieces of a : thermal diffusivity, rr2 s I aluminum of thicknessl9 mm. Along thesewallsweachieved P : coefficient of volumetric thermal acceptableisothermalconditions(within I "C). Thesidewalls expansion,K I were constructed out of plexiglas plate with a thickness of 19 $ = boundary layer thickness,m mm. The sloping wall was maintained at constanttemperaturc y = kinematicviscosity,m2 s I by means of a jacket filled with chilled alcoholsupplied bya was constant temperature bath refrigerator. The refrigerator Introduction capable of controlling the coolant temperaturewithin0.1'C. The object of this note is to report the resultsof a study of The base wall was heated with eight electricstrip heaten steady natural convection heat transfer in an attic-shaped capableof dissipating a maximum power of 1OCIW. (triangular) space.The generalsubject of natural convection The temperature of the aluminum plates was measurd in enclosureshas receivedconsiderableattention; however, using ten Chromel-Alumel thermocouples(fiveon each plae) most of this research has been focused on rectangular en- equally spacedalong the centerlineof the apparatus (Fig.l). closuresIl,2l. Two recentpapersby Flack, Konopnicki, and The thermocoupleswere imbeddedin the aluminum wallsar Rooke t31 and Flack t4l discuss for the firsr time rhe distance of 1.6 mm from the surfacefacingtheinsideof th enclosure. In order to prevent the direct thermalcontrt -lDepartm.nt between the top (cold) and bottom (warm)walls,weadfld of Mechanical Engineering, University of Illinois at Chicago, Ch^icago,lll.60680. the "truncated tip" designshownin Fig. l. 'Department of Mechanical Engineering, University of Colorado, Boulder, The experiments were performed usingdistilledwater rd C o l o . 8 0 3 0 9 ,A s s o c .M e m . A S M E rl in air the triangular cavity. A stat6 sequence of steady C o n t r i b u t e d b y t h e H e a r T r a n s f e r D i v i s i o n f o r p u b l i c a t i o ni n r h e J o u n N e l o r il achievedby varying the electricheat input to thebottom Hrer TneNsret. Manuscript received by the Heat Transfer Division, Sept e m b e r2 0 , 1 9 8 2 . and keeping the top wall at constant temperaturc. IL

The tempe probed using

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fluctuatio successful fomer bet paratus wi insulation plywood (J study, the i walls) was flow: This only I min convection Overall, metic trianl roof is sup under the c sulated and that the flov experiments convection I space (as in e Temperature

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p r e x r g i o spst o l e heo'ler

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Flg.1 Schematlcof experlmental apparatus

t_E-'...l--l

T e m p e r o l u (r "eC ) Flg.2 {verage temperature dlstrlbutlon and standard devlatlon, Ra = 4.7 x 10o

fluctuations of the voltage supplied to the electric heaters were successfully reduced by installing a constant-voltage transfomer between the voltage supply and the heaters. The apparatus was insulated on all sides with 200 mm of fiberglass insulation supported by an outer container of l3-mm thick plywood (Fie. l). During the flow visualization phase of this study, the insulation covering the side windows (the triangular walls) was removed in order to be able to photograph the flow: This operation had no effect on the t'low, as it lasted only I min, which is "very fast" relative to transient natural convection in a cavity filled with water. Overall, the experimental apparatus simulates an asymmetic triangualr space of the kind found in an attic whose roof is supported structurally by a vertical wall positioned under the crest. However, since the vertical side wall is insulated and sincethe flow is driven by the sloped wall, it is felt that the flow and heat transfer picture revealed by the present experiments is also an adequate description of the natural convection phenomenon occurring in a symmetric triangular space(as in an A-shaped attic). Temperature Measurements The temperature field inside the triangular enclosure was probed using a variable depth Chromel-Alumel thermocouple

Journalof HeatTransfer

of 0.4-mm bead diameter. The thermocouple was mounted at the end of a stiff stainless steel capillary tube which was lowered to any desired depth through six access ports. The thermocouple voltage was measured with a digital voltmeter yielding temperature measurementsaccuratewithin 0.05'C. Extensive temperature measurements were made at one of the highest Rayleigh numbers achieved in the apparatus, Ra : 4.7 x 108: this appears to be the highest Rayleigh number studied experimentally in connection with natural convection in triangular enclosures.The flow regime at this high Rayleigh number was found to be turbulent; this finding agrees with Flack's observations of natural convection in an air-filled triangular enclosure, at much lower Rayleigh numbers (7.5 x 104 < Ra < 106). The time-averaged temperature at each point was determined after taking measurements at time intervals of 3 s over a period of 5 min. Figure 2 shows the temperature variation under each accessport. Both the timeaveraged temperature and the standard deviation are reported. The basic feature of the temperature field is the presenceof an approximately isothermal core which covers most of the local height. Near the top and the bottom, the temperature distribution departs abruptly from the isothermal pattern presentin the core, indicating the presence of thermal boundary layers. The temperature fluctuation range is considerably smaller in the core than near the walls,

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showing that the turbulence intensity decreasesaway from each wall. Flow Visualization To visualize the flow we used the thymol blue pH indicator technique developed by Baker [5]. This technique was used successfully in a number of natural convection experiments 16-91.In the presentstudy, we lowered a vertical stainlesssteel wire (welding rod, 0.8-mm dia) through each accessport. We then applied 6 V between the wire (cathode) and the nearest aluminum plate. As shown in Fig. 3, a dark streak was generated at the exposed (1.8-mm long) tip of the vertical wire. Since the flow was turbulent, we studied the local characteristics of the flow at discrete points, rather than trying to measurecomplete velocity profiles under each port. Figure 3 shows the evolution of the dark streak released

from the tip of the wire at a distanceof 4 mm from the bottom wall, under port no. 5. The photographs were taken at time intervals At = 2s. The evolution of the streak revealsthe existenceof a turbulent jet moving to the right (away from the tip of the enclosure).The "hill-valley" flow structure shown in Fig. 3 proves the existenceof a periodic fluid motion which consists of an "eruption away from the wall followed by inrush towards the wall" [0]. The mechanism responsiblefor eruptions and inrushes is still open to question; however,it is often suggestedthat the eruptions are produced by an "instability" of the boundary layer region. The large scale features documehted in Fig. 3 are also encountered in the transition to turbulence in boundary layer flow. The bursting period, I, (time between two consecutive bursts or in-rushes at a fixed point on the wall), can be calculated and cast in a dimensionless form as a Strouhal number 91= -iTsV

(la)

where 6, V are the boundary layer thickness and the velocity of the top of the dark streak. From Fig. 3 we found

st-_

I

(lr)

6.1

This result matches perfectly the growing volume of measurements on the bursting frequency of turbulent boundary layers, (the experimental measurementsare usually reported as T"V/6 = 5, which is predicted by the buckling theory of inviscid flow [0]). The Reynolds number for the boundary layer flow was

pg: Y :41

(2)

The size of the boundary layer thickness, 6, used in equations (l) and (2) was taken as approximately equal to the amplitude of the sinuous fluid motion (Fig. 3). The bursting period,Is, appearing in equation (l) was estimated as the wavelengthof the sinuous streak divided by the speed of the wave (to the right in Fig. 3). It is important to note that the sinuousstreak shown here in Fig. 3 is a property of the boundary layer,asa flow region of thickness, 6. It is observed that the sinuous character of the streak does not depend on the specific position of the probe, as long as the distancebetweenthebluegeneratingtip of the probe and the wall is lessthan 6. More flow visualization results have been compiled in I U. Heat Transfer Results Fig. 3- Streak pattern in lhe bottom iet ai accoss port no. 5, Ra = 4.7 x 1 0 o , A t= 2 s

The heat transfer results are reported in Fig. 4 as a plot of Nusselt number versusRayleigh number

For r the t moc( arith used propl evalu botto electr throu inclut tempr

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the ir perim the e acros Inc we ra worki space the Ri sidera to the The hr of ord good a results differe other I Raint notatio Nu=0.

wherear

Nu=

The fa correlat H/L, P Flack [z Ra. The Ir filled wi Nu :0.(

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I ro' Ro Fig. 4 Summary of heat transfer moasurements: Dlrection of experiments: o o IncreasingQ; o r decreasingQ

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This Nr Rayleigt between approxir attic spa range of slightly approxir data. Finalll the heat graduall5 heaters Figure 4 on the d paratus).

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(3)

Acknowledgment

Ra:ggll3 ATl (dv)

(4)

This research was supported by the National Science Foundation, through grant no. ENG78-20957. The authors thank Prof. Jorg Imberger of the University of Western Australia for his part in defining the present researchtopic. Messrs Michael Hacker, Karl Rupp, and Richard Cowgill constructedthe apparatus and its instrumentation.

Nu:QH|

For each steady state achieved in the apparatus, we measured the top and bottom plate temperatures using the ten thermocouples(five on each plate) located as shown in Fig. l. The arithmetic means of the measured local temperatures were used as the top and bottom wall temperatures. The physical properties appearing in the definitions of Nu and Ra were evaluated at the arithmetic mean temperature of the top and bottom walls. The net heat transfer rate, Q, was measured electrically, by recording the voltage across the current through the bottom heaters. Thus, the measured Q value includes the effect of radiation, which is negligible in the temperature range of this experiment. A separate experiment showed that the heat leak through the insulation is negligible (roughly 1.5 percent); in this experiment there was no coolant in the top wall jacket, so that the electric heat input was balanced by the leakge of heat acrossthe fiberglass insulation surrounding the apparatus. In order to check the validity of our high Ra measurements, we ran the same heat transfer experiment with air as the working fluid: heat transfer data for an air-filled triangular space have already been reported [4]. In our air experiment the Rayleigh number range was 106-107, which is still considerablyhigher than the regime testedin [a] (7.5 x 104 < Ra < 2 x 105). In l4l, H/L : 0.577 was the aspectratio closest to the aspect ratio of the present experiment (H/L : 0.207). The heat leak for the present air experiment was manageable, of order 25 percent. Figure 4 shows that our results are in good agreement with Flack's [4]. It should be noted that the results of [a] were reported based on the overall temperature (Tn + Tt)/2(in d i f f e r e n c eA T = T H - I , , , w h e r e T ^ : other words, Flack's A7'is half the A?nused to report Nu and Ra in the present experiments, equation (3, a)). In the present notation, Flack's data of Fig. 9 are correlated by N u : 0 . 2 2 5 R a 0 ' 3 (, H / L : 0 . 5 7 7 , a i r , 7 . 5 x l 0 a < R a < 2 x 1 0 5 ) (5) whereasthe present measurements for air follow the curve N u : 0 . 3 4 5 R a 0 . 3 ,( H / L : 0 . 2 0 7 , a i r , 1 0 6< R a < 1 0 7 ) ( 6 ) The fact that the numerical coefficients in the two Nu correlations (5, 6) are similar indicates that the aspect ratio, H/L, plays only a minor role in the Nu function. Note that Flack [4] found that decreasing H / L increasesNu for a given Ra. The Nusselt number measurements obtained with the cavitv filled with water are correlated within 4 percent by Nu :0.00038Ra0'5e,(H / L:o.207 , water, 2 . 5 x 1 0 8< R a < 6 . 5 x 1 0 8 )

(7)

This Nu result shows a relatively strong dependence on Rayleigh number. Recalling the Prandtl number difference betweenwater and air, equation (7) can only be regarded as an approximate guide for heat transfer calculations concerning attic spaces filled with air. However, since in the air-water range of Pr numbers the Nusselt number is influenced only slightly by Pr, the high-Ra water data represent a better approximation of Nu than the extrapolation of the low-Ra air data. Finally, we checkedthe repeatability of our data by running the heat transfer experimentsin the reversedirection, i.e., by gradually decreasing the power dissipated in the bottom heaters while keeping the coolant temperature constant. Figure 4 shows that the Nu-Ra measurements do not depend on the direction of the experiment (the history of the apparatus).

Journalof HeatTransfer

References I Ostrach, S., "Natural Convection in Enclosures," Advances in Heat Transfer, Vol.8, 1972,p. 16l. 2 Catton, I., "Natural Convection in Enclosures," Keynote Paper, Proceedings of the 6th Internatinal Heot Transfer Conference, Toronto 1978, V o l . 6 , 1 9 7 9 ,p p . l 3 - 4 3 . 3 Flack, R. D., Konopnicki, T. T., and Rooke, J. H., "The Measurement of Natural Convective Heat Transfer in Triangular Enclosures," ASME J o u n N e r o r H E l r T R l x s F r n , V o l . l 0 l , N o . , 4 , N o v . 1 9 7 9 ,p p . 6 4 8 - 6 5 4 . 4 Flack, R. D., "The Experimental Measurement of Natural Convection Heat Transfer in Triangular Enclosures Heated or Cooled from Below," ASME Jounxlr or Henr TuxsrEn, Vol. 102, Nov. 1980, pp.770-772. 5 Baker, D. J., "A Technique for the Precise Measurement of Small Fluid Vefocities," Journal of Fluid Mechanics, Vol. 26, 1966, pp. 573-575. 6 Imberger, J., "Natural Convection in a Shallow Cavity With Differentially Heated End Walls, Part 3: Experimental Results," Journal of Fluid Mechonics, Vol. 65, 1974, p. 246. 7 Kimura, S., and Bejan, A., "Experimental Study of Natural Convection in a Horizontal Cylinder With Different End Temperatures," Internotional Journal of Heat and Mass Transfer, Vol. 23, 1980, pp. lllT-1126. 8 Sparrow, E. M., Husar, R. B., and Goldstein, R. J., "Observations and Other Characteristics of Thermals," Journal of Fluid Mechanics, Vol. 41, 1970,pp. 793-800. 9 Vedhanayagam, M. Lienhard, J. H., and Eichhorn, R., "Method for Visualizing High Prandtl Number Heat Convection," ASME Jounxlr or Hrlr TR.eNsrEn,Vol. l0l, 1979, pp. 57 l-573. l0 Bejan, A., Entropy Generstion Through Heat and Fluid Flow, Wiley, New York, 1982,p. E7. . ll Poulikakos, D., "Natural Convection in a Triangular Enclosure Filled With Newtonian Fluid or Fluid-Saturated Porous Medium," Ph.D. thesis, University of Colorado, Boulder, May I 983.

Variable Fluid Property Effects on Transport in Pure Water Around the DensityExtremum R. D. Padlogr and J. C. Mollendorf2 Nomenclature b,c : similarity functions, G/4x and G Ce : fluid sPecific heat d - temperature difference, To - T* lt : nondimensional stream function, {t vc : nondimensional streamfunction at "f(0) the surface f (o) : nondimensional streamfunction far from surface g : gravitational acceleration G : modified Grashof number, t -

4lGr,/4 Gr, = local Grashof number, gx3 Ap,/ v2 hit = latent heat of melting (hit : 79.77 cal/gm)

-f Turbocompressor e*tni""l Engineer, Division,JoyMachinery Company, Bu,ffalo, N.Y. 14225

'Associate Professor, Department Mechanical and Aerospace of Engineering, State University of New York at Buffalo, Amherst, N.Y. 14260. MEM. ASME Contributed by the Heat Transfer Division for publication in the Jounxlr or Hser TRAxsrnn. Manuscript received by the Heat Transfer Division November 8, 1982.

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