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ATOMIC ENERGY COMMISSION
HEAT TRANSFER IN SPHERE BEDS
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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
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.
~ ~< ~<>
HEAT TRANSFER IN SPHERE BEDS
By Robert Bentley, Frederick Johnssn, Pd Roert Maurer
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
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
The heat tranter coefficient his defined (see MUC-GY-28) by the equation
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
where A C is the dierenee in heat ecmaentrati corresponding to the temperature dif-
ference AT. The heat ccentrati C is defined by the
rnd the relation between hI, and H1 is
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
= B1~F 6T
H.: I *~I -
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.
.... 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"*
H-:: S. t'S
Ii The Stanton
I 1 the Schmidt
i*,v : ." *
" : .. .
.. : I. ::. *
gp = *p*
ru* ..:, *
i3 .: 11
.:* 11: '. *v
.' **,'" ..
ZiiSl: .. -
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
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.
Benzoic acid and water were chosen as the materials to be used in the experiment.
^s. 'T llMDDC T^:** w-G90;.
Benzoic acid is slightly soluble in water and the> concentration is easily determined b
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.
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.
Two 4.5 inch diameter beds 14 and 21 inches in
5 inh eer bds 1
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
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.
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.
The total weight of acid which dissolved
into the water stream was calculated from the concentration of acid and the total weight of
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
i i F ;1 ';:E ",;
ii """I,; i
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
.iH~ ~ *
. 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
: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
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.
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
If the results of the present work are compared with the results of other workers, it
is found that equation (9) predicts smaller values for the heat transfer velocities.
have arrived at tIh following e4ations for heat transfer from
:+ : '
F ur Ha a
Saunders and Ford
O.8 Rp~-'O.5 SC'06
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
benzoic acid in water is of the order of 1000.
This large difference in the
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
Lta o nss
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
for Reynolds numbers between 350 and 4000.
The exponent of the Schmidt nun
@XAK K^ K*vv ^^^^^K X K K
^ K ::.-.f .: ^ iKKKIK KX KK KK
___ M ^ ^ K-t *_^K y^x ^ xxx^xx
j~~~k ~ N B!k^i ^U-^Hk^B iUy^UB^-^U^vXv-^
(12) < .:;: :
K1 KK KK^K"KKKN KK
S *'- ". -
be w as ^ *'. 1
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
Sand Saunders and Ford4
gases to beds of broken solids
..H!*H ; I'
r.H*r~ .1 *
Sphere Bed Data
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
; jj j"
1' IC cbs
a *~ *
InM C 0
Summary of Data
AC x 103
I ~ "
Diffusion coefficient of benzoic acid in water
D= 1.lx 10o-5 cm2- at 250C (6)
= 8.3 x 10-6 cm2
(Calculated; see CP-2883)
1.14 x 10- poise at 15C
Calculation of Reynolds and Schmidt numbers performed for
a temperature of 15C.
4. V r ~~III~I
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S*-:k. *f J. *C *. < Cf aS -- *-.
S^M~s <^l- 'Q~l '*'V
IRON. CYLINDER .
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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.
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.
Table 4 is a
*^ i*. h i '
K KKKK KKJK/KK K..:
aL SiGJ. =fi -
in the at-&Z
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
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
Ut acid the surface was appreciably roughened. The excellent agreement of equations (13)
and (15) must therefore be considered somewhat fortuitous.
= 2.30 x 10-2 Re
rangement illustrated in Figure 3 (b).
diameter as the acid pipe units.
The details of procedure for the determination
>" *** *****^***^ "i.w:""i.
Summary of Results for Pipe Experiment
Cross sectional area of pipe:
Total Length of acid sections:
Initial area of acid:
Final area of acid:
AC x io3
Re ( -
Diameter of .pipe:
m. -- .. :a
a- .- -
>~> ~< 4
V 0~ >~0<~
1< 4 ~0
: 15 MDDC-990
>i.--: Materiul Transfer in Pinps
A. Hixon and S. Baum, Ind. Eng. Chem.,
33, 478, 1433, (1941)
C. Furnas, Ind, Eng. Ehcm.,
Saunders and Ford, Jour. Iron and Steel Inst.,
CXLI, No. 1, 297, (1940)
5. B. Gamson, G. Thodos,
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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UNIVERSITY OF FLORIDA
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