Transient Natural Convection Experiments in Shallow Enclosures

R. Yewell D. Foulikakos A. Bejan Assoc. Mem. ASME Department of Mechanical Engineering, University of Colorado, Boulder, Colo. 80309

Transient Natural Conwection Experiments in Shallow Enclosures This paper reports experimental observations on transient natural convection in enclosures at high Rayleigh numbers (1.28xl09, 1.49xl09) and low aspect ratios (0.0625, 0.112). The phenomenon consists of the establishment of thin intrusion layers along the horizontal adiabatic surfaces; in time, the intrusion layers exchange heat with the isothermal core of the cavity, leading to the thermal stratification of the core. The approach to steady state is gradual, contrary to the theoretical prediction of Brunt-Vaisala wave motion (Patterson and Imberger [6]). The measured durations of the observed transients agree very well with theoretical estimates.

Introduction The objective of this paper is to report a series of experimental observations on transient natural convection in a shallow enclosure with heated vertical ends and adiabatic horizontal walls. The most recent review article on natural convection in enclosures [1] shows that the research effort on this topic is focused almost exclusively on the steady-state regime of the phenomenon. With few exceptions [2, 3], the transient regime has been overlooked, despite its fundamental role in the establishment of any steady state. Furthermore, many engineering applications of the enclosure problem operate not in the steady state but in the transient regime (e.g., solar collectors, attics and other closed spaces in buildings, the discharge of thermal pollution into shallow bodies of water). In this paper we report experimental results documenting the transient regime in the parametric domain of high Rayleigh numbers (109 - 1010) and low geometric aspect ratios (H/L = 0.0625, 0.112). As pointed out in the preceding paragraph, the present transient experiments bridge a gap in the research on convection in enclosures. The present experiments add also to the relatively scarce information available on the steady state in the (high Ra, low H/L) domain: the only steady-state experiments in this domain have just been reported [4, 5]. An important stimulus for present experimental study has been the recent theoretical paper by Patterson and Imberger [6], to which the authors of this report have had early access. Patterson and Imberger relied on pure scaling arguments to piece together complete "scenarios" for the evolution of the bouyancy-driven phenomenon in the transient regime. Patterson and Imberger were able to test some of their predictions by simulating the transient flow numerically in the parametric domain 10"1 < Ra < 1.4 x 105,H/L = 1 andPr = 2, 7. Inasmuch as the scaling scenarios constructed by Patterson and Imberger [6] are essential to understanding the theoretical foundation of the transient regime, it is important to verify them experimentally in the parametric domain in which they have not been tested. The First Transient Experiment We carried out two transient experiments in two separate enclosures, using water (Pr = 6) as the working fluid. The positions of the two experiments are labeled A, B on the (aspect ratio - Rayleigh number) chart of Fig. 1. On the same chart we show the regions occupied by four of the transient scenarios (regimes II-V) described in reference [6]. We also show some of the characteristics of each scenario. In addition Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division March 1, 1982.

Journal of Heat Transfer

to the present experiments (A,B), Fig. 1 shows the location of the numerical experiments (runs 2-5) reported in reference [6] for a square cavity filled with water. The present experiments (A,B) lie in a parametric domain which has not been investigated previously. We performed the first experiment (A) in a shallow enclosure 2.44-m long, 15.2-cm tall and 76-cm wide (Fig. 2). The main construction details of this apparatus are reported in reference [5]. For the present experiment we increased the number of access ports through the upper wall, from 8 to 30; through these ports we lowered thermocouples and velocity probes into the cavity. In addition, we used a special temperature probe consisting of eleven bead-in-glass (1-mm dia) thermistors aligned in the vertical direction. Details of this special probe are presented in Fig. 2. The probe was mounted close to the geometric center of the cavity, with the thermistor stems oriented perpendicularly to the two-dimensional flow (i.e., normal to the enclosure cross-section shown in the top half of Fig. 2). Through this thermistor array we were able to monitor the history of the vertical temperature profile across the enclosure. The experiment consisted of instantaneously changing the end-temperatures of the apparatus, at a time when the water space was isothermal (T0=24.0°C) and in equilibrium with the ambient. The cold-end temperature was lowered to TL = 15°C, by suddenly circulating precooled fluid (from a constant-temperature-bath refrigerator) through the cooling jacket of the apparatus. The warm-end temperature was raised to TH = 35°C, by suddenly turning the power on and dissipating 300 W in the heaters embedded in the warm end. The selection of the proper temperature levels, TL and TH, formed the subject of a series of preliminary experiments in which the power dissipated in the electric heaters and the refrigerated bath temperature were adjusted until the resulting average temperature (TH + TL)/2 matched, as closely as feasable, the initial temperature of the fluid. Thus, we were able to reproduce in the laboratory the experiment theorized in [6], where both ends of the enclosure experience step temperature changes of equal magnitude. In the experiment, the end temperatures reached TH, TL not suddenly, but over a period of order 15 min. However, considering the time scale of the transient flow observed in the cavity (hrs, Fig. 3) we regard the temperature boundary conditions imposed on the apparatus as adequate to simulate the experiment of [6]. We monitored the experiment by recording the temperature at points under the access ports and at the eleven points defined by the thermistor array (Fig. 2). In all cases, the approach to the steady-state temperature occurred gradually. Within the accuracy of our instruments (±0.1 °C), we did not detect any temperature oscillation, regardless of the position

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RUN 2

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REGIME V

REGIME IV

steady state by decay of internal wave motion horizontal intrusion layers become viscous

REGIME III

REGIME II

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distinct vertical boundary layers; steady state by horizontal layering (no internal wave motion)

id 10

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Ra Fig. 1 Aspect ratio (H/L) - Rayleigh number chart, showing the position of the present transient experiments. The circles represent the numerical runs reported in [6],

of the temperature probe and the stage in the approach to the steady-state. In Fig. 3(a) and 3(b), we report the most representative part of this record, supplied by the thermistor array. Figure 3(a) shows the temperature history of each point tested in the center region of the enclosure; Fig. 3(b) displays the same information in a manner which emphasizes the evolution of the vertical temperature profile. The gradual approach to the steady state occurs in two distinct phases. In the first phase, which lasts approximately 4 hrs, the fluid achieves a thermally layered structure. The second phase lasts approximately 18 hrs, as is due to the slight discrepancy between the average temperature (TH + TL)/2 and the ambient T0. In the second phase, the enclosure "as a whole" reaches a new thermal equilibrium with the ambient: the duration of this process is governed by the conductance of the thermal insulation surrounding the apparatus. In principle, this second phase could be avoided by slightly increasing the Joule heating rate or by slightly increasing the temperature of the refrigerated bath; the experiment documented in Fig. 3(a) and 3(b) represents the best we were able to do to achieve this ideal condition. Comparison With Theory Comparing the experimental results with scenario V envisioned by Patterson and Imberger [6], we find agreement as well as one important discrepancy. The main feature of transient regime V is the approach to steady-state via thermal layering by diffusion between the two intrusion layers (wall jets) lining the horizontal walls of the cavity. This prediction is confirmed strongly by experiment. For example, Fig. 3(b) shows the formation of thin thermal wall layers in the early stages of the transient (0-lhr). These thin layers surround a region of fluid which is essentially isothermal. In time, the two wall layers are cooled and, respectively, heated by the

inner region. In the steady state the fluid sandwiched between the two intrusion layers is linearly stratified. The final thermal layering and the intrusion layers are shown with great clarity in Figs. 4 and 5. Figure 4 was drawn by first measuring the vertical temperature profile under each of the 30 access ports and, on this basis, by constructing the temperature surface T(x,y) prevailing over the cavity cross section. Figure 5 shows a sequence of velocity profiles measured under twelve of the thirty access ports. The velocity measurements were based on the thymol blue pH indicator method which has been used in a number of free convection experiments [7, 8]; the actual velocity probes and the velocity calculation procedure on which Fig. 5 is based are described in [8]. The velocity profiles of Fig. 5 and the isotherms of Fig. 4 show that at the end of the transient the cavity is filled by thermally stratified stationary fluid bounded above and below by wall intrusion layers. These results agree with scenario V described in [6], which predicts the presence of distinct horizontal intrusion layers in the steady state. The duration of the thermal layering of the transient experiment, Fig. 3, is also predicted correctly by the scaling arguments presented in [6]. According to the theory, the horizontal layering should be complete when

(?)

(1)

where t* is the dimensionless time / t* =

H2/a

(2)

and the sign " ~ " stands for "is of the same order of magnitude as." The time estimate t* is based on the argument that vertical end boundary layers are well established and of order 5 ~ HRa ~1/4; the duration of the transient, t, is obtained

Nomenclature D

g H

k L N

vertical dimension of the second experimental apparatus, shown in the upper half of Fig. 6 gravitational acceleration vertical dimension thermal conductivity horizontal dimension Brunt - Vaisala frequency, equation (7)

Pr = Prandtl number Ra = Rayleigh number, equation (3) t = overall duration of the transient phase, equation (2) t* = dimensionless time of transient, equation (1) TH = warm end temperature TL = cold end temperature TQ = initial temperature AT = temperature difference,

x,y = horizontal and vertical coordinates a = thermal diffusivity /3 = coefficient of thermal expansion 8 = thickness of the vertical boundary layers v = kinematic viscosity w = frequency of internal wave motion, equation (6)

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STATION NUMBER

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TEMPERATURE PROBE NO.

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Fig. 2 Top: longitudinal vertical cross section through the first experimental apparatus: bottom: details of the aggregate temperature probe

by dividing the thermal inertia of the entire cavity, (k/u) HLAT, by the end heat-transfer rate, kHAT/8. For the transient experiment of Fig. 3, we have Ra =

gW\TH-TL)

= 1.28xl0 9

(3)

which, combined with H/L = 0.0625 and expressions (1, 2) yields ?~3.7hr (4) This estimate agrees very well with the duration of the approach to thermal stratification shown in Fig. 3(a); for example, the highest probe (no. 0, bottom of Fig. 2) records the peak temperature in the upper intrusion layer at t = 4 hr. The same time constant governs the evolution of the temperature measured by the remaining thermistors in the array. The important discrepancy between experimental results and theory [6] concerns the absence of oscillations in the approach to steady state. Patterson and Imberger argued that for Rayleigh numbers in excess of

(

Pr

V

(5)

one should observe an internal wave motion with a frequency of order TV. [l+(H/L)2]'A

(6)

where TV is the Brunt-Vaisala frequency [9] TV~-

(a»Ra)' /2

H1

(7)

According to criterion (5), the internal wave motion is to be expected in domains IV and V on Fig. 1. For the first transient experiment (A), expressions (6, 7) yield 7V-u~0.55s-

(8)

The corresponding period of the oscillation is of order 11.5s. Although a fluctuation of this type is well within our measuring capability, we were unable to detect it. As mentioned in the preceding section, the temperature of all the points sampled in the cavity evolved smoothly. The discrepancy between the "wave" prediction and the "no-wave" observation could be attributed to the fact that experiment (A) lies relatively close to the wave/no-wave boundary (5) (frontier between scenarios III and IV on Fig. 1). To check the validity of this explanation, we conducted a second transient experiment which, parametically, is situated further from the wave/no-wave boundary (5). The Second Transient Experiment We performed the second experiment (B) in a water-filled horizontal cylindrical enclosure with different endtemperatures. Details regarding the construction of this apparatus were given in an earlier paper on steady-state natural convection in a horizontal pipe [8]. The enclosure has a 14-cm dia and a length of 1.25 m, hence, the geometric aspect ratio (D/L = 0.112) is almost double compared with that of the first experiment. We consider this enclosure adequate for testing the two-dimensional theoretical arguments [4] because, as shown in [8], in the high Ra regime the end (vertical) boundary layers as well as the horizontal

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Fig. 4 Pattern of isotherms at the end of the transient experiment: top, Ra = 7.67 x 10 8 ; bottom, Ra = 1.82 x 10 9 . The numbers on the isotherms represent ( 7 - TL)/(TH - TL).

intrusion layers are thin relative to the third dimension of the enclosure (D, perpendicular to the sketch shown in the upper half of Fig. 6). At a high enough Ra, the horizontal cylinder behaves very much like a horizontal two-dimensional cavity.

This opinion is the result of the earlier experiment [8] which sought and was unable to document the effect of wall curvature. It was found that at a Rayleigh number of order 109 the vertical end layers have a thickness of order £>Ra"l/4 — 1

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station

no.6

no.12

no.13

no.14

no.15

no.16

no.17

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VELOCITY SCALE

Fig. 5 Horizontal velocity profiles at twelve longitudinal positions along the cavity (Ra = 7.67 x 10s); see Fig. 2 for the numbering sequence of the access ports (stations)

!

PROBE I

the first experiment. The pertinent data describing this second transient are

2 3

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EXPERIMENTAL

APPARATUS

4

r0=27.5°C

COLD END

r H = 42°C,T L = 12°C

cc r-

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The end temperatures were established within 7 min by dissipating 115 W in the warm-end electric heaters, and by connecting the cooling jacket of the cold-end to a constanttemperature-bath refrigerator. The water temperature was monitored using chromel-alumel (type K) thermocouples, which were lowered into the cavity through the access ports. Since we were primarily interested in temperature fluctuations, the output from the thermocouples was recorded on tape or, if desired, displayed on the screen of an oscilloscope. The observed transient behavior of the cavity is similar to the behavior documented in experiment A (Figs. 3-5). Once again, we die/ not detect any temperature fluctuations. Figure 6 shows the temperature history at four places in the cavity, labeled 1-4 on the apparatus cross section. The steady state is achieved in approximately 2 hrs: this time interval is in very good agreement with the theoretical calculation