HEAT TRANSFER IN BEDS OF ORIENTED SPHERES D . E. BALDUVIN, J R . , ’
R . B . B E C K M A N , a R. R R O T H F U S , A N D R . I . K E R M O D E
Carnegie Institute of Technology, Pittsburgh, Pa.
Rates of heat transfer have been studied experimentally a t various points within regular cubic and dense cubic beds of approximately 2-inch spheres. The heat transfer medium was pressurized water flowing at Reynolds numbers between 3000 and 70,000. An unbaffled jet of liquid was allowed to strike the first row of spheres. Local j-factors in the first few rows were strongly influenced by the jet. Once the jet was dissipated, the j-factors deep within the lattice followed the generalized correlation of Gupta and Thodos. Average j-factors for whole beds can b e similarly correlated when the initial jet is absent or partially broken up by screens.
MCADAMS has pointed out (a),it is difficult to measure the heat transfer coefificient between a flowing fluid and the particles of a packed bed by means of ordinary steady-state techniques. Consequently, a number of investigators have carried out mass transfer experiments and have obtained heat transfer coefficients through analogy, while others have used methods involving unsteady-state heat transfer; in a few cases direct, steady-state measurements have been made successfully (3,7). Using the steady-state method, Denton (5) investigated the transfer from a “live” copper sphere to air in a randomly packed bed of glass spheres. A significant wall effect was observed a t ratios of bed diameter to particle diameter less than 20. Above that ratio, and for Reynolds numbers between 500 and 50,000, the results can be summarized by the relationship
A
s
Baumeister and Bennett (2) heated a whole bed of metallic spheres by means of a n electrical field, thus generating in each particle a n amount of heat equivalent to the induced current. For Reynolds number!; between 200 and 10,400, their results can be expressed as
variations from sphere to sphere due to position within the bed or the effect of entrance conditions. With the exception of Denton’s work, the data d o not include Reynolds numbers greater than approximately 10,000. Scope of Investigation
I n the present work, an internally heated, l1:/l6-inch, aluminum sphere was moved from point to point within a bed of like spheres, so that heat transfer coefficients might be obtained as functions of both longitudinal and transverse positions. Two widely different packing arrangements were used-dense cubic (e = 0.26) and regular cubic ( E = 0.48). T h e heat transfer medium was pressurized water a t about 240” F. T h e Reynolds number range was from 3000 to 70,000, thus extending upward the data of previous studies. T o establish the effect of a strongly nonuniform initial velocity distribution on the local heat transfer within the bed, the entering jet of water was allowed to impinge directly on the first row of spheres. T h e decay of the jet as it passed through the bed was then studied in terms of local heat transfer coefficients. T o minimize the wall effect, partial spheres were placed adjacent to the container. Disturbance of the packing arrangement was therefore minimal and the situation approximated that of a bed having a large ratio of shell size to particle size. Experimental Equipment and Procedure
T h e test section was situated in a bypass of a hydraulic loop which supplied pressurized, deionized water a t regulated rates and temperatures. Flow rates were measured by means of standard orifice meters and bulk water temperatures were obtained by means of shielded thermocouples connected to a . N R ~= DoG/p (3) continuous recorder. T h e shell of the test unit was a square duct constructed of to account for the porosity, E , of the bed have not been fully successful. They observed, however, that the product, ( ~ j ~ ) ~ , . , steel plates. T h e inside cross section was 12 by 12 inches and the length was 63 inches in the direction of flow. Flat flanges is essentially a function of Reynolds number alone. T h e data on each end of the unit joined the shell to the bypass line, of several investigators (2, 4,6, 9 ) were empirically correlated which was 2l/2-inch pipe. The axis of the test section was horizontal, so for convenience in arranging the packing a over the Reynolds number range from 20 to 10,000 by the brass liner 1l 7 / 8 inches square and 12 inches long was closely relationship fitted within the shell. T h e spheres were packed in the liner outside the shell and the whole assembly was then slipped into its proper position (Figure 1). (4) T h e bed consisted of I15/l6-inch aluminum spheres turned to a tolerance of 1 0 . 0 0 2 inch on the diameter. T h e two packing T h e j-factors in these equations represent averages taken arrangements, the use of partial spheres a t the walls of the over the whole bed. Consequently they d o not account for shell, and the method of specifying the position of a sphere within the bed by numbers are all shown in Figure 2. Spheres with corresponding numbers occupy the same position in Present address, Continental Oil Co., Ponca City, Okla. Present address, University of Maryland, College Park, Md. successive rows of the regular cubic lattice. I n the dense cubic Gupta and Thodos (7) noted that attempts to modify the ordinary Reynolds number
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Figure 2. Orientation of spheres as seen by approaching fluid Numbers indicate method of specifying transverse position
Figure 1.
Detail of test section
arrangement, however, the positions are the same only in alternate rows. T h e single “live” sphere contained a n internal heater and a surface thermocouple. T h e heater was a 120-volt, 85-watt cartridge unit, l l / z inches long and inch in diameter, closely fitted in a 3/g-inch hole drilled in the solid sphere. A second hole ‘/8 inch in diameter was drilled from the surface to a point in the 3/8-inch hole about ‘/4 inch from the open end. An iron-constantan thermocouple insulated with glass fiber was placed in the smaller hole and the junction was secured to the surface by aluminum solder. The lead wires for the thermocouple and heater emerged from the sphere through the 3/8-inch opening which was plugged with a n epoxy resin. This arrangement made it possible to draw the electrical leads through aligned holes in the spheres directly downstream from the test sphere. Thus the lead wires were never in the flowing fluid within the bed, being brought out of the shell through sealed glands behind the packed section. Contact between neighboring spheres contributed negligible heat loss a t the low temperature differences used in the experimental runs. T h e live sphere was placed a t any one of the six definitive positions in any of the first five rows of the six row lattice. After the liner was inserted and the system sealed, the flow rate was established and the water was permitted to circulate until the spheres and fluid reached the same temperature. T h e electrical power was then sent through the heater in the test sphere and when steady state was established data were recorded. Since the surface of the heated sphere was not more than 5’ F. above the bulk water temperature in any case, the properties of the water were taken a t the bulk temperature for purposes of correlation. With the rate of heat loss, q , established by electrical measurements, the local heat transfer coefficient for the heated sphere was calculated in the usual manner-namely, (h)loc
= (q/AAt),oc
(5)
where A is the surface area of the sphere and At is the temperature difference between the surface of the sphere and the bulk 282
I&EC PROCESS DESIGN A N D DEVELOPMENT
temperature of the flowing fluid. While the surface temperature was measured a t only one point on the sphere, this point was randomly aligned with the direction of flow, although always in contact with fluid. Thus the total data for any one position in the bed yield a meaningful value of the heat transfer coefficient for the sphere occupying that position. Results and Discussion
There are six positions in each row a t which whole spheres are uniquely situated in the geometrical sense. T h e row also has to be specified (Figure 2). Consequently it is convenient to designate the position of the test sphere through a double numbering scheme. T h e first number represents the row, the second number the position in that row. Thus, for example, when the sphere is a t position 2-4 it is a t the fourth position in the second row from the upstream face of the bed. Figure 3 illustrates the nature of the experimental results and the degree of precision typically attained. The points refer to spheres deep enough within the beds to be free of entrance effects. The influence of the entering jet is essentially limited to the first four rows of the regular cubic lattice and the first two rows of the dense cubic lattice. This is to be expected, since free passages for the fluid extend through the former arrangement but are blocked in the latter case. The local j-factors for spheres lying beyond the realm of the jet are independent of the transverse position in the bed (Figure 3). T h e experimental data can be expressed satisfactorily by means of the simple equations (Regular cubic) (jx)loo= 0.99 N (Dense cubic)
(jrr)ioc
= 0.94
R ~ ~ . ~(6) ~
N R ~ ~ . ~(7)
over the investigated range of Reynolds number. Although the local j-factors of Figure 3 apply to individual spheres, it is of interest to compare them with the average
5-4
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Figure 3.
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Local heat transfer coefficients for spheres deep within bed
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I00,000
N Re
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------- DENSE
CUBIC
1,50(
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1,50C
.i\" 1,ooc
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b
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z'
u
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Y
50
50C
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Figure 4. spheres
0
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DISTANCE FROM CENTER (IN,) Effect of jet on heat transfer in first row of
j-factors for whole, ransdomly packed beds. Consequently, the ordinate of Figure 3 is formed as suggested by Gupta and Thodos and the line representing Equation 4 is included. Even though the experimental values of the ordinate retain an effect of porosity, Equations 1 and 2 indicate that the average j-factors of Denton (E := 0.37) and of Baumeister and Bennett (E = 0.35) differ by the same amount as the present data. T h e
Figure 5. spheres
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DISTANCE FROM CENTER (IN) Effect of jet on heat transfer in second row of
influence of the porosity on the local j-factors is therefore no greater than the spread of average factors taken a t the same porosity. Oriented spheres can have the same porosity but different free passages for the fluid moving through the lattice. Therefore, porosity by itself is not a sufficient measure of the pertinent geometry. O n the other hand, the lattices studied in the VOL. 5
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present work have widely different arrangements of interstices. I t appears, therefore, that there is general significance in the agreement between the present results and the correlation of Gupta and Thodos, a t least within the precision of currently available data. I t can be concluded on this basis that the local j-factors for spheres deep within the bed approximate the average j-factors for whole beds of similar randomly packed spheres and can be calculated by means of Equation 4 with the same degree of certainty. T h e correlations of Denton and of Baumeister and Bennett are equally valid within the spread of the available data, as are Equations 6 and 7 also, but Equation 4 effects a good compromise. Extrapolation to Reynolds numbers greater than 100,000, however, is not recommended. T h e jet of water entering the test section from the 21/2-inch supply line impinged on the center of the first row of spheres. As might be expected, its presence accentuated the influence of packing arrangement and transverse position over the first few rows. T o show the effect of the jet, graphs of the experimental local j-factors against Reynolds number a t constant position in the bed were cross-plotted as illustrated by Figures 4 and 5 . T h e ordinate was made to include the Reynolds number merely to spread the numerical values and thereby make the graph = NNu 1Vpr-1‘3, so the easier to read. By definition, j”Re ordinate is a commonly encountered quantity. I t can be seen in Figure 4 that the center sphere in the first row (position 1-1) of the regular cubic lattice feels the jet strongly and its j-factor is increased substantially as a result. T h e rest of the spheres in the first row are not much affected, however, and this suggests that the high velocity portion of the jet striking the bed is still roughly the same size as the supply pipe. Figure 5 indicates that the center sphere in the second row (position 2-1) is effectively shielded by the sphere ahead of it. T h e influence of the jet, split by the first row, appears to be strongly transmitted outward from the center of the second row. Thus the maximum j-factors are associated with the spheres twice removed from the center one a t low Reynolds numbers. At higher Reynolds numbers, the jet remains more consolidated and maximizes thej-factors a t the spheres adjacent to the center. Although not shown, the data for the third row of the regular cubic lattice indicate the same trend: T h e jet spreads still farther and its effect is lessened until finally it is more or less completely dissipated in the fourth row of spheres. I n the case of the dense cubic lattice, no sphere was situated a t the exact center of the first row. This can be seen in Figure 2. T h e dzia of Figure 4 indicate that the j-factors for the spheres surrounding the center point are increased by the jet, since these spheres lie within its path. For some reason, they appear to shield their immediate neighbors farther from the center very effectively. Since the hydrodynamic pattern is unknown, no explanation of the phenomenon can be established from the present data. Figure 5 shows that the sphere a t the exact center of the second row is not shielded by the first row as in the regular cubic lattice. This is to be expected, for the sphere in the 2-1 position “sees” the oncoming jet to some extent. T h e data of Figure 3 indicate that the influence of the jet is essentially destroyed by the time it enters the third row of the dense cubic bed. Separate experiments showed that beds protected by upstream screens have average j-factors only a little lower than the local values shown in Equations 6 and 7 . I n turn, this means that Equation 4 can be used satisfactorily as the basis for obtaining the average j-factor over the whole bed in such cases.
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Conclusions
Heat transfer data for beds of spheres oriented in regular cubic and dense cubic lattices indicate that the generalized j-factor correlation of Gupta and Thodos can be extended to Reynolds numbers as high as 100,000, Local j-factors for spheres deep within the bed can also be computed by means of the generalized correlation. When the upstream face of the bed is struck by a narrow jet of fluid, local j-factors in the first few rows of spheres are strongly influenced by transverse position. T h e jet is readily dissipated within the bed, especially when free passages are blocked by the spheres in the lattice. When the entering jet is broken u p by screens ahead of the bed, the Gupta and Thodos correlation offers a satisfactory means of calculating the average j-factor for the whole bed. A rough estimation of the effect of bed depth on the average j-factor for randomly packed beds indicates that the effect is probably no greater than the uncertainty in the Gupta and Thodos correlation. Nomenclature
A = surface area of sphere, sq. ft. Cp = heat capacity of fluid a t constant pressure, B.t.u./ (lb. mass) (” F.)
Do
G h j,
= diameter of sphere, ft.
superficial mass velocity of fluid, lb. mass/(hr.) (sq. ft.) heat transfer coefficient, B.t.u./(hr.) (sq. ft.) (” F.) Colburn j-factor for heat transfer, (h/CpG) ( C ~ f i / k ) ~ / ~ , dimensionless k = thermal conductivity of fluid, B.t.u./(hr.)(ft.)(” F.) niNu = Nusselt number, hDo/k, dimensionless Npr = Prandtl number, C p p / k , dimensionless NRe = Reynolds number, DoG/p, dimensionless q = rate of heat transfer, B.t.u./hr. At = temperature difference between surface of a sphere and bulk of the fluid, ” F. E = porosity (void fraction) of bed, dimensionless p = viscosity of fluid, lb. mass/(hr.) (ft.) = = =
SUBSCRIPTS av loc
= average value over whole packed bed = local value restricted to a single particle
literature Cited (1) Baldwin, D. E., Jr., “Heat Transfer in Beds of Oriented
Spheres,” Ph.D. thesis (chemical engineering), Carnegie Institute of Technology, Pittsburgh, Pa., 1961. ( 2 ) Baumeister, E. B., Bennett, C. O., A. 1. Ch. E. J . 4, 69 (1958). ( 3 ) Bradshaw, R. D., Bennett, C. O., Zbid.,7 , 4 8 (1961). ( 4 ) De Acetis, James, Thodos, George, Znd. Eng. Chem. 52, 1003 (1960). ( 5 ) Denton, W. H., Proceedings of General Discussion on Heat Transfer, Institution of Mechanical Engineers and American Society of Mechanical Engineers, Institution of Mechanical Engineers, London, p. 370, 1951. (6) Glaser, M. B., Thodos, George, A. Z. Ch.E.J. 4, 63 (1958). ( 7 ) Gupta, A. S., Thodos, George, Zbid.,8, 608 (1962). ( 8 ) McAdams, W. H., “Heat Transmission,” 3rd ed., McGrawHill, New York, 1954. (9) McConnachie, J. T. L., M. S. thesis, Northwestern University, Evanston, Ill., 1961. RECEIVED for review July 27, 1965 ACCEPTEDJanuary 28, 1966 Based on material submitted in partial fulfillment of the requirements for the degree of doctor of philosophy at Carnegie Institute of Technology (7). Project sponsored by the United States Atomic Energy Commission. One of the authors (D.E.B.) received educational support through fellowships from the Allied Chemical Co. and the Standard Oil Foundation.
