Thermal Convection by Air in Small Cells Heated from Below Albert C. Kent* and James M. Bowyer, Jr.' Kansas State University, Manhattan, Kansas
Experimental results fot convective heat transfer by air in small cells between horizontal planes are presented for cell depth-to-width ratios from 0.5 5 1/D 5 2.0. Data were obtained by steady-state measurement on a vacuum-compatible guarded hot plate apparatus. Correlations of Nusselt and Rayleigh numbers are given in the form used previously for horizontal fluid layers. Comparisons are made with previous experiments on liquids and gases.
H o n e y c o m b materials have found increased use in structures for aircraft and in other applications where high strength-toweight ratios are desirable. I n many cases, these materials also need to be good insulators. An analysis by Dunkle, et aZ. (1958), and studies by Kendall, et aZ. (1963), on the apparent thermal conductivity (conductance) and the thermal response, respectively, of metallic honeycomb-core panels indicated that convection Py air in the cells could be neglected for most typical cases. However, many honeycomb cores are made of laminated resin-impregnated paper or glass materials; for such cores, there is much less heat conduction through the solid walls of the cells. Kendall, et al. (1963), concluded that, for nonmetallic honeycomb-core materials, convection by gas in the cells may be the dominant mode of heat transfer a t moderate temperatures. This observation was also made by Sauer and Nevins (1964). Thus, an accurate knowledge of the internal convective conductance of gases in such cells should be obtained. This paper reports on the experimental determination of thermal convective heat transfer by air enclosed in cells with low-conductivity, vertical walls. The effect of cell depth-towidth ratio upon convective heat transfer by air in hexagonal and square cells with LID ratios of l/z to 2 is presented. Thermal convective heat transfer is herein defined as heat transfer due to convective motion of a fluid between horizontal parallel plates. The convective motion is due to buoyancy and is driven by having the lower plate hotter than the upper plate. This type of convective heat transfer has received considerable attention in experimental and analytical investigations of undivided fluid layers whose horizontal extent greatly exceeded the distance between plates. Before 1966 very little experimental data (Kroutil, 1963) existed for convective heat transfer and onset of convection in fluids confined within cells between horizontal plates heated from below. Theoretical studies (e.g., Malkus and Veronis, 1958; Ostrach and Pneuli, 1963; Pellew and Southwell, 1940; Pneuli, 1964a,b) had predicted upper and lower bounds for onset of convection as a function of the depth-to-width ratio (L/D) for an enclosed fluid layer. However, most of these
* Present address, Thermal and Environmental Engineering Department, Southern Illinois University, Carbondale, Ill. 62901.
Present address, Defense Department, Gulf General Atomic,
San Diego, Calif. 92112.
solutions used mathematical models which did not match all the physical boundary conditions of the real situation. In 1967, Catton and Edwards presented experimental results showing the effect of vertical partitions in a horizontal liquid layer upon onset of convection and convective heat transfer. Their experimental results were compared with analytical results by using the concept of an equivalent wave number and the Malkus-Veronis power integral technique. Catton and Edwards (1967) reported on results based upon an extensive study by Catton (1966) on liquids in honeycomb cells. Davis (1967) developed a theory for predicting onset of convection in a threedimensional rectangular box of fluid heated from below. Edwards (1968) has presented an analysis for the suppression of three-dimensional cellular convection (in a horizontal fluid layer) by parallel vertical walls with finite conductance. Onset is shown to be a function of the wall conduction parameter H*. Limited experimental data for a silicone oil were obtained to provide experimental verification of the theory. Sun (1970) developed solutions which predict onset of convection by fluids in rectangular and round cylinders of arbitrary wall conductance. Onset is shown to be a function of H*, LID, WID, and, for transparent gases, H**. Design of the Experiment and Test Procedure
Experimental determination of the Nusselt numbers requires accurate measurement of the heat transfer due to natural convection. In each cell or enclosure of a cellular matrix, e.g., a honeycomb core, heat is transferred by gaseous conduction and/or convection, wall conduction, and intracellular radiation. Assuming that the gaseous conduction/convection may be separated from the effects of intracellular radiation and wall conduction, the gaseous conduction/convection may be determined from the energy balance &gas = &total - &walls - &radiation. The total heat transfer may be determined a t any pressure and temperature by using a suitable guarded hot plate apparatus. If the cellular test matrix is subjected to the same temperature difference a t high vacuum, the heat transfer is &hv = &walls -I- &radiation* The solid-solid radiation component of the total heat transfer should remain essentially constant if the temperature difference is the same as in the unevacuated test and the gas is nonabsorbing and nonradiating a t these temperatures. If the wall conduction is assumed to be the same regardless of the Ind. Eng. Chem. Fundam., Vol. 1 1, No. 3, 1972
3 19
Table 1. Summary of Tesf Samples Used* Test sample
Cell height,ln.
cmrssectien shape
designation
* * *
0.500 0.005 0.500 f 0.003 1.000 0.010 1.246 0 003 1.625 0.002
Cell' width, in.
* *
LID
Sample source
1.00 0.005 0.50 Hand-made 5-50 Hexagonal 0.437 0.010 1.14 Hexcel A-100 Hexagonal 1.000 f 0.020 1.00 Hand-made SQ-100 Sqnare 1.250 + 0.060 1.00 Hexcel H-100 Hexagonal 0,750 and H-200 Hexagonal 0.782 mixed 2.25 Hexcel 0.869 f 0.005 0.437 0.010 2.00 Hexcel A-200 Hexagonal 1.975 + 0.016 1.000 0.030 2.00 Hand-made SQ-200 Square *For hexagonal cells, this is the disktnce &cross flats. b Cell walls in every sample consisted of resin-impregnated paper.
*
* *
The effect of the parameter H* was not studied here, but the value of H* in the samples tested varied from 0.017 to 0.128. This implies that the walls were essentially insulating walls (Catton and Edwards, 1967). The presence of radiation, however, will cause the parameter H to have values between one-tenth and one for the samples tested. The effect of L / D was studied by using cells of the same cross-sectional shape with varying L / D . Table I gives B summary of the samples tested. Tesi Apparatus
Figure
1. Photograph of the g u a r d e d hot plote a p p a r a t u s H Center plate support
A Lifting solenoid 0 Rotary rolenoid
I Counter heater
C Latching bars
J Radiation shlelds
D Supportrods
K Hot plate thermocouples
E Coollnp plats
L Cellular test sample
F Center hot phte
M Cold plate thermocauples
0 Guard heatlng ring
I
,
I
F i g u r e 2. Cross secfion view paratus
Ill I
of the guarded hot plate ap-
gas pressure, with all other conditions the same, QBaa = - Qhv = h,AAT. The value of Nusselt number may then he calculated from Nu = h,L/k, For an ideal gas, the Rayleigh number may be defined as (Thompson and Sogin, 1966) Ra = gATLapzcp/TmagkR2. For a given cellular matrix, Ra can be varied by varying the pressure of the enclosed gas while holding AT and T, constant.
QWh*l
320 Ind.
The samples were tested in the apparatus shown in Figures 1 and 2. A central heater (F) consisting of a chromium-plated plate with ring heaters soldered to it supplied heat to the test sample. Four thermocouples (K) were imbedded in the plate to within 0.01 in. of the surface contacting the sample; these permitted measurement of the hot plate temperature. The hot plate was guarded by a ring heater (G) which minimized edge losses from the hot plate. A differential thermopile was used to measure the temperature difference across a n 0.080-in. air gap between the hot plate and guard heater. The counter-heating plate (I) was placed parallel to and 1 in. below the hot plate and guard heater. A differential thermopile was used to measure the temperature differenee between the lower surface of the hot plate and the upper surface of the counter-heating plate. Copper radiation shields (J) were soldered to the counter-heating plate to provide complete shielding of the hot plate from the surroundings. The cold plate (E) consists of a 6/8-iu.thick hollow chromium-plated bronze plate. Water was circulated through the plate to maintain a prescribed cooling plate temperature which was measured by two thermocouples imbedded on the plate surface. Thermocouples were also installed on the cold plate inlet and outlet water lines. The sample to be tested was inserted into the guarded hot plate between the hot and cold plates. The cold plate was suspended above the test sample by the latching bars (C). All heaters were turned on and adjusted for a hot plate temperature of about 340°F. The desired chamber pressure (10-5 to 1000 Torr) was obtained and temperatures and temperature differences were checked. Temperature differences between guard surfaces and the center hot plate were less than 0.5'F, while a temperature difference of 26C-280"F was maintained across the test sample. The test sample between the guard heater and cold plate served as a sample guard seetion. An additional 2 in. of fibrous glass insulation surrounded the test sample during tests. Finally, the cooling plate was lowered onto the sample and final temperature adjustments were made.
Eng. Chem. Fundam., Vol. 11, No. 3, 1972
Symbol
L/D
Sample
Rayleigh Number Figure 3. Heat transfer through air in small cells heated from below
The heat transfer was determined by measuring the power input t o the center hot plate heater. Power from a regulated dc power supply was measured within 1 0 . 0 4 R by precision voltmeter and ammeter. Under the test conditions used, the K'usselt number has ail uncertainty of less than 10% and the Rayleigh number has an uncertainty of less than 5y0. Results
Some general results of this study are compared with other correlations in Figure 3. The plain air layer data, which served to verify the test apparatus, are in excellent agreement with previous correlations. These results indicate that, for 0.5 5 L / D 5 2, the effect of lateral walls is to suppress onset of convection and to reduce heat transfer, as compared to a plain layer of air, for Ra
