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Nuclear Technology
ISSN: 0029-5450 (Print) 1943-7471 (Online) Journal homepage: http://www.tandfonline.com/loi/unct20
Convective Cooling of Simulated Core Debris Beds
H. S. Kim & S. I. Abdel-Khalik
To cite this article: H. S. Kim & S. I. Abdel-Khalik (1985) Convective Cooling of Simulated Core
Debris Beds, Nuclear Technology, 69:3, 268-278, DOI: 10.13182/NT85-A33610
To link to this article: http://dx.doi.org/10.13182/NT85-A33610
Published online: 13 May 2017.
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Download by: [University of Missouri-Columbia]
Date: 27 October 2017, At: 02:16
CONVECTIVE COOLING OF
SIMULATED CORE DEBRIS BEDS
NUCLEAR
SAFETY
H. S. KIM* and S. I. ABDEL-KHALIK University of Wisconsin-Madison
Department of Nuclear Engineering, Madison,
Wisconsin 53706
Downloaded by [University of Missouri-Columbia] at 02:16 27 October 2017
Received March 13, 1984
Accepted for Publication January 2, 1985
The molten core debris will then be ejected into the
sodium coolant, where it is likely to freeze and fragNatural convection heat transfer in simulated core
ment upon quenching.2"6 The particulated fuel debris
debris beds has been examined. The debris beds are
will then settle on various horizontal surfaces in the
simulated using electrically heated packed tube bundles
reactor vessel. They will generate heat by radioactive
arranged in either a square or staggered lattice with
decay of the fission products and actinides; the heat
porosities varying between 0.31 and 0.95. The effects
generation rate depends on the reactor operating
of bed height, heat generation rate, particle size,
power level history before shutdown. The decay heat
porosity, overlying liquid layer height, and top surface
can be removed by conduction, convection, or local
boundary condition on the downward and upward
boiling of the sodium. If the heat generation rate
power fractions and Nusselt numbers have been deterexceeds the cooling limits of the bed, it will eventually
mined. Flow patterns within the bed and overlying
dry out and melt to form a molten debris pool. The
fluid region have been visualized using particle tracing
molten pool can then flow downward by attacking
techniques.
neighboring supporting structures and causing them to
fail.
Correlations for the downward and upward Nusselt numbers, NuB and NuT, as functions of the interThe objective of postaccident heat removal assessnal Rayleigh number have been developed. In all cases,
ments is to examine the capability of achieving a stable
the beds are bounded from below by a cooled isothercoolable material configuration following the postumal surface. When the overlying fluid is bounded
lated accidents. Therefore, it is important to underfrom above by a cooled solid isothermal surface, the
stand the heat transfer processes taking place in core
Nusselt numbers are given by NuB = 0.424 Ra0226 and
debris beds. Debris bed investigations conducted so far
NuT = 1.61 Ra0 220. When the upper surface of the
consist of both in- and out-of-pile experiments, as well
overlying fluid is free, the downward Nusselt number
as theoretical model developments. Most of these
is given by NuB = 0.503 Ra0180. These correlations
investigations
are "integral" in nature focusing primarare valid for the ranges 102<Ra < 107 and 0.1 <
ily on the evaluation of the dryout limit and its depen7) <1.0, where ?j is the ratio between the heights of the dence on the different variables, including particle
overlying fluid layer and the bed.
diameter, bed height, bed stratification, coolant type,
and boundary conditions.
Several studies dealing with single-phase natural
convection within debris beds have been reported in
the literature. The majority of these use particle beds
I. INTRODUCTION
with adiabatic bottoms. Heat addition is accomplished
by either joule heating of the liquid coolant or direct
A postulated loss-of-flow accident with failure to
heating of the particles. The earliest experimental
scram in a liquid-metal fast breeder reactor will lead
study of the joule heating type was reported by
to coolant voiding and, eventually, to core melting. 1
Burreta and Berman7 who observed discontinuities in
the Nusselt-Rayleigh relation at Rayleigh numbers of
70 and 200 for 3- and 6-mm-diam particles, respec*Present address: Korean Advanced Energy Research Institively. The data for the Rayleigh number ranges of 30
tute, Dae Jun, Chung Nam, Korea.
to 70 for 3-mm-diam particles and 30 to 200 for 6-mmdiam particles were correlated by (see Nomenclature
on p. 277)
Nu = (Ra/32)
0237
,
(1)
where the Nusselt and Rayleigh numbers are defined
as
N u ^ QvLi/2km{TB
- Tt)
(2)
and
_
[email protected]
(3)
bed height at constant bed power. Their data were correlated by
Nu = (Ra/14) 0.5
Lipinski et al. reported the results of in-pile
experiments on natural convection heat transfer in
debris beds. Their data were obtained from the D-l,
D-2, D-3, and D-4 series of experiments, 15 ' 16 conducted at Sandia National Laboratories using sodiumcooled, fission-heated, U 0 2 particle beds. In these
experiments, the height of the overlying coolant layer
was not specified. The data were correlated by
Downloaded by [University of Missouri-Columbia] at 02:16 27 October 2017
2 k^oifVf
Nu = (Ra/0.76) 0.34
respectively. Data for the Rayleigh number ranges of
70 to 900 for 3-mm-diam particles and 200 to 900 for
6-mm-diam particles were correlated by
Nu = (Ra/38) 0.553
(4)
Subsequent experimental studies reported by Sun8 did
not show any discontinuities in the Nusselt-Rayleigh
relation over the same range of variables.
Hardee and Nilson 9 carried out an analytical
study on convective heat transfer in a porous medium
with volumetric heating of the coolant and rigid
boundary walls. A correlation of the Nusselt-Rayleigh
type was obtained using the convective roll cell
model 7,10 :
Nu = (Ra/32) 0.5
(5)
11
Rhee et al. examined the effect of the presence
of an overlying coolant fluid layer on the heat transfer correlations for induction-heated particle beds.
Correlations of the Nusselt-Rayleigh type were
obtained for several values of the overlying fluid layer
height ratio rj = Lf/Lp. The data indicated that for
the same Rayleigh number, the Nusselt number
increases as rj increases with no further increase
beyond rj = 1.0.
The effect of the bottom surface boundary condition on the heat transfer correlations for inductionheated particle beds was examined by Cherng et al. 12
Their data were correlated by
Nu = (Ra/31.5) 0 55
for isothermal bottom
with rj = 1.0
(6)
for adiabatic bottom
with 7j = 1.0 .
(7)
and
(8)
15
(9)
I.A. Objectives
The above discussion points to the large discrepancies among the results reported in the literature for
single-phase natural convection in debris beds. Therefore, the aim of this study is to further investigate the
heat transfer characteristics of single-phase naturalconvection-cooled, volumetrically heated debris beds.
Specific objectives are:
1. to extend the range of parameters covered in
earlier investigations, namely, particle size, bed
porosity, bed height, heat generation rate, and
overlying fluid layer height so that generalized
correlations of the Nusselt-Rayleigh type valid
over a wide range of parameters can be obtained
2. to visualize the flow pattern and measure the
coolant velocity distributions within the bed
and overlying fluid layer.
The data obtained in this investigation can be used to
validate predictions of three-dimensional natural convection codes, such as COMMIX-1A (Ref. 17).
The experimental apparatus and procedure are
described in Sec. II. The results are presented and discussed in Sec. Ill, and conclusions and recommendations are given in Sec. IV. Additional details regarding
this work, as well as preliminary comparisons between
the data and predictions of the COMMIX-1A code,
can be found in Ref. 18.
II. EXPERIMENTAL APPARATUS AND PROCEDURE
Nu = (Ra/12) 0 69
Barleon and Werle 13 ' 14 performed an extensive
experimental study on natural convection cooling and
dryout of induction-heated particle beds. Their data
showed discrepancies with the results of Hardee and
Nilson 9 and Rhee et al. 11 They also reported that for
bottom-cooled beds, the downward power fraction
was relatively small ( - 2 0 % ) and decreased with
increasing bed power at constant bed height and with
A photograph of the entire experimental setup
appears in Fig. 1. The test cell housing in which the
simulated debris bed is placed (Fig. 2) is a 15.5- x
10.2- x 29.0-cm rectangular box bounded from the top
and bottom by water-cooled, nearly isothermal brass
plates. The height of the top plate is adjustable while
the bottom plate is fixed. The four vertical surfaces of
the test cell are made from 0.63-cm-thick tempered
glass plates supported by four aluminum pillars with
a 2.5- x 2.5-cm cross section. The pillars are fastened
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Fig. 1. Overall view of the experimental setup: 1 = test cell, 2 = light source, 3 = data logger, 4 = dc power supply, 5 = ac
variac, 6 = flowmeters, 7 = amp and volt meters, 8 = current control panel, 9 = constant temperature bath, 10 =
positive displacement pumps, and 11 = three-dimensional sliding table.
t o the b o t t o m water-cooled plate. For experiments not
involving flow visualization, all f o u r vertical surfaces
are covered with 2.5-cm-thick s t y r o f o a m insulation.
T h e t o p a n d b o t t o m plates are constructed f r o m
1.27-cm-thick, 10.2- x 15.5-cm brass plates machined
t o p r o d u c e a d o u b l e spiral cooling water flow channel providing alternate h o t a n d cold legs of the flow
p a t h t o o b t a i n nearly isothermal surfaces. E a c h plate
is fitted with five 30-gauge iron-constantan thermocouples placed in individual wells drilled to within 0.05 cm
of the plate f r o n t s u r f a c e . T h e t h e r m o c o u p l e s are
cemented in their respective wells by silicon glue. T o
assure isothermal conditions, t h e differences between
the t h e r m o c o u p l e readings o n each plate are kept
below 0.3 °C; this can be accomplished by a d j u s t i n g
the coolant flow rates so that the differences between
the c o o l a n t s ' exit a n d inlet t e m p e r a t u r e s are < 5 ° C .
The debris bed is simulated using p a c k e d t u b e
bundles with d i f f e r e n t t u b e diameters, n a m e l y , 3.2,
6.4, 9.5, and 11.1 m m . The thin-walled tubes are m a d e
of stainless steel and are arranged in a square p a t t e r n
with 1.27-cm pitch in b o t h the vertical a n d horizontal directions. In addition, two triangular grid bundles
with 9.5- and 11.1-mm-diam tubes and a pitch of 1.27
cm have been tested. T h e square grid c o n f i g u r a t i o n is
m a i n t a i n e d using 0.15- or 0.32-cm-diam brass rods
welded to t h e tubes in the f r o n t a n d back planes in
such a way as t o connect the d i f f e r e n t c o l u m n s of
tubes in series in order t o limit the current required t o
directly heat the debris bed. The 0.15-cm rods are used
f o r the 3.2- a n d 6 . 4 - m m - d i a m t u b e bundles while the
0.32-cm r o d s are used f o r the 9.5- a n d 11.1-mm b u n dles. The staggered tube bundle configuration is maintained using 0.3-cm-thick Plexiglass f r o n t a n d rear
plates with drilled holes in a staggered pattern t o hold
the tubes. All individual tubes are t h e n connected in
series using 0.15-cm-diam brass rods. All t u b e bundles
are 15 cm wide a n d 10.2 cm long with heights varying f r o m 3.2 t o 9.5 cm.
Direct heating of t h e simulated debris bed is
achieved by passing dc current through the thin-walled
stainless steel tubes. A 200-A, 17- t o 40-V welding
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Fig. 2. Photograph of the test cell: 1 = simulated debris
bed for very high porosity, 2 = overlying fluid
layer, 3 = top isothermal plate, 4 = bottom isothermal plate, 5 = inlet coolant flow for top plate,
6 = inlet coolant flow for bottom plate, 7 = top
plate holder, 8 = power lines, and 9 = thermocouple lines.
power s u p p l y 3 is used f o r t h a t p u r p o s e . T o c o n t r o l
t h e power dissipated within the simulated debris bed,
a n a d j u s t a b l e resistor b a n k is connected in series with
t h e b u n d l e . P o w e r i n p u t is determined by m e a s u r i n g
the current a n d voltage d r o p across the b u n d l e . T h e
current is m e a s u r e d by m e a n s of a calibrated 100-A,
50-mV s h u n t ; the voltage d r o p s across the shunt a n d
test b u n d l e are measured with an accuracy of ± 0 . 1 %
using a M o n i t o r L a b s 9300, 40-channel d a t a logger.
a
Miller model CP-200.
A d d i t i o n a l experiments using either b o t t o m heating of the bed or joule heating of the coolant have also
been c o n d u c t e d . Details of these experiments can be
f o u n d in R e f . 18.
T h e p o w e r removed at the t o p and b o t t o m surfaces is determined by measuring the coolant flow rate
a n d t e m p e r a t u r e rise f o r each plate. Cooling water is
supplied independently to each plate by a positive displacement M a s t e r f l u x variable speed p u m p . A constant inlet t e m p e r a t u r e ( ± 0 . 1 ° C ) is maintained by a
F o r m a Scientific 2160 constant temperature bath.
F l o w integrators are placed d o w n s t r e a m f r o m the
p u m p s t o reduce the cyclic variations in the flow rates.
Coolant flow rates are measured for each circuit using
t w o M a t h e s o n G a s P r o d u c t s R7630 series rotameters.
M e a s u r e m e n t of the inlet and outlet water temperatures f o r each plate is m a d e using 30-gauge ironconstantan thermocouples. The temperature difference
is also measured directly using two thermocouples connected in series.
T h r e e to seven 30-gauge iron-constantan thermocouples, depending o n the height of each tube bundle, are carefully placed between the tubes to monitor
the vertical variation of coolant temperature along the
centerline of the test cell. Vertical variation in the overlying fluid layer region is measured using a thermocouple p r o b e m o u n t e d o n a c o m p o u n d micrometer stand
t h a t could b e indexed to m e a s u r e p r o b e positions
along the vertical centerline. Outputs f r o m all the thermocouples used in this experiment are measured using
t h e M o n i t o r L a b s 9300, 40-channel d a t a logger with
an accuracy of ± 0 . 1 °C. F o r a given experiment, variations of the coolant t e m p e r a t u r e s and upward and
d o w n w a r d power f r a c t i o n s with time are monitored
using the d a t a logger. Steady-state conditions are
reached when these quantities stabilize and the energy
balance error is minimized, typically within 2 to 3 h.
A t t h a t point all d a t a are r e c o r d e d . The steady-state
energy balance error is typically about - 5 % for experiments with a n isothermal u p p e r b o u n d a r y and about
- 1 0 % f o r experiments with a free upper surface.
E x p e r i m e n t s have been conducted f o r different
t u b e bundles, i.e., t u b e diameter, bed height, and
porosity, with various p o w e r inputs, overlying fluid
layer heights, a n d t o p surface b o u n d a r y conditions.
T h e ranges of d i f f e r e n t experimental variables are
listed in T a b l e I. F o r experiments with a free upper
s u r f a c e , the t o p cooling plate is positioned ~ 1.0 cm
a b o v e the overlying coolant surface. A total of 389
experiments have been conducted: 193 experiments use
a solid i s o t h e r m a l u p p e r surface while the remaining
196 experiments use a free u p p e r b o u n d a r y .
In a d d i t i o n to the a b o v e experiments, 16 experiments have been repeated without the outer styrofoam
insulation layer t o allow flow visualization and measurement of the velocity distribution. These include ten
experiments with a solid isothermal upper surface and
six free surface experiments. Flow visualization within
TABLE I
hT=qT/\AT{Tc-TT)\
Experimental Variables
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a
Top surface boundary
Free, isothermal solid
Particle size (mm)
3.2, 6.4, 9.5, 11.1
and
(11)
hB =
qB/[AB(Tc-TB)]
where
Bed porosity
0.951, 0.804, 0.558, 0.490
0.399, 0.306 a
Bed height (cm)
3.2, 5.7, 8.3, 9.5
Overlying fluid height
ratio ( L f / L p )
0.1, 0.2, 0.5, 1.0
Power input (W)
25, 50, 100, 200, 300, 400
Grid structure
Square and staggered
a
QT-> QB = u p w a r d a n d d o w n w a r d heat flow rates
AT,AB
= top and b o t t o m surface areas
TT, Tb = t o p and b o t t o m average surface temperatures
Tc = m a x i m u m coolant t e m p e r a t u r e within
the bed.
T h e Nusselt numbers defined above differ f r o m those
normally used f o r volumetrically heated beds with adiabatic b o u n d a r i e s [see E q . (2)]. T h e effective thermal
conductivity km is evaluated using the K a m p f a n d
Karsten m o d e l 1 9 :
Porosities for staggered grid.
the t u b e b u n d l e and overlying fluid layer is accomplished using particle tracing techniques. A planar light
source is used t o illuminate any vertical plane within
the test cell by placing the latter o n a power screwdriven cross table. T h e flow pattern in the illuminated
plane is m a d e visible by placing a small a m o u n t of fine
a l u m i n u m p o w d e r (15-/im diameter) in t h e coolant.
The flow p a t t e r n is p h o t o g r a p h e d using a N i k o n F-2
camera with an 85-mm, f 1.2 macrolens. Nikon m o t o r
drive M D - 2 is used, together with a Nikon intervalometer M T - 1 , t o o b t a i n a series of flow p a t t e r n p h o t o graphs with a 1-s exposure time a n d periods of 1.0 to
30.0 s between consecutive p h o t o g r a p h s . T h e p h o t o graphs are t a k e n a f t e r steady-state conditions are
reached, as evidenced b y the u n c h a n g i n g values of
d o w n w a r d a n d u p w a r d power fractions a n d minimization of the energy balance e r r o r .
k
/V™
— Nk- R
,2/3
•
1 + e 1/3
K
f
(12)
- 1
T h e u p w a r d a n d d o w n w a r d heat f l o w rates qT
a n d qB are c o m p u t e d f r o m the m e a s u r e d values of
coolant flow rate and temperature rise f o r the top and
b o t t o m plates. The m a x i m u m temperature Tc is determined f r o m the measured temperature profiles within
the bed and overlying fluid along the center line. Typical t e m p e r a t u r e variations are shown in Fig. 3; these
c o r r e s p o n d to a simulated bed height of 9.5 cm, an
overlying fluid layer thickness of 4.8 cm {r\ = 0 . 5 ) , a
particle diameter of 11.1 m m , a n d a porosity of 0.40.
III. RESULTS AND DISCUSSION
T h e p r i m a r y goal of this investigation is t o determine t h e e f f e c t of d i f f e r e n t p a r a m e t e r s o n t h e d o w n w a r d a n d u p w a r d p o w e r f r a c t i o n s a n d Nusselt
n u m b e r s f o r b o t t o m - c o o l e d simulated debris beds.
H e r e , the Nusselt n u m b e r s are d e f i n e d as
Nur =
hTLp/km
(10)
and
Nu
B=hBLp/km
6
where
Lp = bed height
km = effective t h e r m a l conductivity
h r , h B = average heat t r a n s f e r coefficients at t h e
t o p a n d b o t t o m surfaces, respectively,
defined b y
8
10
Height (cm)
Fig. 3. Temperature variations along the centerline within
the bed and overlying fluid regions: line A = top
plate level for experiment with a free coolant upper
surface, and line B = liquid level for both solid isothermal and free top experiments.
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T h e p o w e r input f o r the experiment with a solid isothermal upper surface is 300 W , while that f o r the free
upper surface experiment is 100 W . For both cases, the
t e m p e r a t u r e increases m o n o t o n i c a l l y within t h e bed
region, a n d , except f o r a thin b o u n d a r y layer near the
upper surface, remains nearly u n i f o r m within the overlying fluid layer. This is a result of vigorous convection within the overlying fluid, as indicated by the flow
p a t t e r n picture shown in Fig. 4. Similar t e m p e r a t u r e
profiles h a v e been obtained f o r d i f f e r e n t bed porosities and overlying fluid layer height ratios rj. For low-??
values, the position of the m a x i m u m coolant tempera t u r e shifts slightly d o w n w a r d .
Figure 5 shows a variation of the d i f f e r e n c e between Tc a n d t h e average b o t t o m plate t e m p e r a t u r e
with power input f o r experiments with either a free
upper surface or a solid isothermal boundary. D a t a are
presented f o r t w o beds with t h e highest a n d lowest
porosities examined, namely, 0.95 and 0.31, respectively. T h e high-porosity bed is 8.26 cm high with
Fig. 4. Typical photograph showing the convective flow
pattern within the overlying fluid layer region:
Lp = 9.5 cm, T? = 1 . 0 , e = 0.56, d = 9.5 mm,
power = 200 W, solid isothermal top, 1-s exposure
time.
Fig. 5. Difference between the maximum coolant temperature and bottom surface temperature for experiments with high
and low bed porosities with either a solid isothermal or a free upper boundary.
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3.18-mm-diam tubes arranged in a square pattern;
the low-porosity bed is 8.25 cm high with 11.1-mmdiam tubes in a staggered arrangement. The results in
Fig. 5 indicate that because of convective mixing the
maximum temperature difference is significantly lower
than the conduction value. As expected, the temperature difference does not increase proportionally to the
heat generation rate but tends to saturate at highpower inputs, i.e., high Rayleigh numbers. Surprisingly, however, the maximum temperature difference
increases only slightly as the void fraction is significantly decreased. The temperature increase is caused
by the reduction in permeability as the porosity
decreases. Figure 5 also shows that, for the same
power input, the maximum temperature difference for
beds with a free upper surface is significantly higher
than that for beds with an isothermal upper surface.
This result is not surprising because of the significantly
reduced rate of heat loss from the upper boundary in
the former.
Figures 6 and 7 show variations of the downward
power fraction with the power density and overlying
fluid layer height ratio for beds with a free upper surface or a solid isothermal upper boundary, respectively. The downward power fraction for the free
surface case decreases perceptibly as the power density increases. The reason is that the pool temperature
rapidly approaches the boiling point so that evaporation from the free surface becomes the dominant heat
removal mechanism rather than downward convection
to the lower boundary. For beds with an isothermal
upper surface, however, the downward power fraction
gradually decreases with power input and approaches
a nearly constant value of -0.15. For both types of
upper boundary conditions, downward heat transfer
is enhanced by increasing the depth of the overlying
fluid layer. This can be seen in Fig. 8 where data simi-
lar to those presented in Fig. 7 are cross-plotted. The
downward power fraction increases rapidly as the
overlying fluid layer thickness increases with little
increase beyond 77 = 1.0. This trend is in agreement
with the results of Rhee et al. 11
Variations of the downward heat transfer coefficient with power density and overlying fluid layer
height ratio for beds with a free upper surface or a
i
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Fig. 6.
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-
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2
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I
1
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1
Qtotal
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/a
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/
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A-200
/
300
A
L p = 9.525 cm
e = 0.3987
d = 11.1 mm
*
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l
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105
2
Power Density, Q'" (W/m3)
• 1.0
*
Results similar to those in Fig. 6 for experiments
with a solid isothermal upper surface.
0.4
e %
a
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LD = 8.25 cm
1
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5
o
0.55
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•
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-
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3
Power Density, Q" (W/m )
•
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0.1
o
0.1
0.2
0.5
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CO
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Staggered
*
e = 0.4899
e Lp = 8.25 cm
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Solid Top
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Variation of the downward power fraction fdOW„
with power density Q'" and overlying fluid layer
height ratio 17 for experiments with a free upper
surface.
0.1
Fig. 8.
i
i
1
1
1
0.2
0.4
0.6
0.8
1.0
Overlying Fluid Height Ratio, Lf/L p
l
1.2
Variation of downward power fraction with overlying fluid layer height ratio and power input for
experiments with a solid isothermal upper surface.
Downloaded by [University of Missouri-Columbia] at 02:16 27 October 2017
solid isothermal upper boundary are shown in Figs. 9
and 10, respectively. As expected, the heat transfer
coefficient increases as the power density increases
because of the increase in Rayleigh number [Eq. (3)].
Downward heat transfer is enhanced as the overlying
fluid layer depth increases; this effect is more pronounced for the case of a solid isothermal upper
boundary.
The effect of bed porosity on the upward heat
transfer rate is illustrated in Fig. 11. Here, the ratio
between the experimentally measured average upward
heat flux and a theoretical prediction (q'e Xp /qS) is
plotted as a function of power input for five different
bed porosities. The theoretical value q(y is calculated
using a correlation for the Nusselt number in a
horizontal fluid layer bounded at the top and bottom
and heated from below 20 :
1
I
i
1
1
300
ID
o
o _
ia 200
Q) O
.
-
M—
V) CN
•
S-l
H?
& CD 100
\
1
-
-
c
5
o
Q
50
40
i
5
1
10"
-
*
3
*
^
Free Top
Staggered
E = 0.4899
L p = 8.25 cm
d = 9.5 mm
i
.
s
10
2
V
e
«
o
•
0.1
0.2
0.5
1.0
i
.
:
'
106
Power Density, Q'" (W/m3)
Fig. 9.
Variation of downward heat transfer coefficient
with power density and overlying fluid layer height
ratio rj for experiments with a free upper surface.
Nu = 1 + 1.44 1 -
1708 V
+
Ra o )
Rap /3
140
1 - In
+ 2.0
1
'
R a
§:E
«
ra5 100
03
T
J- -C
•g
•
a•
9
o
1
Isothermal Top
-
Staggered
-
e = 0.4899
CO
3
c
S
o
O
•
»
8
50
40
10
i
<0
<u
I
^ 8 j f A T L }
(14)
1
Lf/Lp = 1
Solid Isothermal
"
1
e
1
o 0.804
7 0.5582
O 0.4899
• 0.3987
"
A 0.3057
—
o o
V V
o
•
•
*
0
°
V
•
•
A
•
a
V
A
to
g
a
1.5.
£1
A
•
2.0
O 0.5
• 1.0
L p = 8.25 cm
i
>
1
5
105
2
Power Density, Q'" (W/m3)
(13)
Composite plots of all the experimental data
obtained in this investigation are shown in Figs. 12,
13, and 14. Figures 12 and 13 show variations of the
upward and downward Nusselt numbers with the
2.5
u
e 0.1
0 0.2
Ra 1/3
140
III.A. Correlation of Experimental Data
1
• •
o o
e
% e
_ ,
The parenthesis with an asterisk in Eq. (13) is set equal
to zero if the argument inside the parenthesis is negative. The temperature difference AT in Eq. (14) is set
equal to the difference between the bed/overlying fluid
interface temperature and the upper boundary temperature.
Figure 11 shows that the ratio q'eXp/qo decreases
as the bed porosity decreases. This is caused by a
reduction in the "coupling" between convection within
the bed and overlying fluid layer. For high-porosity
beds, the induced flow within the bed can easily penetrate into the overlying fluid layer and cause more
active convection than for the case of a horizontal
layer heated from below. It is interesting to note, however, that even for the lowest porosity examined,
namely, 0.31, there is significant coupling between
convection within the bed and overlying fluid layer
that results in the doubling of the upward heat flux
( q : x p / q o ~ 2 m Fig. 11).
3.0
•
f
OLfVf
300
o
o
200
® b
M
\5830/
where the Rayleigh number Ra 0 is defined by
3.5
i
(
1
100
1
200
1
300
-
1
400
Total Power Input, Q (W)
106
Fig. 10. Results similar to those in Fig. 9 for experiments
with a solid isothermal upper surface.
Fig. 11. Variation of the ratio between actual upward heat
flux and that for a viscous fluid layer heated from
below for different bed porosities and power inputs.
Downloaded by [University of Missouri-Columbia] at 02:16 27 October 2017
10"
105
106
107
Internal Rayleigh Number, Ra
103
Fig. 12. Variation of the upward Nusselt number with
Rayleigh number for volumetrically heated beds
bounded from above and below by cooled solid
isothermal surfaces.
10"
105
106
107
Internal Rayleigh Number, Ra
Fig. 14. Variation of the downward Nusselt number with
Rayleigh number for volumetrically heated beds
bounded from below by a solid isothermal surface
with a free upper coolant surface.
Ck =
1—e
ln1
1 — (1 — e)
l + (l-e)2.
(16)
F o r the same porosity, the c o n s t a n t Ck calculated
using E q . (16) is considerably higher t h a n the p a c k e d
spheres bed value of 5.0. For low values of e, however,
the Ck value f o r a bank of cylinders is nearly equal to
t h a t f o r the p a c k e d spheres b e d .
Based o n the d a t a in Figs. 12, 13, a n d 14, the following correlations have been o b t a i n e d :
!
103
10"
105
106
107
Internal Rayleigh Number, Ra
108
1. For beds bounded from
cooled solid isothermal
above and below
surfaces:
by
N u r = 1.61 R a 0 2 2 0
(17)
N u s = 0.424 Ra'0.226
(18)
and
Fig. 13. Variation of the downward Nusselt number with
Rayleigh number for volumetrically heated beds
bounded from above and below by cooled solid
isothermal surfaces.
internal Rayleigh number for beds with a solid isothermal upper b o u n d a r y . Figure 14 shows N u s versus the
Rayleigh n u m b e r f o r the case of a free u p p e r surface.
T h e Nusselt n u m b e r s in these figures are defined by
E q s . (10) a n d (11); the internal Rayleigh n u m b e r is
defined by E q . (3). T h e permeability P in E q . (3) is
given by the K o z e n y - C a r m a n 21 e q u a t i o n as
P =
bounded from below by a solid
surface with a free upper
coolant
N u s = 0.503 R a 0 . 1 8
(19)
E q u a t i o n s (17), (18), and (19) represent t h e m a i n
results of this w o r k . These equations are valid f o r the
ranges 10 2 < Ra < 10 7 a n d 0.1 < rj < 1.0. T h e d a t a
used to develop these correlations pertain t o beds with
porosities ranging f r o m 0.31 t o 0.95.
IV. CONCLUSIONS
D
36C k (1 - e )
2. For beds
isothermal
surface:
2
(15)
where Ck is a c o n s t a n t depending o n p o r e structure.
H e r e , Ck is evaluated using equations f o r flow perpendicular t o a b a n k of cylinders 2 2 :
Natural convection heat transfer in simulated core
debris beds has been examined. The debris beds are
simulated using electrically heated p a c k e d tube b u n dles arranged in either a square or staggered lattice
Downloaded by [University of Missouri-Columbia] at 02:16 27 October 2017
with porosities varying between 0.31 and 0.95. The
effects of bed height, heat generation rate, particle
size, porosity, overlying liquid layer height, and top
surface boundary condition on the downward and
upward power fractions and Nusselt numbers have
been determined. Flow patterns within the bed and
overlying fluid region have been visualized using
particle tracing techniques.
The results indicate that significant coupling exists
between the convective flow within the bed and overlying fluid layer even at the lowest bed porosity. The
results also indicate that the upper coolant surface
boundary condition has a significant impact on bed
behavior.
Correlations for the downward and upward Nusselt numbers, Nu B and Nu r , as functions of the internal Rayleigh number have been developed. In all cases,
the beds are bounded from below by a cooled isothermal surface. When the overlying fluid is bounded
from above by a cooled solid isothermal surface, the
Nusselt numbers are given by Nu B = 0.424 Ra 0 226
and N u r = 1.61 Ra 0,220 . When the upper surface of
the overlying fluid is free, the downward Nusselt number is given by N u s = 0.503 Ra 0 1 8 0 . These correlations are valid for the ranges 1 0 2 < R a < 107 and
0.1 < rj ^ 1.0, where »j is the ratio between the heights
of the overlying fluid layer and the bed.
<7o
= upward heat flux for a viscous fluid layer
heated from below
Qv
= volumetric heat generation rate
Ra
= internal Rayleigh number for a volumetrically heated particle bed, defined in Eq.
(3)
Ra 0
= Rayleigh number for a viscous fluid layer
heated from below, defined in Eq. (14)
TB, TT, 7} = bottom, top, and interface temperature,
respectively
af,am
= fluid and effective thermal diffusivity,
respectively
|8f
= thermal expansion coefficient
rj
= ratio between overlying fluid layer depth
to bed height
Vf
(LF/LP)
= kinematic viscosity of the fluid
ACKNOWLEDGMENTS
The assistance of W.P. Chang in conducting the experiments is gratefully acknowledged. Financial support for
this work was provided by the Fast Breeder Project,
Nuclear Research Center, Karlsruhe, Federal Republic of
Germany.
NOMENCLATURE
AT,AB
= top and bottom surface areas, respectively
REFERENCES
CK
= constant used in Eq. (15), defined in Eq.
(16)
g
= gravitational acceleration
hT, hB
= heat transfer coefficients for upward and
downward directions, respectively, defined in Eq. (11)
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km
— effective thermal conductivity of the bed
[Eq. (12)]
kf
= thermal conductivity of liquid coolant
ks
= thermal conductivity of solid particles in
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LF,LP
= height of overlying fluid layer and bed,
respectively
Nufl,Nu7- = bottom and top Nusselt numbers, respectively, defined in Eq. (10)
Nu
= Nusselt number, defined in Eq. (2)
P
= permeability of porous media, defined in
Eq. (15)
Qb>Qt
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Porous
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