Heat transfer in sphere beds

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Title:
Heat transfer in sphere beds
Series Title:
United States. Atomic Energy Commission. MDDC ;
Physical Description:
16 p. : ill. ; 27 cm.
Language:
Undetermined
Creator:
Johnson, Frederick
Bentley, Robert
Maurer, Robert
U.S. Atomic Energy Commission
Publisher:
Technical Information Division, Oak Ridge Operations
Place of Publication:
Oak Ridge, Tenn
Publication Date:

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Genre:
federal government publication   ( marcgt )
non-fiction   ( marcgt )

Notes

Statement of Responsibility:
by Frederick Johnson, Robert Bentley, and Robert Maurer.

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 005023531
oclc - 495231303
SBRT5016334
System ID:
AA00008529:00001

Full Text
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UNITED STATES

ATOMIC ENERGY COMMISSION

OAK RIDGE

TENNESSEE












HEAT TRANSFER IN SPHERE BEDS



by



Frederick Johnson

Robert Bentley

Robert Maurer


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published for use within the Atomic Energy Commission. Inquiries for additional copies

and any questions regarding reproduction by recipients of this document may be referred

to the Technical Information Division, Atomic Energy Commission, P. O. Box E, Oak Ridge,
Tennessee.


Inasmuch as a declassified document may differ materially from the original classified

document by reason of deletions necessary to accomplish declassification, this copy does


not constitute authority for declassification of classified copies


*may bear the same title and authors.


Date of Manuscript:


July 4 1945


: Document Declassified: May 19,


1947


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ABSTRACT


The rate of solution of benkoic acid in water has been determined for a system consist-

ing & a bed of acid spheres with water flowing through the interstices of the bed. It is pro-

posed to use the results to estimate the heat transfer between a sphere bed and a liquid


coolant.


The equation proposed for this purpose is


= 2.36 x 10-1 Re-0"3 Sc"-06


where St, Re and Sc are the Stanton, Reynolds and Schmidt numbers.









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HEAT TRANSFER IN SPHERE BEDS


By Robert Bentley, Frederick Johnssn, Pd Roert Maurer


INTRODUCTION


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This report represents an attem to estimatethe heat transfer properties


ed More specifically the problem is as follows. A bed of metal or oxide sphereswith
which bt is being generated, is to be cooled by water'movimgthrough the interstices


the bed. For design purposes it is necessary to know the flux of heat from the s]
the water as a function of the temperature difference between spheres and water


a function Of the water velocity and bed geometry


(See MUC -GY-31). The term,


etry, includes such factors as sphere diameter, bed porosity, and overall bed


da


tmetew.i


The direct measurement of heat transfer coefficients forsach a system is a dif
and tedious task. Mr. Young suggested that preliminary estimates might be made by the
measurement of material transfer coefficients and the use of this data to calculate heat
transfer.


THEORY


The heat tranter coefficient his defined (see MUC-GY-28) by the equation


U


where F. is the flux of heat across the solid-liquid interface, expressed in calories per
square centimeter-send. AT is the difference between the temperature of the interface
and the average temperature of the liquid. A transfer velocity Hn may also be defined by
the equation


= HrAC


A


where A C is the dierenee in heat ecmaentrati corresponding to the temperature dif-


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ference AT. The heat ccentrati C is defined by the


*0


eqcratic


-c PZT


rnd the relation between hI, and H1 is


= e/Pc


fhe:specfie heatof the liquid is c, and Pis its density. For the analogous situation of a


material dssolvingrin a liquid stream from a solid surface, the e


quafl


0x8


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= B1~F 6T




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Fm = ELmAC


may be written. Here Fm is the flux of material across the solid-liquid interface, expressed


in grams per square centimeter-second. The difference between the concentration of mater-
ial in the liquid at the interface and the average concentration of material in the liquid
stream is AC. It is assumed that the film of liquid in contact with the interface is saturated
with the material in question.
ma


.... The transfer velocities for heat and material defined by equations (2) and (5) will be
iP. functions of the properties of the system. If it is assumed that the pertinent variables are
i:::'.the viscosity, density, and velocity of flow of the liquid, the diameter of the spheres, and
.i!.:i the diffusion coefficient for heat (or material) in the liquid, then a dimensional analysis
i':JIiadicates that the relationship between these variables and the transfer velocity may be

::.:~*: written in the following form.
*:.: il "M"*
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H-:: S. t'S


Ii The Stanton
I 1 the Schmidt
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number, St, is (H/V). Re is the modified Reynolds number ( V P d/f). Sc is
number (p /PD). In the above definitions


V = velocity of flow of the liquid


d = diameter of the spheres


p = viscosity of the liquid


D = diffusion coefficient of heat (or material) in the
liquid


9 If one works with a given fluid and dissolved material at a fixed temperature the Schmidt
number is a constant. The diffusion coefficient for heat in a liquid may be related to the
- thermal conductivity of the liquid by the equation.


D = k/Pc (7)


. The experimental problem is to determine explicitly equation (6). The assumption is
Then made that the relationship between the Stanton, Reynolds, and Schmidt numbers which
is obtained experimentally for material transfer may be used to calculate the heat transfer
velocity if the diffusion coefficient for material is replaced by the diffusion coefficient for
heat in the Schmidt number.


EXPERIMENTAL


Benzoic acid and water were chosen as the materials to be used in the experiment.


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Benzoic acid is slightly soluble in water and the> concentration is easily determined b


titration.


The solubility6 of benzoic acid in water has been determined as a function of


temperature near room temperature.


The diffusion coefficient of benzoic aid


has been determined at


Spheres of benzoic acid were cast from C.P. material in a brass mold. TI
were obtained by having the acid at 140C and the mold at room temperature.


Linwa~r
>0


beWre


The cast


spheres had a smooth, white surface. The average diameter of the spheres was 0.635 inches.
... ,, .. ^ _


The form of the sphere bed is shown in Figure 1.


The spheres were held in a cylinder


f galvanized iron closed at top and bottom by wire screen. It was found that the porosity


of the bed, which is the fraction of the bed volume which is void space,


could not be varied


but was fixed by the diameter of the ed.
legt wer exmnd n" nhdimtrbd 8Ice I egh a sd


length were examined. One 8 inch diameter bed, 18 inches in length, was used. I
cerning the beds are given in Table 1.

The velocity of flow of the water through the bed was regulated by the head of


the area of the orifice, 0, at the bottom of the bed.


The water-velocity,


Two 4.5 inch diameter beds 14 and 21 inches in
5 inh eer bds 1


)ata con-


tat
Tfn^^ilBy1' N "'
N~ JBg ^ kk
f ^^ ^^^ ^H ^^^ H^f^


V, is defined as the


quotient of the volume of water passing through the bed per unit~ime and the cro
area of the cylinder containing the bed. The volume of water passing through the
unit time was measured by weighing the water issuing from the bed in a known in
water was used in all the experiments.
S i c t h w a e v e o i y de o a e h cm d ub r e emn t o


SSince the water velocity, V, does not e
dependence of the transfer velocity, Hm,

Stantop number and the Reynolds number.

ceded in the following fashion. Water, pa
collected and weighed. From this data the


SS-


enter the Schmidt number, a determination
upon V, determines the relationship betwe

The determination of Hm as a function oi


missed through the bed for a timed interval,


e water velocity,


V, was calculated.


-sectional







Swas


The concen-


tration of benzoic acid in the water after passage through the bed was determined by titra-


tion with 0.01 molar sodium hydroxide, using phenolpthalein as an indicator.


were made on the tap water used in the.experiments.


Blank titrations


The total weight of acid which dissolved


into the water stream was calculated from the concentration of acid and the total weight of


water.


The flux, F, may then be calculated from the above data, the known total area of


sphere surface, and the time of contact of the water with the bed.


The concentration of dis-


solved acid was in all cases negligible by comparison with the solubility of the acid so that


the concentration difference A C was taken as the solubility,
ot the acid in water.


, A sample of the data from a typical experiment is given i


at the temperature of the water,


STable 2. In Table 3o


ledted a summary of all the experimental data. A lgtog- plot o the transfer velocity
the water velocity is given In Figure 2. rom this data one qbts the equation


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Hrf7.1 x 10-4 V0.70


for Hm and V.

In experiments of this type it is difficult to determine the functional dependence of the
Stanton number upon the Schmidt number. This is so because none of the quantities which
enter the Schmidt number may be readily varied. Heat transfer experiments upon liquids
flowing in pipes indicate that the Stanton number is inversely proportional to the Schmidt


number raised to the six-tenths power1


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. Experiments upon heat and material transfer


from solids to agitated liquids indicate an inverse square root dependence of the Stanton


number upon the Schmidt number2


. For flow of liquids normal to cylinders the Stanton num-


-her is, inversely proportional to the seven-tenths power of the Schmidt number1


.We have


:assumed that the Stanton number is inversely proportional to the Schmidt number raised to

the six-tefiths power and have rewritten equation (8) in the form


St = 2.36 x 10-1 Re


-0.3 Sc-0.6


Since the data for the three beds, of varying height, diameter, and porosity are fairly
well represented by the single equation (8), there is reason for belief that these variables
are not of primary importance.
I I

DISCUSSION
The assumptions involved the use of equation (9) for the calculation of heat transfer
The assumptions involved in the use of equation (9) for the calculation of heat transfer


velocities must be kept in mind. The first assumption concerns the manner in which the
Schmidt number enters the equation. There is little doubt that the inverse six-tenths po
is approximately correct. A better representation of the behavior of the system might be
obtained, however, by a change in the exponent of the Schmidt number of +20%. Such a
change would be of some importance since the diffusion coefficients for heat and benzoic
Said in water differ by a factor of approximately one hundred (See MUC-GY-28, p.3). In
passing by the use of equation (9) from material to heat transfer, a 20% change in the ex-
ponent of the Schmidt number changes the calculated heat transfer velocity by a factor of
approximately two.


If the results of the present work are compared with the results of other workers, it


wer


is found that equation (9) predicts smaller values for the heat transfer velocities.


FurasS






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MDDC-990


have arrived at tIh following e4ations for heat transfer from
:+ : '


F ur Ha a


Saunders and Ford


O.8 Rp~-'O.5 SC'06


--.p"~-0.


i each case, the exponent of the Schmidt number has been assumed. In this conr
is worth noting that the Schmidt number for heat in gases is of the order of unity


iect it
ai or


benzoic acid in water is of the order of 1000.


This large difference in the


magni


Schmidt number may make a comparison of the above equations with our results
The Schmidt number for heat in water is approximately seven. In addition the da


have been taken for Reynolds numbers of the order of 200.


The work of Gamson


tUde ofltk.

Lta o nss

and hi


co-workers indicates that this is in the transition region between streamline and turbulent


flow. Extrapolation of equations (10) and (11) into the region of turbulent flow
not permissable. In the present work, the range of Reynolds number which w
extended from 200 to 3000.

Gamson, working with heat transfer from a sphere bed to a gas, found the


= 1.064 Re


-0.41 g-0.67


for Reynolds numbers between 350 and 4000.


The exponent of the Schmidt nun


is there.fore
Is covered







(12)



ber was
@XAK K^ K*vv ^^^^^K X K K
^ K ::.-.f .: ^ iKKKIK KX KK KK
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assumed. Equation (12) leads also to larger heat transfer velocities than our results do.
The work of Gamson was performed upon very shallow beds which may account for the
discrepancies.


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Sand Saunders and Ford4


gases to beds of broken solids


(10)


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TABLE 1


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Sphere Bed Data


Bed 1


Diameter of Sphere Bed


Cross-sectional.Area of Bed


Number of Spheres


Area of Sphere Surface


Height of Sphere Bed

Volume of Spheres


Volume of Sphere Bed


Porosity


1L4


102.6


1,002.


8,010.


Bed 2


Bed 3


20.3 cm


102.6


1,416.


11,580.


322.1 cm2


4,330.


35,400.


45.7 cm


2,125.


3,647.


3,105.


5,460.


95,00.


14.720.


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TABLE 3

Summary of Data


Fx 106


AC x 103


H xl04


Bed 1


A'.!'
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1.69
2.49
6.86
8.53
8.79
11.97
16.44


1.07
1.95
4.50
6.75
7.95
14.2
21.1


Bed 2


1.25
4.60
6.71
13.6
15.4


Bed 3


1.07
2.32
4.44
8.44
14.7


Diffusion coefficient of benzoic acid in water

D= 1.lx 10o-5 cm2- at 250C (6)
see


= 8.3 x 10-6 cm2


at 15C


(Calculated; see CP-2883)


1.14 x 10- poise at 15C


Calculation of Reynolds and Schmidt numbers performed for
a temperature of 15C.




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VALVE


GALVANIZED
IRON. CYLINDER .







BENZOIC ACID
SPHERE BED












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APPENDIX
Hatrnfr"""" crs se.n a

SHeat transfer to liquids flowing in smooth walled pipes of circular cross section has
been extensively investigated and for the region of well developed turbulent flow jRe> 10,
the following equation is well established.


v:5:h


-0.2 S-0.6


Material transfer velocities were measured in a. circular pipe as a function of


water velocity to determine if an equation similar to (13) existed.


Figure 3 (a) is


oi the benzoic acid pipe used in this experiment. A brass core was centered in a
by means of end caps and benzoic acid cast in the space between the core and thi


The brass core was then withdrawn.


Two such sections were used


The water was led from a reservoir to thu


acid pipes- by means of a brass pipe approximately six feet long and of the same


transfer velocity were the same as in the sphere bed experiments.


LI


Table 4 is a


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irass tube
*^ i*. h i '


K KKKK KKJK/KK K..:

aL SiGJ. =fi -
)rass tUbe


Souter "
in the at-&Z
e benzoic
internal
aEthe
summartt
UA' i~C'/^lic'^^ ***^*V p =
^^^^ ^ ^ ^ ^ ^ f A A r ^^ H / A K ^ V : :
^A^ -. K K KK^KKK.KK


of the data. Solution of the benzoic acid in the water during the experiment increased ap-
preciably the diameter of the pipe section. Assuming that the change in diameter was


constant for each run, a corrected diameter was used in computing the Reynolds number
and surface area for each experiment. In Table 4 are given the initial and final areas


the benzoic acid surface.


Figure 4 is a log-log plot of the material transfer velocity versus


the linear water velocity. The straight line of Figure 4 represents the equation


= 1.23 x 10-4 V0"8


Making the usual assumption concerning the dependence of the Stanton number upon the
Schmidt number, equation (14) may be rewritten as


V



21
C


St= 2:04 x 1Q-2 Re-0-2 Sc"0"6


Cbmparison of this result with equation (13) for heat transfer indicates satisfactory agree-
ment. This result lends considerable weight to the proposed use of material transfer data
for the estimation of heat transfer performance.

It is rather surprising that the data of Figure 4 fit a straight line plot of slope 0.8 as
well as they do. For Reynolds numbers less than 10,000 the heat transfer data indicate a
greater slope. A partial explanation may lie in the fact that in the course of solution of


C
V 4"
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4<4,0<


Ut acid the surface was appreciably roughened. The excellent agreement of equations (13)


and (15) must therefore be considered somewhat fortuitous.


Thi~a&


= 2.30 x 10-2 Re


brass shell.


rangement illustrated in Figure 3 (b).


diameter as the acid pipe units.


The details of procedure for the determination


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TABLE 4


Summary of Results for Pipe Experiment


0.95 cm


Cross sectional area of pipe:
Total Length of acid sections:


Initial area of acid:
Final area of acid:


I a.

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.709 cm2
58.5 cm


174.2 cm2
237.1 cm2


s 106


AC x io3


Hz io4


Re (PVd)
Re ( -


sec


:1.90
.8.25
.1.00


2.53


3.98


15.55
26.20
31.01
53.20
73.10
110.0
113.4
86.60

72.10
43.10

29.10
15.41


123.3
188.9


2,390
3,350
5,260
11,600
17,900
25,700
28,000

18,800
14,100
7,410
4,930
3,500


147.9


Diameter of .pipe:




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BRASS -






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REFERENCES


W. McAdams,


"Heat Transfer"


, (1942)


A. Hixon and S. Baum, Ind. Eng. Chem.,


33, 478, 1433, (1941)


C. Furnas, Ind, Eng. Ehcm.,


22, 26,


(1930)


Saunders and Ford, Jour. Iron and Steel Inst.,
CXLI, No. 1, 297, (1940)


5. B. Gamson, G. Thodos,


Chem.


and 0. Hougen, Trans. Am. Inst.


Eng., 39, 1, (1943)


6. A. Hixon and Wilkens


, Ind. Eng. Chem.,


25, 1196, (1933)


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