EXPERIMENTAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER TO MERCURY D. V. JULIA”
AND R. G. AKINS
Department of Chemical Engineering, Kansas State University, Manhattan, Kan.
66602
Laminar natural convection heat transfer from a vertical flat plate in mercury and water was experimentally investigated. Temperature profiles on the surface of the plate, isotherms in the boundary layer near the plate, and the boundary-layer thickness are presented. The results are in agreement with the similarity and integral solutions of the boundary-layer equations. The dimensionless profiles also showed the trends, with position up the plate, predicted by the first-order perturbation analysis.
XPERIMEKTAL temperature profiles for free convection
E heat transfer from a vertical flat plate have been measured
in both air and water. The results (available in the literature) have verified the boundary-layer assumptions (used to reduce the continuity, motion, and energy equations) for materials whose Prandtl number is near 1. However, no experimental verification has been made for liquid metals such as mercury (Pr = 0.03), where the thermal conductivity and the boundary layer thickness are large compared with air or water. Therefore, the purpose of this study was to obtain experimental temperature profiles and to compare the results with the available analytical solutions. Profiles were taken in water to check the accuracy of the equipment. There are many analytical solutions available for steady, laminar, natural convection heat transfer from a vertical flat plate. Possibly the first theoretical analysis was presented by Lorenz (1881). He assumed that the fluid flow was upward and parallel to the plate, that flow perpendicular to the plate was negligible, that the fluid temperature far from the plate was constant, and that the temperature a t any point in the fluid depended only on the distance from the plate. His results, obtained by making a heat balance on a differential section of flowing fluid and integrating, were in fairly good agreement with experiments except for lorn Prandtl number fluids. In 1904 Prandtl simplified the Navier-Stokes equations by dividing the flow into two regions: a thin boundary layer along the solid surface where viscous effects were important and the velocity gradient normal to the wall mas large, and a bulk flow region where viscous effects could be neglected (1904). Sparrow (1955) used the approximate Von KarinanPohlhausen integral method to solve the boundary-layer equations. This involved substituting the equation of continuity into the equations of motion and energy and then integrating the resulting equations over the boundary-layer thickness. This assumed that a common velocity and thermal boundary-layer thickness could be used. Sparrow and Gregg (1956) solved the boundary-layer equations, without the need for assumed profiles, by finding a similarity transformation which reduced the boundary-layer equations t o a pair of ordinary differential equations, which were solved numerically for Prandtl numbers of 0.1, 1, 10, and 100. Chang et al. (1964) extended Sparrow and Gregg’s solutions to include Prandtl numbers of 0.01 and 0.03 (liquid metal range). T o improve the above similarity solutions, which apply only for thin boundary layers (Grashof numbers approaching Present address, Procter and Gamble Co., Cincinnati, Ohio
infinity), Chang, Akins, and Bankoff (1966) extended the perturbation annlysis for the constant wall temperature (Yang and Jerger, 1964) to the case of constant heat flux. They found, for low Prandtl numbers, that the profiles no longer fell on a single line (as with the boundary-layer solution), but changed with position on the plate and Grashof number. Saunders obtained some experimental temperature profiles for free convection from a constant temperature vertical plate in mercury and in water (1939). The geometric configuration used was that of a flat plate; however, it was constructed so that it contained no leading or trailing edges-that is, it was a heated portion of a wall. This configuration was difficult to analyze and the data lack the necessary accuracy to test the present analytical solutions. However, Saunders’ work appears to be the only fundamental experimental investigation of natural convection in liquid metals. Dotson (1954) studied natural convection heat transfer to air from a vertical aluminum sheet. Temperatures, in the uniformly heated plate, were recorded with 63 thermocouples soldered t o the back of the plate. He experienced some difficulties due to conduction and radiation loss. Local Kusselt numbers were calculated. Schecter and Isbin’s (1958) experiments in water showed that the boundary-layer assumptions used to reduce the equations of continuity, motion, and energy produced results in excellent agreement with experiments. Since these equations are dependent on the Prandtl number and the Prandtl number for water and air is around 1, it mould be important to check the solutions for liquids metals where the Prandtl numbers are one hundredth as large. -Also, two of the assumptions, small boundary layer and negligible conduction up the plate, will not apply as well for liquid metals. The boundary-layer thickness for mercury is approximately seven times that for water and the thermal conductivity is more than 15 times greater. Experimental Equipment and Instrumentation
The experimental equipment was babically a container to hold the fluid, a heated vertical flat plate, a device to measure the temperature profiles, and a constant temperature bath to regulate the bulk temperature of the fluid under investigation. The container which held the experimental fluids (water or mercury) was 7 inches wide, 9 inches long, and 13 inches deep, and was made of Type 410 stainless steel. This size was selected because it was felt that the fluid could be considered practically infinite (compared to the boundary-layer VOL.
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were minimized by having the thermocouple pointing upstream (downward). The plate was heated electrically using direct current. The output from the thermocouple was recorded on a Sanborn, Model 296, two-channel oscillographic recorder. The sensitivity of this instrument was 2 pv. per mm., thus allowing temperature difference measurements to within 0.03' C. Experimental Procedure
The thermocouple was located vertically at the point of interest on the plate. The probe was driven away from the plate with the synchronous motor a t a constant rate of 1/16 inch per minute. The profile on the chart remained constant initially and then started to decrease, indicating that the thermocouple had left the plate. The temperature continued to decrease as the thermocouple moved through the fluid near the plate. As the probe continued out into the bulk, the temperature change gradually ceased. When the temperature stopped changing, the motor was reversed and the probe driven back to the plate. An indication that the system was at steady state was the point at which the same profile was obtained in both directions. This also showed that the direction of probe movement had no noticeable effects on the profile. The apparatus was dismantled and set up several times and the shape of the probe tip was changed, with no change in the data being observed. Results for Water
The experimental temperature profiles were put into dimensionless form, so that they could be compared with Sparrow and Gregg's (1956) similarity solution of the
a thm coatinn ol a ulasticized acrvlic resin to eliminate electrical leakage. The vrohe uositioninn mechanism consisted of a 7I16-inch thick, 8 by inch base plate which supported the various other parts. A slot was cut in the base plate, over which a slide moved. The slide was positioned by a 1/2 inch32-precision pitch screw and held down by tension screws. The slide moved ,the thermocouple horizontally toward and away from the plate. Mounted on the slide was a support for a second 1/2 inch-32-precision pitch screw which moved a l/Cinch stainless steel tube vertically. The thermocouple was mounted in this l/Cinch tube and t,herefore could he moved vertically and horizontally. To obtain continuous horizontal motion, a synchronous 2-r.p.m. motor was mounted on the screw that moved the slide horizontally. Two heavy fittings welded to the base plate served &s feedthroughs for the electrode assembly, which consisted of two, 1/2-inch-diameter, 3-EootAong copper bus bars surrounded with alumina insulation and contained in 3/4-inch stainless steel tubes. At the bottom of each copper bus bar was a copper projection 2 inches high, 1 inch long, and 1/4 inch thick, which was used to hold the vertical flat plate (Figure 1 ) . The temperature profiles were measured with a BLH Electronics copper-constantan thermocouple which was enclosed in a 2-inch long by 0.014-inch-diameter, Type 302 stainless steel sheath. The end of the sheath was flattened (0.008 inch thick by 0.021 inch wide) onto the junction of the thermocouple to increase the response time. The time constant was approximately 50 milliseconds. Any disturbances in the flow that were caused by the thermocouple Y
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many experimental runs with water at various power levels and positions up the plate. The average Prandtl number was 5 , with a maximum deviation of &1/2. The dashed line is our analog computer solution to the similarity equations for Prandtl number of 5. The solid lines are Sparrow and Gregg’s analytical solution for Prandtl numbers of 1 and 10. The spread in the data was entirely due to random experimental error and does not indicate any trends. The reference temperature for the physical properties was calculated, as recommended by Sparrow (1956) , as a weighted average of 70% of the wall temperature and 30% of the bulk temperature. The density, viscosity, thermal conductivity, expansion coefficient, and Prandtl number were computed a t this reference temperature. The electrical resistance of the plate was computed a t the plate temperature and the heat flux was calculated as the square of the current times the plate resistance. Local Kusselt numbers Ivere calculated from the data, which were taken a t several power levels and positions on the plate and plotted against modified Grashof number in Figure 3. The least-square fit of the experimental data was about 795 higher than predicted by the similarity solution of the boundary-layer equations (Sparrow, 1956). Figure 4 shows the experimental boundary-layer thickness as a function of position on the plate, for a heat flux of 1790 I3.t.u. per hour sq. foot. These data were obtained from the temperature-0s.-position profiles. The boundary layer was taken as the distance from the plate where 95% of the total temperature change, between the plate and the bulk, had occurred. Also plotted is the assumed and calculated boundary-layer thickness from the Von Karman-Pohlhausen
integral method (Sparrow, 1955). The experimental and theoretical values of the boundary-layer thickness are in good agreement. The boundary-layer thickness increased for increasing distances up the plate, except in the vicinity of the top of the plate. The temperature a t the top of the plate was not the
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Boundary-layer thickness for water Heat flux. 1750 B.t.u./hr.rq.ft. Prandtl number. 4.5
Standard
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Figure 3.
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Experimental Nusselt numbers for water Heat flux, B.t.u./hr.rq.ft. 389
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0 0 0
DISTANCE
FROM FRONT
OF
PUTE,
inch08
713 1565 2364 3280
Figure 5. Isotherms for laminar natural convection from uniformly heated vertical plate in water Heat flux. 1 7 5 0 B.t.u./hr. rq. ft. Bulk temperature. 33.5’ C.
Prandtl number. 5 Positions. 3, 1, 14, and 13 inches from leading edge
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maximum, as theory predicts, because of the effect of plate thickness. Figure 5 presents the isotherms for laminar natural convection heat transfer from a uniformly heated vertical flat plate, The data presented were obtained from profiles taken a t 11 positions on the plate and five positions over the top of the plate. The heat flux was 1790 B.t.u. per hour sq. foot. The isotherms verified that the plate was vertical, since they were symmetric about the top of the plate, and that the plate was flat, sinre there were no unusual changes in the shape of the isotherms near the plate. Erratic behavior, seen in the isotherms near the top of the plate, is believed to be caused by the flow discontinuity and heat loss a t the top of the plate. The abscissa scale mas expanded to 20 times the ordinate scale and the plate thickness was drawn to scale (it was only 0.004 inch thick). Scatter in the data would be multiplied tremendously because of this expansion in the scale of the abscissa. In the development of the boundary layer equations the heat conduction up the plate (z direction) was neglected compared to the conduction away from the plate ( y direction). Figure 5 shows that the assumption was valid for water, since d T / d z was over 70 times larger than d T / d y . The experimental dimensionless temperature distributions as well as the heat transfer coefficients for water agreed well R ith the boundary-layer solutions (Sparrow and Gregg, 1956). Since these solutions have been experimentally verified, it was concluded that the experimental apparatus mas giving accurate results. Results for Mercury
Figure 6 presents the dimensionless temperature profiles for mercury. The experimental points, for Prandtl number of 0.022, can be compared with the dashed line, which is our analog computer solution to the similarity equations, and the solid lines which represent the boundary-layer solutions for Prandtl numbers of 0.01 and 0.03. There was excellent agreement between the analytical and experimental values. However, the experimental data appear to be slightly low. This was observed in all the runs, which would eliminate errors such as location of the probe up the plate, the verticalness of the plate, impurities in the mercury, or in the shape of the probe, because all these were changed several times throughout the experiments with no apparent change in results. Therefore, the error must have been caused by something that did not change when the experimental equipment was completely taken apart and set back up. Julian (1967) showed that the thickness of the probe (0.008 inch) could cause the experimental data to be slightly low when presented on a dimensionless plot of temperature us. distance. However, this should also show up on the profiles for water, which was not the rase. Other possible explanations are that the finite container size affected the results or that the boundary-layer equations did not fully represent the physical situation. The latter would seem most likely to be the case. The increased spread in the data points for mercury was partly due to the small temperature difference across the boundary layer, 3" C. for mercury compared to 6" C. for water. Furthermore, this small temperature difference was spread over a much larger distance, 0.5 inch for mercury compared to 0.07 inch for water. The perturbation solution predicts that the temperature profiles should depend on the vertical distance up the plate as well as on the Grashof number. To check the perturbation predictions, two sets of measurements were analyzed for 644
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0
Figure 6. mercury
2
6
4
8
1 0 1 2 1 4
Dimensionless temperature
distribution for
Heat fluxes. 530 to 501 0 B.t.u./hr. sq. ft. Prandtl number. 0.022 Positions. $, 1, 1 and 19 inches from leading edge
3,
i,
Grashof numbers of 4.0 X los and 1.0 X lo9. A fifth-order polynomial least-square fit of the dimensionless temperature profiles was made a t 1/4 and 13 inches from the bottom of the plate. The results tend to confirm the theoretical predictions of the perturbation solution, but the random experimental error was too large to observe more detail about these trends. There were no observable trends for the set of runs with Grashof number equal to 1.0 X lo9. Figure 7 presents the local Nusselt numbers, plotted against the modified local Grashof number. Data computed from runs made a t several heat fluxes and positions on the plate gave the experimental expression Gr,*a.188/Nu z
- 5.1
with a standard deviation of 0.021. This can be compared to the similarity solution, which predicted Grz*o.2/Nuz= 6.3 for Prandtl number of 0.022. Thus the experimental heat transfer coefficients were in excellent agreement with the analytical solution to the boundary-layer equations. A wider range of experimental conditions (larger heat fluxes and more positions on the plate) caused the standard deviation for mercury to be slightly larger than for water. Also plotted on Figure 7 are the only other available heat transfer data for mercury. These data, taken by Saunders in 1939, compared fairly well with the present experimental study. The bwndary-layer thickness (again, using 95% of AT) for mercury was approximately seven times thicker than for water. Figure 8 is a plot of the experimental and theoretical
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7. Experimental Nusselt numbers for mercury
Figure 8.
Heat flux, B.t.u./hr.sq.ft.
Prandtl number. 0.022 Positions. i,i, 1, 13, 19, 1 $, and
4,
l&
0.4
0.5
inchrl
Boundary-layer thickness for mercury Heat flux. 21 00 B.t.u./hr.sq.ft. Prandtl number. 0.022
0
530 A 1140 0 1690 0 2120 2790 A 3460 5010 X Sounders' data
03
I '
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inches from leading edge
boundary-layer thickness as a function of the position up the plate for a heat flux of 2100 B.t.u. per hour sq. foot. The constant temperature lines in the boundary-layer adjacent to the vertical flat plate generating heat at a uniform rate of 2100 B.t.u. per hour sq. foot, are presented in Figure 9, which was constructed in a manner similar to Figure 5 . This graph again shows that the plate was both vertical and flat. The increased amount of scatter, compared to Figure 5 , was due to a small temperature difference across a larger boundary layer and to the hydraulic jump a t the top of the plate which was more important in mercury because of its higher density. The abscissa is expanded 4 times the ordinate. This figure shows that neglecting the conduction in the vertical direction compared to the conduction in the horizontal direction is reasonable for liquid metals, since dT/dx was 15 times larger than d T / d y . Temperature stratification was noticed in the bulk fluid for runs in both water and mercury. This was more noticeable as the heat flux of the plate was increased. In no case was the temperature difference in the bulk more than 10% of the minimum temperature difference between the plate and the bulk. All calculations at one vertical position on the plate were made using the bulk temperature a t that same verticle position. This stratification was due to the fact that that the fluid container was not infinite in regard to the heat added by the plate. Therefore, as the plate dissipated heat to the fluid, the fluid temperature rose until steady state was reached. The warmer fluid tended to collect in the upper portion of the
DISTANCE
FROM
FRONT O F
PUTE, inchor
Figure 9. Isotherms for laminar natural convection from uniformly heated vertical plate in mercury Heat flux. 2100 B.t.u. hr. sq. ft. .Bulk temperature. 40" C.
fluid container, causing the stratification. Runs without stratification were made by recording the profiles immediately after turning the power on and before the heat from the plate affected the whole fluid. This seemed to make no difference in the results, so the temperature stratification was decided t o be preferable to the transient nature of the other approach. VOL.
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Conclusions
The boundary-layer assumptions, used to reduce the coupled equations of motion, energy, and continuity which describe laminar natural convection heat transfer from a vertical plate, have been experimentally shown to be valid for both water and mercury in the moderate (IOs to lo9) Grashof number range. The experimental dimensionless temperature profiles agreed with the similarity solution and with the more recent first-order perturbation analysis. The plate temperature varied as the 1/5 power of the distance up the plate, as predicted by Sparrow and Gregg. The Kusselt number for both water and mercury agreed well with theory. The boundary-layer for mercury was about seven times thicker than for water and could be fairly accurately predicted by the Von Karman-Pohlhausen integral solution to the boundarylayer equations. Nomenclature Q
h, Gr,* k
Nu, Pr 4
T
T,
T, x
Y
= thermal diffusivity, sq. ft./sec. = coefficient of thermal expansion,
a
= gravity force. ftJsec.2 = local heat-transfer coefficient, B.t.u./hr. sq. ft. F. = modified Grashof number based on 2, gpp4/kv2,
dimensionless = thermal conductivity, B.t.u./hr. ft. F. = local Xusselt number, h,X/k, dimensionless = Prandtl number, v/a,C,/k, dimensionless = heat-flux rate, B.t.u./hr. sq. ft. = temperature, O F. = wall temperature, O F. = bulk temperature, O F. = coordinate measuring distance along plate from leading edge, ft. = coordinate measuring normal distance from plate, ft.
@
O
6( 5 ) 7) V
p
= = = =
-
(l/p)(ap/aT),
F.-I
boundary layer thickness, ft. similarity variable, y/z[Gr,*/5]1’5, dimensionless kinematic viscosity, lb., sq. ft./sec. density, lb. m/ft. see3.
literature Cited
Chang, K. S.,Akins, R. G., Bankoff, S. G., IND. EXG.C H m . F U N D A M E ~6,T26-37 A L S (1966). Chang, K. C., Akins, R. G., Burris, L., Bankoff, S. G., “Free Convection of a Low Prandtl Number Fluid in Contact with Uniformly Heated Yertical Plate,” Argonne National Laboratory, ANL-6836 (1964). Dotson, J. P., Heat Transfer from a Vertical Plate by Free Convection,” h1.S. thesis, Purdue University, 1954. Julian, D. V.,“Experimental Study of Natural Convection Heat Transfer from a Uniformly Heated Vertical Plate Immersed in hlercury,” Ph.D. thesis, Kansas State University, 1967. Lorenz, L., Wiedemanns Ann. 13, 582 (1881). Prandtl L., “Uber Flussigkeitsbewegun bei sehr kleiner Reibung,” Proceedings of Third International hlathematical Congress, Heidelberg, 1904; reprinted in Tier Abhdl. zur HydroAerodynamik, Gottingen, 1927; NACA TN-462 (1955). Sauriders, 0. A., Proc. Roy. SOC.(London) A172, 55-71 (1939). Schecter, R. S., Isbin, H. S., A.Z.Ch.E. J . 4, 81-9 (1958). Sparrow, E. hI., “Free Convection with Variable Properties and 1-ariable \Tall Temperatures,” Ph.D. thesis, Harvard University, 19FiB. Sparrow, E. M., Laminar Free Convection on a Vertical Plate with Prescribed Konuniform Wall Heat Flux or Prescribed Nonuniform Wall Temperature,” NACA TN 3608 (1955). Sparrow, E. hl., Gregg, J. L., Trans. A S M E 78, 435-40 (1956) Yang, K. T., Jerger, E. W.,J . Heat Transjer, Trans. A S M E C86, 107-15 (1964).
RECEIVED for review J d y 31, 1968 ACCEPTEDJune 23, 1969 Work partially supported by N.S.F. Grant GK-2114.
P A R T I C L E DEPOSITION F R O M T U R B U L E N T STREAMS BY M E A N S OF T H E R M A L F O R C E R. L E E B Y E R S A N D S E Y M O U R CALVERT’ Center for Air Environment Studies, The Pennsylvania State University, University Park, Pa. 16802 Particle deposition from turbulent streams of hot gases was studied to determine what factors control particle collection. Collection efficiency was found as a function of particle size over the range of 0.3 to 1.3 microns. Diffusion was not a contributing factor in particle deposition; the only significant mechanism proved to be thermophoresis. When the mean free path to particle radius ratio, the Knudsen number, increased from small to large values, collection efficiency rose gradually at first and then increased more rapidly at about a Knudsen number of 0.2. This phenomenon i s explained by the increasing contribution of the free molecular thermal force as the Knudsen number approaches unity. The effects of flow parameters and size of collector were determined. A mathematical model, based on existing theories of thermophoresis, was developed to predict collection efficiency. Agreement between the model and experiment was good, considering the accuracy with which the slip flow coefficients and constants are known. The transition thermal force is related theoretically to the free molecular thermal force by an exponential term containing the inverse of the Knudsen number and constant, T . Results suggest that T i s not a constant but is temperature-dependent, because of its basic relationship to the momentum accommodation coefficient.
HE motion of small particles suspended in a gas containing Ta thermal gradient has been the subject of many investigators. The force responsible for this motion, called the thermal force, has been studied primarily by observing the movement of a particle suspended in a stagnant gas. This technique has permitted the experimental testing of various Present address, University of California, Riverside, Calif. 646
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proposed theories. As a result, the thermal force is fairly well understood. It has been applied in the development of a dust sampler which permits sampling a small quantity of dust-laden gas at essentially 100% collection efficiency. The use of these samplers has been discussed by Fraula (1956), Gordon and Orr (1954), and Kitto (1952). The design of this device is such that the gas passes through
