Natural convection heat transfer from a trombe wall geometry to a rectangular enclosure
by Peng-Cheng Lin
A thesis submitted in partial fulfillment of the requirements for the degree of MASTERS OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Peng-Cheng Lin (1982)
Abstract:
A 1/18th scale experimental model was developed to characterize the natural convection heat transfer
in passive solar heating with a Trombe wall geometry. Steady natural convection heat transfer was
measured from an isothermal heated Trombe wall (inner body) with various gap sizes, 2.54 cm gap,
0.635 cm gap, and zero gap, to a rectangular enclosure (outer body). Temperature profiles within the
test space were obtained and fluid flow patterns were observed. Dow Corning 20 cs fluid was utilized
as the working fluid with Prandtl numbers which ranged from 125 to 277. The inner body with 2.54 cm
gap appears to give a highest heat transfer from the Trombe wall, especially in the low temperature
range, due to the exchange of mass between the Trombe wall space and living space. However, when
the gap size was increased the temperature stratification in the living space was also increased. A
circulation cell around the upper portion of the living space was observed. The recommended empirical
equation describing the heat transfer using two independent parameters was Nuh = 1.6106 RaH0.1760
(H/Hi)-0•2159 for 6.2x10 8 < RaH < 1.5x10 9 with an average percent deviation of 3.01. STATEMENT OF PERMISSION TO COPY
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Date
9 / I7 / S' T-
NATURAL CONVECTION HEAT TRANSFER FROM A TROMBE WALL
GEOMETRY TO A RECTANGULAR ENCLOSURE
by
PENG-CHENG LIN
A thesis submitted in partial fulfillment
of the requirements for the degree
of
MASTERS OF SCIENCE
in
Mechanical Engineering
(fk-sjLjl
Chairperson, Graduate Committee
Head, Major Department
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
September, 1982
iii
' ACKNOWLEDGEMENT
The
author
wishes
to
express,
his
thanks
and
appreciation to the following for their contribution to.this
investigation:
Dr. R. L. M u s s u l m a n and Dr. R. 0. W arrington, for their
guidance, advice and instruction.
Dr. A. Demetriades, for serving as a c o m m i t t e e m e m b e r
and reviewing this thesis.
Pat V o w e l l
and Gordon W i l l i a m s o n
for
their helpful
assistance in the construction and maintenance of the heat
transfer
apparatus.
The Mechanical Engineering Department of Montana State
University,
for funding of this investigation.'*'
The Theater Arts Department of Montana State University
for providing lighting equipment for this investigation.
Lastly,
encouragement,
yet
foremostly,
S u e - M a y , for
and for typing this thesis.
her
constant
TABLE OF CONTENTS
Chapter
v i t a ......................................................
... ii
ACKNOWLEDGEMENT.......^..................................lll
LIST OF TABLES............... . .......... ..................... V
LIST OF FIGURES............. . ........................... . . . .vi
NOMENCLATURE......
viii
X
ABSTRACT........... . ....... ................. ........ .
I.
II.
III.
IV.
INTRODUCTION...... ...... .............................. . .1
LITERATURE REVIEW.................. ....... ............ 5
EXPERIMENTAL APPARATUS AND PROCEDURE..... ........ .,.19
DISCUSSION OF RESULTS...... .................... .
.37
V. , CONCLUSION AND RECOMMENDATIONS............. ..... ...... 72
APPENDICES. . . . ............................... ............... .76
APPENDIX I
APPENDIX
II
APPENDIX III
HEAT TRANSFER DATA REDUCTION PROGRAM.... ...77
PARTIALLY
REDUCED
D A T A ................ .....87
EFFECTIVE HEAT TRANSFER AREA. .......... . ... .89
BIBLIOGRAPHY..........
95
V
LIST OF.TABLES
Table
^aae
4.1
Range of. Dimensionless Parameters ..............
39
4.2
Correlations for LHS Test Space ........
66
4.3
Correlations for RHS Test Space ................
4.4
Overall Heat Transfer Correlations ........
.67
69
LIST OF FIGURES
Figure
Page
1.1
Solar Passive Heating with aTrombe
3.1
Heat Transfer Apparatus ..........
3.2
Schematic of the Heat Transfer Apparatus ...........22
3.3
Interior of the Inner Body .....................
3.4
Heat Loss by Edge Conduction
4.1
Geometric Effect for All Gaps
(OVERALL HEAT TRANSFER) ............................
4.2
4.3
wall Geometry ...3
.21
26
................ .34
Geometric Effect for All Gaps
(HEAT TRANSFER TO LEFT HAND SIDE SPACE)
Geometric Effect for All Gaps
(HEAT TRANSFER TO RIGHT HAND SIDE SPACE)
40
............ 41
.... .
42
4.4
A Sketch of Flow Pattern for the Inner Body with
Zero Gap, AT = 30.80°K (55.44°R) , Ra11 = 4.79xl09 ...44
4.5
A Sketch of Flow Pattern for the Inner Body
with 2.54 cm Gap, AT = 31.33°K (56.390R ) ,
Rafi = 4.89xl09 ................... ...... .45
4.6
(a)
Photographs of Flow Pattern in the Living Space
for Zero Gap
Upper Left
..................
(b)
Upper Right ................ ...... .......... ........ 47
4.7
(a)
Photographs of Flow Pattern in the Living Space
for 2.54 cm Gap
Upper Left
.... .......... ...... ...............48
(b)
Upper Right .................. ........... ........... 49
4.8
Comparison of Temperatures for All Inner
Body Gaps ............................
46
52
vii
Pjigiu;.e
4.9
4.10
Page
Variation of Temperature Profile with AT, Inner
Body with Zero Gap ...... ....... .......... ...... ..
54
Variation of Temperature Profile with AT, Inner
Body with 0.635 cm Gap ..........................
55
4.11
Variation of Temperature Profile with At , Inner
Body with 2.54 cm Gap ............. ...... .56
4.12
Comparison of Present Data with Raithby et al.
(Zero-Left) ..... ..... ;...................... ....... 61
4.13
Flow Patterns in the Trombe wall space for Zero-Left
(a) Rag = 4.8x10® (b) Rag = 9.0x10®........... ...62
4.14
Comparison of Present Data with Raithby et al.
(Zero-Right) .................... ........ ............ 64
4.15
Comparison of Heat Transfer Results between
LHS and RHS for Zero Gap .................... ........ 65
A3.1
Unvented Inner Body ...... ...........................90
A3.2
The Effective Heat Transfer Area for the Left
face of the Unvented Inner Body .... ........... .....92
A3.3
The Effective Heat Transfer Area for the Right
. face, of the Unvented Inner Body .....................93
viii
NOMENCLATURE
Symbol
A
Aspect ratio, H/L
Heat transfer area on the inner body with zero
gap, 929 c m 2 (1.0 ft2)
(3-7)
AP
Aperture ratio, the ratio of central
the enclosure height
opening
to
Empirically determined constants
Universal function of Pr for laminar flow
(2-10)
Specific heat at constant pressure
Ct
Universal function of Pr for turbulent flow(2-11)
f
Denotes arbitrary function
9
Accele r a t i o n of gravity, 9.81 m/sec2
(32.17ft/sec2)
Gr
Grashof number, g$p2X 3 AT/y2
Hi
Height of the inner body. Figure 1.1
H
Height of the outer body. Figure 1.1
h
Heat transfer coefficient
k
Thermal conductivity
L
The test rectangular enclosure total length
L1
Left hand side space length. Figure 1.1
L2
Right hand side space length, Figure 1.1
NUg
Nusselt number, hs/k
Pr
Prandtl number, yCp/k
ix
Symbol
Description
^cond
Heat transfer by conduction
Qconv
Heat transfer by convection
^rad
Heat transfer by radiation
Qtot
Total amount of heat transfer,
Qtot = Qconv + Qcondv + Qrad
Ras
Rayleigh number, Cpggp2S3 AT/yk
s
Some characteristic length
Tc
Average cold surface temperature
Th
Average hot suface temperature
Tf
Film temperature,
T*
Dimensionless temperature,
AT
Temperature difference, Tj1 - Tc
x
Horizontal distance from the south facing wall
of the enclosure
(Tj1 + Tc )/2
(T - Tc )/(Tj1 - Tc )
Dimensionless length, x/L-j
Distance along vertical flat plate from leading
edge
Vertical distance from the enclosure top
Dimensionless length, y/H
6
Thermal expansion coefficient
U
Dynamic viscosity
P
Density, of the fluid
X
ABSTRACT
A 1 / 1 8 th scale e x p e r i m e n t a l model wa s developed to
characterize the natural convection heat transfer in passive
solar heating with a Trombe wall geometry.
Steady natural
c o nvection heat transfer was m e a s u r e d from an isothermal
heated Trombe wall (inner body) with various gap sizes, 2.54
c m gap, 0.635 c m gap, a n d z e r o gap, to a r e c t a n g u l a r
enclosure (outer body).
T e m p e r a t u r e profiles within the
test space were obtained and fluid flow patterns were
observed.
I)ow Corning 20 Cs fluid was utilized as the
working fluid with Prandtl numbers which ranged from 125 to
277.
The inner body w i t h 2.54 cm gap appears to give a.
highest heat transfer from the Trom b e wall, especially in
the low tempe r a t u r e range, due to the exchange of mas s
b e t w e e n the T r o m b e wall space and living space.
However,
w h e n the g a p s i z e w a s i n c r e a s e d the t e m p e r a t u r e
stratification in the living space was also increased.
A
circulation cell around the upper portion of the living
space wa s observed.
The r e c o m m e n d e d empir i c a l equation
describing
the h e a t t r a n s f e r u s i n g t w o i n d e p e n d e n t
parameters was
N u h = 1.6106 Ra110 '1760 (HZHi)
for 6.2x10** < Ra g
deviation of 3.01.
< I.S x l O 9
with
an
•2159
average
percent
CHAPTER I
INTRODUCTION
The study of natural c o nvection heat transfer within
enclosures has been receiving increasing attention in recent
years.
in
Despite this, there is still a dearth of knowledge
this
field.
The
study
of
natural
convection
wit h i n
enclosures has not advanced sufficiently to the point where
an analytical or a numerical investigation can be developed
without simplifying the governing equations,
boundary
conditions/
assumptions.
or
As a result,
applying
the
experimental
idealizing the
boundary
studies
layer
are needed
to tackle the p r o b l e m of natural convection heat transfer
within enclosures. . This is particularly true for predicting
fluid flow b ehavior from a body to an enclosure.
has
been
reactor
cabin
concerned
technology,
design,
and
with
solar
design,
nuclear
cooling electronic devices,
aircraft
numerous
heating
Interest
other
applications.
Most
recently applications in passive solar heating have required
a better understand of natural convection within enclosures.
Since 1967, when the prototype roTrombe W a l l ro house was
built and monitored in Odeillo, France, several experimental
and analytical investigations have, been presented pertinent
to this
geometry
in passive
solar heating.
The passive
solar heating with a Trombe wall geometry is illustrated in
2
Figure
1.1.
Two
disadvantages
studies w e r e observed:
has a long conductive
dependent
changes
heat
too
evaluate.
wall
time constant
height
by
problem,
mak i n g
Nevertheless,
only a few seconds,
full-scale
building
(I) the thick concrete Trombe wall
transfer
rapidly
of
resulting in a time
(2)
full-scale
solar
irradiation
data difficult to
the convective time constant is
which is calculated by dividing Trombe
airflow
velocity
[30].
T h u s , steady
convective data are appropriate for application to the time
dependent heat transfer behavior of a Trom b e wall system.
Since passive solar heating designs, including the Trombe
wa l l itself,
are b e c o m i n g e c o n o m i c a l l y c o m p e t i t i v e w i t h
conventional
heating
systems,
there
is a need to obtain
steady convective data for application to a Trombe wall
system.
These considerations led Mussulman and Warrington
[1] to propose to build a 1/18 scaled experimental model to
elim i n a t e
the above two disadvantages.
The criteria for
designing
this
be
chapter
experimental
mod e l
will
described
in
III.
The objective of the present
study was to develop a
1/18th scale experimental model to characterize the natural
conve c t i o n heat transfer in
Trombe wall geomety.
passive solar heating with a
In order to accomplish this the heat
Trombe Wall Space
Top Duct Gap
4% x
Living Space
Th
X
H
T
C
H
L2
lI
South
Double
Glazing
Figure 1.1
---Trombe Wall
—
Bottom Duct Gap
Solar Passive Heating with a Trombe Wall Geometry
4
transfer from an isothermal heated Trombe wall geometry with
three different gap sizes to a rectangular enclosure w as
measured.
Temperature profiles within the Trombe wall space
and.living space were obtained and the fluid flow
in
the
living
(silicone)
space
fluid
were
with
a
observed.
kinematic
Dow
patterns
Corning
viscosity
of
200
20
c e ntistokes at 32°C (90 °F) was utilized as the test fluid.
The Rayleigh number based on the enclosure height varied
from 6.2x108 to 1.5x10-*-®.
Moreover,
empirical
equations to
d e t e r m i n e the heat transfer w i t h i n the living space we r e
determined for several different combinations of parameters.
Chapter II
LITERATURE REVIEW
The p h e n o m e n o n
driven
effect,
of natural convection, is a buoyancy-
that
density differences
fluid.
is,
external
infinite
fluid
studies
[9-20]
motion
results
from
of analytic and experimental
convection heat transfer have been made
over the last 70 years.
of
fluid
caused by temperature gradients in the
A considerable number
studies of natural
that
the
The most throughly studied case is
natural
medium
have
convection
[2-8].'
been
So m e
focused
on
from
of
a body
the
more
natural
to
an
recent
convection
within simple rectangular vertical enclosures or rectangular
enclosures
inclined
from
the vertical.
Cylindrical
have also been
and
spherical
enclosure geometries
investigated
[21-23].
The study of natural convection heat transfer from
several different geometric bodies such as spheres,
I
and cylinders to both cubical
also been presented
cubes,
■
[24].
and spherical
enclosures has
Moreover,. several recent studies
of natural c onvection with i n rectangular enclosures have
investigated passive solar heating geometries
The
toward
focus
of
publ i s h e d
the
follo w i n g
articles
on
discussion
natural
is
directed
convection.
discussion is organized into two sections:
plates and rectangular enclosures.
[19-20,24-32].
The
vertical, flat
In addition,
studies in
6
pas s i v e
solar
heating
wit h
a Trombe
wall
geometry
are
presented.
VERTICAL FLAT PLATES
Numerous early works dealt with natural convection heat
transfer from vertical flat plates
[2-8].
Researchers have
developed theoretical solutions to heat transfer problems by
simplifying the governing equations, idealizing the boundary
conditions,
and applying
the boundary
layer
assumption.
These solutions were in a good agreement with experimental
results.
Eckert
and
Jackson
convection, on an
momentum
integral
[4]
analyzed
turbulent
i s othermal
vertical
plate
by
natural
applying
equations to boundary layer theory.
A
heat transfer equation was derived,
Nu = 0.0246(Gr)2/5(pr)7/15[1+o.494(pr)2/3]-2/5
for Grashof
numbers
greater
than
10*0.
Nu,
(2-i)
Gr,. and
Ra
without a subscript are based on the height of the. vertical
flat
plate.
A
comparision
with
the
experimental
heat
transfer results of Jakob [2] and M c A d a m s [3] sho w e d good
agreement.
They also pointed out that the transition region
was between Rayleigh numbers of IO8 to l0*° with air as the
working fluid.
Five years after Eckert and. Jackson's work, Bayley [5]
7
proposed
a theoretical
analysis
of
turbulent
natural
convection from an isothermal vertical pla t e . . Two equations
were presented
Nux = 0.10(Grx Pr)0 •3 3 7 for 2xl09 < (GrxPr) < IO12
(2-2)
and
Nux = 0.183 (GrxPir)0 *3 1 , for IO12 < (GrxPr)
In additionz
an a p p r o x i m a t i o n
of
< IO15 . (2-3)
the heat
transfer
for
mercury (Pr = 0.01) was
Nux = 0.06(Grx )0l23z for 1010 < Grx < IO15 .
(2-4)
With the same geometry, Warner and Arpaci [6] performed
an
experimental
investigation
of
turbulent
natural
c onvection along a vertical i s othermal heated flat plate
with air as the test medium.
profile data near
Several plots of temperature
the hot wall w e r e
obtained.
The heat
transfer results of their study have shown good agreement
with the analytical
correlation of Bayley
numbers up to 1012„
The one third power of this correlation
showed, that when
turbulent free convection
[5]
for Rayleigh
is encountered,
the.local heat transfer coefficient is essentially constant
wit h X.
Vli e t
and Liu
[7]
experimentally
studied
turbulent
natural convection boundary layers for a constant heat flux
surface condition using water as the w o r k i n g fluid.
The
8
isothermal
surface condition data of Eckert and Jackson
[4]
and Bayley [5] have revealed quite good agreement with their
heat transfer data,
but exhibited a much earlier transition
point due to the different heating modes.
A general c o rrelation of laminar and turbulent free
natural convection from an isothermal vertical
developed by Churchill and Chu [8],
surface was
This correlation is
Nul/2 = 0.825+0.387Ral/6/[l+(0.437/Pr)9/16]8/27 (2-5)
which
may
numbers.
be
applied
Although
to
the
entire
t h i s .equation
is
range
oyf
suitable
Rayleigh
for
most
e n gineering calculations, slightly better accuracy can be
obtained for laminar flow by using
Nu = 0.68+0.67Ral/4/[i+(0.492/Pr)9/16]4/9
for Ra < 109.
(2-6)
The tw o resulting equations m a y be applied
for constant heat flux conditions as well as for constant
surface temperature conditions.
RECTANGULAR ENCLOSURES
,
Natural
convection
within
investigated analytically from
However,
the
interactions
enclosures
the middle
of
has
this
been
century.
of the boundary layers of the
enclosure with its core region cause complex flow patterns
making
analytical
Theoretically,
solutions
difficult
to
obtain.
at large Rayleigh numbers the heat transfer
9
across each of the turbulent boundary layers might approach
the heat
medium;
transfer
from
a vertical
plate
in an infinite
this was not found experimentally.
analytical
solutions
for
natural
rectangular
enclosures
dealt
only
The existing
convection
with
the
within
steady
two-
dimensional case.
Such solutions have compared poorly with
the e x p e r i m e n t a l
results for high Rayleigh numb e r s with
aspect
ratios
less
than
one
because
three-dimensional
effects are important. The aspect ratio is defined as the
ratio of the enclosure height to length nor m a l to the hot
wall.
In general,
vertical
walls
horizontal
rectangular enclosures with two opposite
at
different
temperatures
and
insulated
surfaces have been extensively examined.
depth (the distance b e t w e e n the remai n i n g walls)
The
of this
rectangular enclosure was made sufficiently large to assure
two-dimensional flow in the central region of the cavity.
Natural
adiabatic
convection
t op
an d
in
bottom
a rectangular
surfaces
and
cavity
two
with
opposing
i s o t h e r m a l vertical wal l s at different tempe r a t u r e s was
first
invest i g a t e d
analyt i c a l l y
by
Batchelor
[9].
He
expanded temperature and stream functions in power series of
the Rayleigh number and defined various flow
resulting equation was
regimes.
The
10
N u l = 0.48 RaL1/4(A)3/4 for
(RaL/500)
> A.
(2-7)
Eckert and Carlson [9] made an experi m e n t a l study of
natural c onvection in a rectangular enclosure with three
different aspect ratios (2.5, 10, and 20) using water as the
working
fluid.
Temperature
distributions
wit h i n
the
enclosed space w e r e obtained using an i n t erferometer
describe
conduction,
transition,
and
boundary
to
regimes.
Grashof n u m b e r s based on the enclosure height were on the
order of 10® in this investigation.
Flow fluctuation and
wave motions were observed in some cases.
Elder
[11-12]
performed
an
extensive
experimental
investi g a t i o n in a rectangular cavity with aspect
ratios
ranging from I to 60 for laminar and turbulent flow regimes.
Medicinal paraffine,
silicone oil
utilized as test fluids.
(Pr~1000)
and water were
The flow was made visible by using
a l u m i n u m pow d e r suspended in the fluid.
The e x p e r i m e n t s
were conducted especially
insight into the
fluid flow behavior.
the
hot
observed
wall
of
for
8xl08(Prl/2/A3).
the
to gain further
Travelling wavelike motions growing up
slot
Rayleigh
and
dow n
numbers
the
above
cold
wall
were
approximately
These waves developed most readily midway
b e t w e e n the t w o endwalls.
Near Ra = l O ^ O / A 3 an intense
entrainment and mixing process between the region near the
11.
wall and the interior was initiated.
increased,
As the Rayleigh number
the turbulent middle portion of the flow extended
further toward the endwalls.
Numerical
an d
experimental
studies
of
natural
convection b e t w e e n vertical planes with m o d e r a t e and high
Prandtl number
Emery
[13],
s u bstituted
fluids were carried out by MacGregor
The
into
vorticity
and
the governing
stream
function
and
were
equation and a numerical
solution was obtained by using a finite difference technique
for
isothermal
They concluded
and constant
that
the
heat
net
flux boundary conditions.
heat
transfer
wa s
a strong
function of the aspect ratio when correlated with Ra^.
The
correlations for both the isothermal and constant heat flux
conditions obtained from the experiments were:
N u l = 0.42 (A)-0 •30Pr0 •012Rall9 •25 for 104<RaL <107
with aspect ratios from I to 40. However,
(2-8).
the following one
parameter correlation could be used:
N ul = 0.046 Rall1^ 3
for RaL > IO6
Ostrach and Raghaven
study
which
[14]
(2-9)
conducted an e x p e r i m e n t a l
described the effect of stabilizing thermal
gradients on natural convection.in rectangular enclosures
with
aspect
ratios
of
s t r e a m l i n e s and m e a s u r e
I and
3.
In
order
to
track
the
the a p p r o x i m a t e velocity of the
12
fluid,
oils
Pliolite plastic particles were mixed into silicone
with
centistokes
kinematic
at
25
viscosities
C.
Thermal
established by heating,
cooling,
of
10,000
boundary
an d
conditions
2,000
were
or insulating on opposite
walls of the enclosure so that simulataneous horizontal and
vertical, heat flows w e r e achieved.
Different strea m l i n e
patterns wer e observed by varying the ratio of vertical to
h orizontal
Grashof
numbers
and
the
aspect
ratio.
The
results s h o w e d that a stabilizing effect on the flow was
establ i s h e d by heating the upper surface and cooling the
lower surface.
An a p p r o x i m a t e analysis of natural convection with i n
the rectangular enclosures wa s proposed by Raithby et al.
[15].
The resulting, solutions were
Nu l = 0.7SC ^(RallZA)1/4
(2-10)
and for the laminar regime
N ul = 0 .29Ct (Ralj)1Z 3
(2-11)
for the turbulent regime where C %= 0.50/(l+(0.49/Pr) 9/16) 4/9
and Ct = 0.14Pr0'084,
Good agreement was found in comparing
these equations . to the data available up to 1975.
There is
a lack of data at high Rayleigh numb e r s (Ra > 10l0) an<3 low
aspect
ratios
(A < 5).
The
validation
of
predictions
covering this range of p a r a m e t e r s has to rely on further
13
experiments.
Berkovsky
solution for
and Polevikov
natural
vertical slots.
[16]
developed a numerical
convection
heat
transfer
within
The heat transfer solution was
N u l = 0.22(A)“ 0 '2 5 (RaL Pr/(0.2+Pr))0 ’28
(2-12)
for 2 < A < I O 7 Pr < I O 5 and R a L < IO 1 0 .
Cattonf Ayyaswamyf and Clever
in
a rectangular
cavity
at
[17]
various
studied convection
angles
of
tilt.
Of
interest in this analytical investigation was the comparison
of their
results
for
horizontal surfaces.
adiabatic
and perfectly conducting
The results for a vertical slot showed
that the overall heat transfer across the gap is lower for
perfectly conducting horizontal surfaces than for adiabatic
horizontal surfaces.
Examination of the local heat transfer
indicates that at high aspect ratios, there is very little
difference
except
significant
ratios
very
difference
(A = I an d
pronounced thermal
near
can
the
be
A = 0.2).
surface.
seen
at
the low e r
This
is
due
interaction at
to
the perfectly
aspect
the
more
conducting
With
and Holland
[18] obtained an empirical correlation for a
surfaces:
air
slot
with
geometry,
A
boundaries.
vertical
a similar
horizontal
perfectly
Elsherbiny, Raithby,
conducting
horizontal
14
Nu1 = 0.0605 Rali1/ 3
. (2-13)
N u 2 = [l+{0.104RaL 0 *293/(l+(6310/Ra))1 *3 6 }3 ]1/ 3 (2-14)
Nu3 = 0.242(RaL/ A ) ° “272
(2-15)
and the m a x i m u m of Nu^, N u 2 r and Nug w a s r e c o m m e n d e d for 5 <
A < HO
and I O 2 < R a L < 2 x l 0 7 .
More
recently
the
works
of B a u m a n
et al.
[19]
and
Nansteel et al. [20] w e r e m o t i v a t e d by studies of natural
c o nvection heat transfer w i t h i n buildings.
B a u m a n et al.
[19] c onducted an e x p e r i m e n t a l and n umerical study of the
natural convection heat transfer
in a rectangular enclosure
w h i c h s i m u l a t e d a full scale room with an aspect ratio of
0.5.
The
enclosure
consisted
of
two vertical
copper
endwalls at different temperatures and adiabatic plexiglas
horizontal
surfaces.
The Rayleigh numbers achieved,
on the enclosure length,
working fluid.
were about l O ^
based
using water as the
The experimental heat transfer result in the
enclosure appeared somewhat higher than the approximation of
Raithby
et al.
[15].
This ma y have occured,
vertical boundary conditions were
because the
not perfectly
isothermal
and the horizontal surfaces were not well insulated.
With a
similar g e o m e t r y and the same w o r k i n g fluid, Nansteel and
Greif
[20]
focused
on an
experimental
study
of natural
conve c t i o n in undivided and p artially divided rectangular
15
enclosures.
in
the
Rayleigh numbers based on the enclosure length
range
seemed
of 2.3x10^0
to l.lxlQ-^
were
obtained.
It
that no fully developed turbulent flow was observed
w i t h i n the enclosure,
even for R a L as high as IO^1 .
The
recommended heat transfer correlations were
N u l = 0.748 Ap 0 -256 RaL ° *226
.
(2-16)
for a conducting partition and
N u l = 0.726 Ap 0 *473 RaL 0 -226
for an adiabatic partition,
central
opening
to
the
(2-17)
w h e r e A p is the ratio of the
height
of
the
enclosure,
and
is
called the aperture ratio.
From this review of the existing work on heat transfer
by natural convection w i t h i n a rectangular enclosure, the
important dimensionless parameters a r e .
Nu g = hs/k
(Nusselt Number),
Grs = (g3 p2s3 AT)/^2
Pr - (UCp)/k
and
A = H/L
(Grashof Number),
(Prandtl Number),
(Aspect Ratio) ,
where s is some c h a r acteristic length.
grouping
The dimens i o n l e s s
(Grs Pr) is wid e l y used as the Rayleigh number,
which is
Ras = Gr s Pr =' (Cpg 3p2 s3AT)/yk .
In general, the heat transfer by natural convection can be
16
determined in functional form as
Nu s = f (A, Ras , Pr)
„
Since 1967, whe n the prototype T r o m b e wall house was
built and monitored in Odeillo,
and
experimental
France,
investigations
several
have
been
analytical
presented
pertinent to this geometry in passive solar heating,
B a l c o m b et al,
[27-29]
condu c t e d several full scale
e x p e r i m e n t a l studies in passive solar heating with either
unvented
or
vented
T r o m b e - type
thermal
storage
walls.
Temperature variations of the building walls and the ambient
room temperature on a daily or monthly basis were obtained.
The effect of storage capacity on annual energy delivery for
a Trombe-type passive system was also examined.
However, no
convective heat transfer data were obtained, nor could it be
calcu l a t e d from their e x perimental data.
It is probably
difficult to reach the steady state condition for the full
scale building, because the solar irradiation changes too
rapidly and the Trombe wall is rather thick providing a long
c onductive time constant (10 to 15 hours) to transfer heat
from one side to another.
Free convective laminar flow within the Tro m b e wall
space was assumed similar to the flow between two parallel,
infinitely wi d e vertical plates with an adiabatic bottom
17
surface by Akbari
and Borgers
[30]..
By this assumption,
they solved nondimensional boundary layer equations by using
a forward-marching,
technique.
line by line implicit finite difference
Several velocity and temperature
profiles were
obtained for different wall temperatures.
An
experimental
i n v estigation
of
the
Trombe
wal l
passive solar energy system was carried out by Casperson and
Hocevar [31].
The wood-frame test room was constructed with
outside d i m e n s i o n s of 3.66 m x 4.27 m x 3.05 m (12 ft x 14
ft x 10 ft).
The T r o m b e wall consisted of a 30.48 cm (12
in.) thick solid concrete block wall with a movable doubleglazed
Kalwall
cover
unit,
which
allowed
the
wall
gap
(Trombe wall space) to be varied from approximately 2.54 cm
(I
in.)
profiles
to
25.4
cm
obtained
(10
in
in.).
the
Velocity
Trombe
wall
and
temperature
space
with
room
t e m p e r a t u r e at 1 4 . 8 0C (55 °F) and 15.6°C
(60 0F)
indicated
that the flow is likely to be turbulent.
No heat transfer
data were obtained in this study.
Stotts,
Warrington,
to
and
simu l a t e
Mussulman
mod e l
isolated
gain,
systems.
The T r o m b e wall model p e r f o r m a n c e was verified
Trombe
wall
gain,
developed a
computer
and
direct
[25]
passive
indirect
solar
gain,
heating
using data from passive test cells at NCAT (National Center
18
for Appropriate Technology)
[32]?. located in Butte?
Montana.
Radiative interactions between room's interior surfaces were
considered.
C onvection from the wal l s to the air was also
considered.
The s i m u l a t e d globe t e mperature showed good
agreement with the measured values.
The study of natural
convection within
enclosures has
increased rapidly in the last decade.
Results describing
the heat
enclosures
transfer w i t h i n
a pplicable
for
the
rectangular
full-scale
building
heating due to aspect ratio mismatches.
studies
have
been
ma d e
pertinent
to
heating with a Trombe wall geometry,
are not
in passive solar
Alth o u g h several
the
passive
solar
no natural convection
heat transfer data have been published to date.
There is. a
definite need to study
convection
the phenomena
of natural
in passive solar heating w i t h a Trombe wall g e o m e t r y with
either
scaled
mod e l s
or
full-scale
structures.
These
studies w ill help the d e v e l o p m e n t of computer simulations
and the design of Tro m b e wall systems. The intent of this
investigation was to develop a scale model for convective
heat transfer w h i c h s i m u l a t e d a full-scale structure wit h
Trombe wall geometries.
CHAPTER III
EXPERIMENTAL APPARATUS AND PROCEDURE
EXPERIMENTAL APPARATUS
A l/18th
scale p a r a m e t r i c
study of passive solar
heating with a Trombe wall geometry was carried out as part
of
the present
experimental
investigation.
e x p e r i m e n t wa s to simu l a t e a two
room wi t h
length,
The present
story high rectangular
inside dimen s i o n s of 5.5 m in height,
and
dimensionless
5.5
m
in
depth.
parameters
Typical
characterizing
8.2 m in
values
the
of
the
convection
process for this room, filled w i t h air at 21 °C, and with a
maximum of 9 0C temperature difference [26] between a Trombetype thermal storage wall and the inside building are:
A = H / L = 0.67,
Pr = 0.71,
and
R a H = G r 11 Pr < 1.4X101 1 .
The range of Ray, based on the height of room, suggests that
natural
convective
flow within this building can be either
laminar or turbulent,
governed by the specific temperature
distributions on the boundary surfaces and the configuration
of the enclosure.
c o nvection
In order to characterize buoyancy-driven
in this two story room,
the design of a heat
transfer apparatus has to meet the above requirements.
experiment covered the following range of parameters
!•
.
The
20
A=
0.67
124.7 < Pr < 276.9
6.2xl08 < RaH < I.BxlO10
using 20 cs fluid as a working fluid was appropriate because
it permitted typical Rag to be approached in a 1/18th scale
apparatus;.
natural
Raithby
convection
et
al.
processes
[15]
are
have
pointed
insensitive
out
that
to Pr
> 5»
Although the range of present Prandtl numbers is higher than
the Prandtl number of air, the
Nusselt numbers attained in
this experiment, the general flow pattern, and heat transfer
can be expected to be similar to the full-scale building.
A heat transfer apparatus was then designed to provide
the capability to study the heat transfer from an isothermal
heated T r o m b e wall
Moreover,
the
c onvective
flow
investigated.
geometry
temperature
around
this
to a rectangular
profiles
Trombe
an d
wall
the
the
heated
wall,
experimental
test
a
enclosure,
wat e r
system,
a water
cooling
A schematic
consisting
jacket,
system,
controlling and monitoring instruments,
3.2.
were
A photograph of the assembled apparatus and
entire
rectangular
natural
geometry
peripheral c o m p o n e n t s is sho w n in Figure 3.1.
of
enclosure.
and
of
a
a vertical
temperature
is shown in Figure
These will be described in detail below.
21
Figure 3.1
Heat Transfer Apparatus
22
-JI
Cooling
Water
Storage
Tank
Cooling
Water
In
Trcxnbe Wall
Thermocouple
Leads and
(^J
--- Mtph
,
3 LflJl
Figure 3.2
Filter
Chiller
w
Cooling
Water
Out
Pump
Schematic of the Heat Transfer Apparatus
<)
23
The
of
rectangular
30.48
cm
(12
test enclosure with
in.)
in
height,
45.52
inside dimensions
cm
(17.92
in.)
in
length, and 30.48 cm (12 in.) in depth, was fabrica ted from
1.27 cm
(0.50 in.) thick clear sheet plexiglas.
However,
to
provide a higher heat transfer between the Trombe wall space
and
the
outside
constructed
aluminum.
from
of
the
1.27
enclosure,
cm
(0.50
one
endwall
in.) , ty p e
6066
was
sheet
This wa s done to s i mul at e a double glazing on
south facing wall.
The water jacket enclosure,
with inside
d i m e n s i o n s 41.40 cm x 41.91 cm x 57.15 cm, sur rounding the
rectangular test enclosure was also made of plexiglas.
This
• !
water
jacket
consisted
of
a
separate
3.81
cm
wide
rectangular channel for each face of the rectangular test
‘
'
enclosure except the bottom face which was insulated by R-Il
■
•
fiberglas.
i
j
.
I
The two enclosures were fastened together with
machine screws and sealed with silicone sealant.
' 1
j
i
Access to
•
the test chamber was gained by removing a rectangular lid on
I
the
I
top wat er
jacket and an
inner
lid w h i c h
form ed
the
ceiling of the test space. A groove, 1.27 cm wi d e and 0.64
. '
cm deep on the inside surface of each lid, wa s m i l l e d to fit
a 0.24 cm (0.094 in.) thick rubber gasket.
i
' •
'
The top was then
'
secured in place wi t h wing nuts and mac hi ne screws.
'
j
.i
|
arrangement facilitated access to the test space.
■
This
'
j
j
'II
I
24
A vertical electrically heated wall with the same duct
gap size at the top as the bottom, wa s emp l o y e d inside the
rectangular test enclosure as an inner test body to simulate
the Trom be wall.
Three different duct gap sizes, 2.54 cm ,
0.635 c m , and zero cm (no duct, gap) , were studied in this
inves tigat ion
by
stopping
down
the
2.54
cm
gap
to
the
desired size w i t h a l u m i n u m blocks. A front v i e w and an end
view of the inner body are pre sen ted in Figure 3.2.
The
inner body was fabric ate d from two 0.47 cm thick and 30.23
cm square type 6061 aluminum sheets.
Aluminum was selected
for the inner body to minimize any temperature gradients on
the inner body surface.
and bott om
the top
edges of each a l u m i n u m sheet we r e mill ed out
pro vid in g two notches,
length.
To construct the duct gaps,
A 0.7 9 cm
2.54 cm in height and 25.4 cm.
(0.31
in.)
thick
aluminum
spacer
in
was
tightly adhered around the m a r g i n s of the inner surface of
each aluminum sheet using a high temperature silicone rubber
sealant.
This a l u m i n u m spacer wa s 2.54 cm wi d e along the
sides and 1.27 cm wide along the top and bottom of the wall
margin.
together
The
spacers
with
and a l u m i n u m
mac hi ne
scre ws
sealant to form the inner body.
wall,
B:
and
sheets
the
were
high
a s sem bl ed
temperature
The, edges of the assembled
which contacted the water jacket, were covered with a
25
0.038 cm
(0.015 in.) thick rubber gasket to m i n i m i z e heat
losses and to eliminate fluid flow.
Electrical resistance heat tape supplied the heating
for the inner body.
The heat tape, 0.064 cm thick and 0.32
'
cm
wide,
consisted
of
an
electrical
resistance
wire
sandwiched between an adhesive backed fabric and a metallic
ihsulative foil.
The electrical
resistance wire was rated
at 28.87 o h m s / m (8.8 ohms/ft) and a m a x i m u m power of 75.46
Watts/m
(78.53 Btu/hr/ft).
Five sets of these heater tapes
were attached to each inner surface of test body.
Each set
cons is te d of 10 rows of heater tapes, conn ected in series.
High temperature silicone cement was spread over
after
insta llation
to
insure
adhesion
to
the tapes
the
surface.
Figure 3.3 shows the heater tape arrangement, thermocouple
*
"
location, and the interior of the test body. The inside of
the
test
body
was
filled
with
a styrofoam
s h eet
an d
fiberglas insulation to m i n i m i z e internal heat transfer.
Input power
to the tapes wa s
ind ividually controlled by
means of Ohmite Variable Power resistors (0-35 ohms or 0-50
ohms,
150 watts,
2.07
ampere s
or
1.68
amperes
maximum)
connected in series with each set of the heater, tapes.
This
c o m b i n a t i o n a l l o w e d the two sides of the test body to be
maintained
at
essentially
equal
a nd
constant
26
Figure 3.3
Interior of the Inner Body
27
temperatures.
The
■
cooling
system
con sisti ng
of
.
_
a c h i l l e r , pump,
filter, and i nsu lat ed storage tank pro vid ed wat er to cool
the test enclosure.
The cooling water was collected from
the water jacket through a drain manifold system and pumped
through
the chiller,
into the
storage tank,
through
the
filter and a supply manifold system^ and back into the water
jacket.
A uni fo rm flow rate through each channel of the
w a te r jacket w a s a c c o m p l i s h e d by valves w h i c h fed several
inlet ports for each channel.
This arrangement allowed the
te mp er atu re of the b o tt om face of the test c h amb er to vary
while providing a uniform temperature on the remaining faces
of the test space.
cooling wate r
Since the apparatus cannot w i t h s t a n d
pre ssure above 41 KPa
(6 psi),
a regulator
valve and a 122 cm tall stand pipe we re plac ed in the mai n
outlet
from
the
around
13.8
KPa
emergency
water
jackets
(2.0 psi)
to mai nt ai n
under
shutdown conditions.
normal
the pressure
operation
and
Three aircocks were placed
on the wat er jacket lid to release air bubbles
from
the
jackets.
The inside surface temperature of the rectangular test,
space enclosure was monitored by using 27 copper-constantan
thermocouples epoxied 0.08 cm from the inner surface.
There
28
we re
five
t h e r mo cou pl es
per
face
of
the
except the bottom face which had only two.
leads
exited through holes
jackets.
test
enclosure
All thermocouple
drilled in the cooling water
The holes were sealed tightly with epoxy.
The t e m p e ra tur e of the heated vertical test wall was
monitored using 10 copper-constantan thermocouples placed in
each face of the heated wall.
The the rm oco up le leads and
heat tape leads pass ed through holes in the sides of the
spacers,
one lead per hole, and were brought downward inside
slots in the sides of the vertical heated wall.
The holes
and the slots were sealed with silicone rubber caulk.
The
leads were directed toward the plexiglas endwall where they
exited
through
t w o 1.9 cm
inside diameter,
10.2 cm long
stainless steel tubes, whi ch passed through a hole in the
bottom
of the apparatus
and then threaded
into the test
cha mbe r at the corners of ple xi glas endwall.
of
the
tubes,
the
leads
were
sealed
off
At both ends
from
the
test
enclosure with silicone rubber sealant.
Local
fluid
temperatures
in
the
tes t
s p ace
were
measured using seven thermocouple probes to traverse a plane
in the
mid dl e
inner body.
of the enclosure and per pen dicular
to the
These probes consisted of a copper-constantan
t h e r m o c o u p l e coated with m a g n e s i u m oxide insulation and
29
sheathed wi th a 0.32,cm. diameter, type 304 stainless steel
tube.
The region between each thermocouple bead and tubing
end wa s s t r e a m l i n e d wit h an epoxy cone.
This arr angem ent
was intended to limit heat conduction along the probe shaft
from the measuring point.
Three 0.38 cm diameter holes were
drilled in the plexiglas endwall and four holes were drilled
in the a l u m i n u m endwall along the vertical center.
These
holes w e r e co un te r-b or ed to a dia me te r of 0.51 cm to a depth
of one half the wall thickness to allow placement of O-rings
and a l u m i n u m or plexigl as tubes in the holes.
a l l o w e d the probes
the
test
aluminum
be passed through the tubes and into
space w i t h o u t
cooling water.
This design
seepage
of
the w o r k i n g
fluid
or
Four of the probes were posi tione d in=the
endwall
of the enclosure
20.32 cm and 30.30 cm from
at 0.18 cm,
the enclosure top.
10.16 cm,
The other
probes were positioned in the plexiglas endwall at 2.54 cm,
10.16 cm,
and 22.86 cm from the enclosure top.
The rectangular test space enclosure and the inner body
we re painted flat black except for a light source slit the
length of the ch a m b e r located in the center of the ceiling
and
one
clear
side
face
pho togr ap hs to be taken.
tw o
650
Watt,
high
allowing
flow
observations
and
A lighting system equipped w it h
intensity,
tungsten
halogen
lamps
30
provided a thin collimated beam of light necessary for flow
visualization.
EXPERIMENTAL PROCEDURE
The inner body was placed perpendicular to the floor at
2.54
cm
(1.0
in.)
from
rectangular test space.
the
aluminum
endwall
in
the
Hence, the rectangular test space
w a s separat ed into two parts by the inner body as shown in
Figure
3.2.
One
was
the
left
hand
side
Trombe wall space) with aspect ratio 12.
right hand side test
ratio 0.75.
floor wer e
space
test
space
(or
The other was the
(or living space)
wi th aspect
The lead w ir e s from the inner body along the
sealed off from
the test space w i t h
silicone
rubber caulk and.connected as sho wn in Figure 3.2.
Final
assembly involved securing the inner lid and the rectangular
lid on the top wat er
jacket and connecting
the p l umb in g
associated with the chilling system.
Di sti lle d wa t e r wa s added to the system through the
stand pipe until all of the air was forced out of the water
jacket.
The cooling system was activated.
The test fluid,
20 cs silicone oil, was then introduced into the rectangular
test space by grav ity feed.
The gravity fed fluid flow ed
through a filler stem located on the bottom of the apparatus
and
exited
out
of
a
0.79
cm
i.d.
stainless
tube
whi c h
31
threaded into the.ceiling of the test chamber.
The tube was
flush with the insi de surface of the ceiling.
Power wa s
app lied to the heater tapes and adjuste d until the desired
isothermal temperature condition was achieved on the surface
of the inner body.
At this condition, no conduction could
occur b e t w e e n these faces.
flow
Sim ultaneously, cooling water
rates were adjusted to attain an isothermal condition
on all faces but the bottom face of the test enclosure.
establishment
equilibrium
of
the
within
importance.
Once
isother mal
the
tes t
thermal
(approximately six hours)
for
each
run:
(I)
conditions
chamber
and
thermal
of
primary
was
reached
were
equilibrium
the following data were
The
inn e r
thermocouples millivolt readings,
body
The
an d
recorded
out e r
bo d y
(2) The input amperage to
each of the inner body heater tapes,
(3) The input voltage
to each of the inner body heater tapes.
The program used to
reduce the data is presented in Appendix I.
A minimum of 12 heat transfer data points were obtained
wit h each duct gap arrangement.
reduced
data
is pro vid ed
in App endix
.
twelve
temperature
conjunction
with
II.
.
selected
In
addition,
..
profile
thermocouple probes was
A list of the partially
sets
data
were
runs.
<
collected
Each
of
in
seven
inserted into the test space until
-(
32
it contacted the inner body.
It was then withdrawn in small
increments until the probe tip was flush with the endwall of
the
test
along
space,.
the
withdrawal.
.Th$ th e r m o c o u p l e
traverse
were
recorded
readings and position
at e a c h
step
of
the
Positions were measured within 0.05 cm.
*
In
order
to
m i n i m i z e e r r o r s in the data,
all
/
instruments were calibrated before the test.
Comparison of
some of the th erm o c o u p l e s w i t h an O M E G A type 2809 Digital
T h e r m o c o u p l e meter revealed that the standard calibration
w a s accurate, wit h a m a x i m u m error of about plus 1.6°C (2.8
0F).
M e a s u r e m e n t s of power input at the inner body were
accurate to within plus or minus one percent.
All the data
were collected at the condition of room temperature in the
range of 21.1 0C (70 °F) to 26.7 0C (80°F) and pr essure around
one atmosphere.
The percent temperature variation for either the inner
body or the outer body was defined as
Temperature Variation % =
^P^alfmax----local,min x ^QQ
?h -
Tc
(3-1)
where
Tlocalfinax
(minimum)
inner
(Tlocalfinin) r e p r e s e n t s
or outer body temperature*
the
maximum
The average
te mp er a t u r e va ri ati on for the left face of the inner, body
33
was 9.6%,
face
of
with a maximum variation of 15.9%.
the
inner
body,
18.5%, respectively,
and 24.37%,
The
these quan titie s
For the right
were
11.4% and
For the outer body, they w ere 13;65%
respectively.
heat
transferred
by
natural
convection
was
calculated by
^conv = Qtot “ Qcond “ Qrad
(3-2)
Since the 20 cs fluid was opaque to radiation only the heat
loss due to conduction had to be determined.
To det e r m i n e the co nduct ion heat loss, the inner body
was m o v e d to the central pos ition of the test space.
The
test space and the two end wat er jackets we r e then filled
with
st yro foa m
permitted neither
sheets
and
fib e r glas.
convection nor
This
arr an geme nt
radiation to occur within
the test space and l i m i t e d condu ction in the space.
the co m p l e t e
syst em a s s e m b l e d as described,
With
the cooling
flo ws to the end water jackets wer e cut off and a m i n i m u m of
six heat transfer data points w e r e taken over the range of
temperature difference for the zero cm gap (no duct gap) and
2.54 cm
duct gap arrangements.
The temperature difference
was ca lcula ted as the average tem pe ratu re on each face of
the inner body minus the average temperature of the top and
sides of the outer body.
The data are shown in Figure 3.4.
n-
2.54 cm (I in.) gap
O' ZeroSaP
Solid symbols - left face
Open symbols - right face
Figure 3.4
Heat Loss by Edge
Conduction
35
These data we re used.in a least squares curve fit to develop
an equation for each face of the zero gap and 2.54 cm gap.
For the inner body, with zero gap, the resulting equation for
the left face was (QCond in watts and AT in degrees Kelvin)
Qcond = .2574 (AT)1 *0495
with
an
average
percent
(3-3)
dev iat ion
of
5.39
and
maximum
percent deviation of 13.42, and for the right face
Qcond = -2483 (AT)1 -0115
(3-4)
with a 5.82% average deviation and 15.17% maximum deviation.
For
the
in n e r
body
with
a 2.54
cm
gap,
the
resulting
equation for the left face was
0Cond = -188 (AT)1 "0709
(3-5)
with a 6.13% average deviation and 10.32% maximum deviation,
and for the.right face
°cond = -1825 (AT)I"0304
(3-6)
with a 7.1% average deviation and 14.43% maximum deviation.
No heat conduction heat loss data were taken for the inner,
body w i t h 0.635 cm gap,
the heat conduction losses we r e
ca lc u l a t e d from the resulting equations of the case of the
2.54 cm (I in.) gap because the same edge area contacted the
test enclosure.
These equations were used to correct the
heat transfer data in subsequent measurement.
p e r c e n t a g e .of
heat
lo ss
by
conduction
The average
to
the
heat
36
tr ans ferre d by con ve ct io n was 7.99 for the no gap, 6.1 for
the 0.635 cm gap, and 5.8 for the 2.54 cm gap.
Once
the heat
tra nsferred by natural convection was
known, the average heat transfer coefficient wa s obtained
from
h = Oconv / IA i (Th - T c )]
wh e r e Ajy
(3-7)
9 2 9 c m 2 (1.0 ft 2 ), is the surface heat
area of the inner body with zero gap.
transfer
This area is the same
as the surface area on the south facing aluminum wall.
the ease
of
the eng ineering calculations,
For
this area was
chosen for all the inner body gap arrangements.
The fluid
properties were evaluated by ah existing function subroutine
[23] based on the fluid film temperature which is defined by
Tf = (Th + Tc ) / 2.
(3-8)
To observe the flow p h e n o m e n a w i thi n the test space,
tracer
particles
fluorescent
paint
were
made
particles
by
into
spraying
the
20
cs
an d
test
mixing
fluid.
These paint particles were visible when the mid pla ne was
i l l u m in ated by high intensity lights.
condition,
At the steady state
the flow pattern was photographed with a Calumet
4x5 profess ion al c a me ra using Polaroid,
type 52 positive
film.
i
CHAPTER IV
DISCUSSION OF RESULTS
The
heat
transfer
data
dimensionless
parameters
nondimensional
form
momentum,
and
are
which
energy.
The
the ge om e t r i c aspect ratio.
the
are
in
derived
terms
from
of
the
of the governi ng equations of m a s s ,
independent
parameters are the Rayleigh number,
parameter,
correla ted
Nuss elt
d i m e n sio nl ess
the Prandtl number, and
The dependent dim en s i o n l e s s
number,
was
calc ulate d
in
the
following functional forms to fit the heat transfer data by
using a least squares method of curve fitting:
Nu g = Ci RasC2
(4.1)
Nus = C1 RasC2 Prc3
(4.2)
N u s = C1 RasC2 AC3
(4.3)
Nus = C1 Ras^2 Prc3 Ac4
(4.4)
or L 2 )an^ A is
where s is some cha rac ter is ti c length (H,
an
aspect
ratio
(H/L).
For
all
the
data
combined,
the
aspect ratio A wa s replaced by the height ratio Hj/H whi ch
is defined as the ratio of height of the inner body to the
outer body.
All the notations of the inner and outer bod ies .
are given in to Figure 1.1.
Three cha ra cte risti c lengths,
the height H, Trombe wall space length Lyt and living space
length
Lg
of
the
rectangular
enclosure
we re
adopted
to
calculate the Rayleigh and Nusselt numbers. The range of the
38
correlating parameters is provided in Table 4.1.
The
ge o m e t r y
following
on heat
visualizations,
sections
will
transfer.
discuss
Te m p e r a t u r e
the
effect
profiles,
of
flow
and co m p a r i s o n s will be included in each
section as they are applicable.
GEOMETRIC EFFECTS
W h e n the gap of the inner body was dec reased from the
2.54 cm
(I in.) gap to the 0.635 cm
(0.25 in.) gap or the
zero cm gap, there wa s a decrease in the heat transferred.
This decrease was exhibited for both the Trombe wall space
and the living space.
This decrease is most pronounced in
the low Ray lei gh num be r range and bec ome s less in the high
Rayleigh
number
range
sho wn
0.635 cm
4.1.
an
the
Nussel t
numb er
4il4% less than the 2.54 cm gap's average
Nusselt
num be r
and
the zero
gap has
For
heat
3.88% less
the
in Figure
overall
number
transfer,
as
gap has an average
average
Nusselt
than the 2.54 cm gap's average Nusselt
number.
Since each face of the inner body was separately heated
to m a i n t a i n iso th er ma l conditions, a similar analysis was
possibl e for the left hand test space (Trombe wall space)
and right hand test space (living space) as shown in Figure
4.2 and
Figure
4.3.
These
two
separate
test
spaces are
39
TABLE 4.1
RANGE OF DIMENSIONLESS PARAMETERS
DIMENSIONLESS
PARAMETER
nuH
m L2
n u L1
GrH
Gr T
.
MINIMUM
MAXIMUM
55
109
. 73
130
5.0
9.3
2.ZxlO6
I.ZxlO8
5.SxlO6
2.SxlO8
1.3xl03
6.SxlO4
6.ZxlO8
I.4x1O10
. I.SxlO9
3.4xl010
2
Gr.
1
r3H
teL2
Ra7
L1
Pr
3.7x1O5
125
8.ZxlO6
200.0
OVERALL HEAT TRANSFER
□ - 2.54 cm (I in.) gap
A - 0.655 cm (0.25 in.) gap
O ~ Zero gap
nuH
100.0
0.1823
Illll
%
Figure 4.1
Geometric Effect for All Gaps and A Heat Transfer Correlation
with All Data Combined
HEAT TRANSFER TO LEFT HAND SIDE SPACE
200.0
D
A
O
nuH
I
- 2.54 cm (I in.) gap
- 0.635 cm (0.25 in.) gap
- Zero gap
Li
I
ltoH
Figure 4.2
Geometric Effect for All Gaps and A Heat Transfer Correlation
with All Data Combined
I
200.0
HEAT TRANSFER TO RIGHT HAND SIDE SPACE
□ - 2.54 cm (I in.) gap
A - 0.635 cm (0.25 in.) gap
O - Zero gap
%
100.0
Nu^ = 1.5734 R bh
0.1778
50.0
Illll
I I I
10
9
10
10
%
Figure 4.3
Geometric Effect for All Gaps and A Heat Transfer Correlation
with All Data Ccsnbined
43
denoted
LHS
an d
R H S , respectively,
in
the
following
I
discussions.
For the heat transfer in LHS,
and
gap have
the zero
the 0.635 cm gap
average Nuss elt n u mbe rs
3.84% and
4.26% less than the 2.54 cm gap's average Nus se lt number.
For the heat transfer in RHS,
For
the
2.54
cm
gap
the values are 5.57% and 4.3%.
geometry
the
warmer
fluid
rose from the left face of the inner body and pass ed through
the upper gap into the RHS test space causing an exchange of
mass between these two spaces.
Flow visualization data also
support this fact as seen in the flow patterns of Figure 4.4
an d
Figure
respectively.
4.5
for
the
ze r o
ga p
an d
the
2.54
cm
gap,
Photographs of the flow patterns at the upper
portion of the living space are shown in Figures 4.6 and
4.7.
This
exchange
of
mass
may
have
resulted
in
the
higher heat transfer for lower Rayleigh numbers for the 2.54
cm vented inner body than for the unvented inner body within
the test space since no exchange of mass could happen for
the unvented inner body.
small,
Since the 0.635 cm gap is rather
it only shows a slight difference (below 1%) in heat
transfer coefficients compared to the zero gap.
Figure 4.4, which shows the flow pattern in the living
space for the inner body wi th no gap, is des cri bed by the
following;
Figure 4.4
A Sketch of Flow Pattern for the Inner Body with Zero Gap
AT = SO6SO0K (55.44°R), Rafi = 4.79xl09
Weak Motion
Figure 4.5
A Sketch of Flow Pattern for the Inner Body with 2.54 an Gap
AT = 31e33°K (56.39°R), Ra^ = 4.89xl09
46
Figure 4.6 (a)
A Photograph of Flow Pattern in
the Living Space for Zero Gap
(Upper Left)
47
Figure 4.6 (b)
A Photograph of Flow Pattern in
the Living Space for Zero Gap
(Upper Right)
48
Figure 4.7 (a)
A Photograph of Flow Pattern in
the Living Space for 2.54 cm Gap
(Upper Left)
49
Figure 4.7 (b)
A Photograph of Flow Pattern in
the Living Space for 2.54 cm Gap
(Upper Right)
50
1)
The
boundary
vel ocity
regions
cl oc k w i s e
comprise
circu latio n
a relatively
high
ceil A in which
the
st re aml in es closely f oll ow ed the shape of the living
space.
2)
A
clockwise circulation cell B at the upper portion of
the living space resulted from the lower density fluid
rising
from
the
hot
wall
and
interacting
with
the
conducting cold ceiling.
3)
A. st rea mli ne
C was
observed below
s tr ea mli ne C started near
the cell B.
The
the cold vertical boundary
and ended near the hot wall.
4)
W e a k m o t i o n w a s not det ected in the core region until
the
shutter
speed
of
the
cam era
was
reduced
t o . 24
seconds, and outside this region near the wall s again
is a horizontal entrainment flow.
5)
A
c l o c k w i s e eddy D wa s observed in the upper corner
adjacent to the heated wall.
The flow pattern in the living space for the inner body
with a 2.54 cm gap is different from the inner body with no
gap as show n in Figures 4.4 and 4.5.
difference
was
observed
inside
the
The dis tinguishing
gap
region.
The
clockwise circulation of cell A was deformed and interacted
with the counter-clockwise circulation of main cell in the
51
Trombe
wall
space
streamline
into
the
due
which
living
to
the
passed
space
was
existence
through
separated
cl oc k w i s e cir cu la ti on of the m a i n cell.
(perhaps
mo re
than
one)
was
observed
region.
For the flow pattern wit h
of
the
from
the
gap.
upper
the
A
gap
cou nt er ­
A c loc kw ise eddy
at
the
upper
gap
the 0.635 cm gap,
no
distinguishing difference was observed in comparison to the
i
no gap except in the gap regions whe r e there wa s an exchange
of mass.
A c o m p a r i s o n of tem perat ur e profile data with all of
the inner body gaps at a p p r o x i m a t e l y the same tem perat ure
difference
is
shown
in
Figure
4.8.
The
plot
uses
a
dimensionless temperature and dimensionless length which are
defined as
whe re T is the local temperature, mea s u r e d at any vertical
position, Y* = y/H, along midplane within the test space and
X* = x / Lj, whe re x is the local distance from the a l u m i n u m
endwall to the test point in the test space and
is. the
distance b e t w e e n the inner body and the a l u m i n u m endwall.
The temperature profile was plotted in two different scales.
For X* from 2.5 to 17 the smaller scale plot is inset in the
Open Symbols - AT - 30.9eK (SS-SeR), Ra^ - 4.4xl09, 2.54 cm gap
Solid Lines - AT - 30.S8K (55.411), Ra^ - 4.SxlO9, 0.635 cm gap
Solid Symbols - AT - 30.3°K (54.6°R), Ra^ - 4.4xl09, no gap
A-Y
■ 0.01
O - Y* - 0.33
■0
■O
*
V - Y - 0.67
*
D - Y v
O - Y -
0.08
O ’ Y* - 0.33
O - Y* - 0.75
0.99
-Q r
i
I
1.0
r
0.8
Q
A < ______
0.6
A
on
to
fry I
jjz r Ir
Al
0.2
----- BO- QEftin a—
0
0.2
0.4
Figure 4.8
0.6 0.8
1.0
2.5 17
Comparison of Temperature Profiles for All Inner Body Gaps
17.6
17.9
53
right upper portion of the figure.
In order to display the
t e m p e ra tu re di str ibu tio n adj acent to the walls,
a larger
scale plot is p res en te d in the mai n portion of the figure.
When
the gap size
thermal
of
the inner
stratification
body
within
inc reased as shown in Figure 4.8.
vented
inn er
body
t e nd s
to
the
was
increased,
living
space
the
als o
This suggests that the
provide
a larger
thermal
str atif ic at io n than the unvented inner body in the living
space due to the exchanges of mass,
mom en tum,
and energy
between the Trombe wall space and the living space.
The tem pe ra tu re profile along Y* = 0.08 and Y* = 0.33
w i t h i n the living space is increas ed wit h increasing gap
sizes.
This
implies
that
the
larger
gap
size
tends
to
increase convect ive heat transfer to the upper portion of
the living space.
Along Y* = 0.75, the 0.635 c m gap has a
higher temperature profile in the RHS than the 2.54 cm gap.
Throughout the LHS space,
the no gap inner body gives the
highest temperature profiles.
no lower
This is not surprising, since
temperature mass could pass through the lower gap
from the RHS space.
Figures
4.9
through
4.11
present
the
dim ens io nl ess
temperature profile plots at different AT for all inner body
gaps.
The magnitude of mid-range dimensionless temperature
Open Symbols - AT - 57.I K (102.8 R), Pr - 136.2
Solid Symbols - AT - 34.7 K (62.4 R), Pr - 179.1
O1.0
A-Y
0
O
? 0-01
O - Y * - 0.33
V - Y* - 0.67
V-
r
I
O-
D - Y* * 0.99
i
i
r
0 -Y
0.08
2:=:
0.33
0.6
,.,jw&&
0.75
OO Q Q Q O OQ^
r
I
I
I
r
I
I
0.2
in
i 4
0.6
Al
0 00
0.2
vvV
$ s m
O O 0
Trombe Wall
pT
I
0o
n rfll
0.2
0.4
Figure 4.9
0.6
0.8
1.0
&
2.0
4
2.5 17.0
0
O
O
oO a J
0.4
-P*
2.5
17.6
Variation of Temperature Profile with AT, Inner Body with Zero Gap
17.9
Open Symbols - AT = 59.0°K (106.I0R), Pr - 133.8
Solid Symbols - AT - 33.8°K (60.8°R), Pr ■= 181.9
A-Y
•0
1.0
V-
r
I
O
0.6
O - Y* = 0.33
-O
O-
» 0.01
# 66#
iuofiaBfiaae^
'►?»»»
V - Y* - 0.67
D - Y* = 0.99
0.4
O-2CgjSHfflB
0.8
& 866
0.6
A
A
^
6
A
▲
M
$22
A
0
0.2
0.4
Figure 4.10
0.6
0.8
1.0
6
a
°0 a
PfflfiQ B C g
& .n::, » •□ □<m n -Q . ..non,
Ln
A «
Trombe Wall
0.2
Ln
2.5
n n Y
2.0
2.5 I
17
17.6
Variation of Temperature Profile with AT, Inner Body with 0.635 on
17.9
Gap
Figure 4.11
Variation of Temperature Profile with AT, Inner Body with 2.54 cm Gap
57
profiles general ly increased with increasing AT.
Several
features c o m m o n to each of the tem pe ratu re pro files we r e
observed.
These are (I) the temperature profiles are almost
a n t i s y m m e t r i c about the center line either of the Trom be
wall space or of the living space.
The majo r deviations
from a n t i s y m m e t r y are near the hot and cold w a lls where
fluid
viscosities
temperature.
can
significantly
(2) The appearance of
observed
(3)
parts.
con si de re d
relative
throughout
as
one
In
the
a
wall
(4) Near Y * .=
o
and
Tro mbe
sense
in w h ich
to the T r om be
interact.
surfaces
the
the
layers
horizontal in the central region in the living space.
horizontal
the
boundary
by
are
profiles
near
thermal
affected
profiles
The
be
were
wall
thi s
situation
boundaries
space
space
and,
are
have
no
can
be
not
therefore,
thin
they
the tem perat ure distribution is
d o m i n a t e d by the heat lost through the cold ceiling,
nearly linear profiles were
space.
and
observed in the Tro mb e wall
Near Y* = I bel ow the b o tto m vent in and the botto m
por tion of the living space, the tem perat ure distribution
was
nearly
insulated.
Warrington's
inner
constant
because
Si mi l a r i t i e s
of
the
bottom
surface
the tem perat ure profiles
was
to
[23] we r e a steep drop at the inner body, an
transition
region,
a
level
zone,
an
outer
58
transition
endwall.
zone,
and
a fairly
steep
dr o p
at
the c o l d
The five zones are convi ncing ly obs erved only in
the RHS test space.
In the profiles for the LHS test space,
no level zone wa s obs erv ed due to the narrow space b e twe en
the hot and cold w a l l s (i.e., high aspect ratio 12).
HEAT TRANSFER RESULTS FOR EACH INNER BODY GAP
The data obt ained from the inner body with each gap
size we r e co rre lat ed separately for both the left face and
right face of the inner body. This is also referred to the
Trombe
wall
space
and
the
living
space.
Moreover,
a
separate correlation for the overall heat transfer
(the sum
of
data
heat
transfer
presented.
for
the
left
and
right
faces)
is
The notation throughout the remainder of this
discussion will consist of presenting a two-part designation
in w h i c h the gap size will be given first, f o llo we d by the
inner
body's
face
(i.e.,
zero-left,
zero-right).
When
combined data are presented the last term will be all.
The present
experiments
respectively.
data from
are
plotted
the zero-left
in
Figures
and zero-right
4.2
and
4.3,
The recommended equation which correlates the
heat transfer with a single parameter for zero-left data was
N u h = 0.904 Ran0,208
with
an
average
percent
deviation
(4.5)
of
1.09
and
maximum
I
59
percent deviation of 3.28.
The percent deviation at a point
is defined as the absolute difference between the data value
and
the
average
equation
percent
deviati ons
value
divi de d by
dev iation
divided
by
is
the
the
the
sum
number
of
data
of
value.
the
data
The
individual
points..
The
r e c o m m e n d e d best correl ati on for the zero-r igh t data with
one parameter was
N u h = 0.862 RaH0e205
(4.6)
‘ s
with 1.38% average deviation and 5.54% maximum deviation.
The best equation using a single correlation parameter
for all the zero gap data was
N uh =
0.887
Ralj0 '206
(4.7)
with 5.27% average deviation and 8.27% maximum deviation.
A
correl ati on including aspect ratio resulted in an average
deviation of 1.22% with a maximum deviation of 5.49%.
correlation was
Nu h =
This
.
0.847 Ra110l206A0 *038
(4.8)
The Prandtl number of the 20 cs fluid varied fr om 124.7 to
276.9 in this investigation.
The best correlation including
'
the Prandtl number effect was
■'
'
Nu h =
•
'
0.1184 RaH0 *252Pr0ll9:L
I
i
'
I
(4.9)
w i t h 5.25% average dev iat ion and 8.01% m a x i m u m deviation.
The effective heat transfer area was used to calculate
|
J
I
;
60
the heat transfer data for the zero gap only when mak ing a
*i .
c o m p a r i s o n w i t h R a i thby's predication.
The met ho d for
defining
this
Appe nd ix III.
effective
heat
transfer
is
described
in
C o m p a r i s o n to the results of. Raithby et al.
[15] was ma de with a plot of Nu versus Ra based.on enclosure
length, Ljf for the zero-le ft data as shown in Figure 4.12.
It
appears
Raithby's
especially
to give
higher
analytical
in the
average
approach
lower
Nusselt
in the
num be rs
than
laminar
regime,
range.
This is
Rayleig h number
because the heat transfer in the lower Rayleigh number range
was
significantly
conducting boundary.
the
affected
by
bound aries
in the higher
entrainment
Rayleigh number increased,
wall
to
cold
the
upp e r
horizontal
The flow in the Trombe wall space was
dominated
hot
by
wall
flow
Ray lei gh
between
number
range.
vertical
As
the
the direct flow from the vertical
pro bably
was
increased
thereby
lessening the role played by the upper horizontal surface in
the heat transfer.
Figure 4.13 shows that wh e n Rayleigh
number increases, the flow pattern is modified.
Probably
the direct flow from the hot wa ll to the cold wall yie lde d
higher velocities at high Rayleigh numbers than the flow at
lower Rayleigh numbers.
Hence, for the higher aspect ratios
and higher Rayleigh numbers, the heat transfer in the cavity
Zero-Left (A = 12)
O
Lines
- Present Data
- Raithby et al.
Turbulent Regime
Laminar Regime
Figure 4.12
Comparison of Present Data with Raithby et al.
62
C ^
"r-S
Z-^
(a)
Z
,Zn
/
(b)
■0
../ZL.
H _
3
I "
H
2
2
Figure 4„13
\
Za..
Flow Patterns for Zero-Left
(a) Rs h = 4.SxlO9
(b) Ra^ = 9.OxlO9
63
with
the conducting upper horizontal
surface and
insulated
lower horizontal surface tends to be similar to the case of
the adiabatic horizontal surfaces.
For the zero-right/ a comparison with Raithby’s general
a p p r o x i m a t i o n was also ma de as sho wn in Figure 4.14.
The
present heat transfer data give slightly lower results with
Raithby's analytical
prediction.
This is not surprising,
since Raithby's correlation is not claimed to be valid for A
< 5.
Figure 4.15 sh o w s
that the heat transfer
in the LHS
test space is higher than the heat transfer in the RHS test
space.
The average Nusselt numb er
of the LHS
9.84% greater than the average Nuss elt number
space.
space was
of the RHS
This is due to the different flow patterns in the
spaces as mentioned previously.
Table
constants,
4.2
an d
Table
4.3
present
the
calculated
average devition, and maximum deviation for the
co rr elati ons for each inner body gap size in the LHS test
space and RHS test space,
respectively.
The equation using
a single correlation parameter for all three inner body gap
sizes in the LHS test space is
N uh = 1.3657 Ra110 "1894
(4.10).
with 2.37% average deviation and a 8.26% .maximum deviation.
200.0
Zero-Right (A
O
Line
- Present Data
- Raithby et al.
100.0
Laminar Regime
Figure 4.14
Comparison of Present Data with Raithby et al.
200.0
Solid Symbols : LHS
Open Symbols : RHS
nuH
100.0 -
.8?
8"
°
8
60.0
I l l l l
I l l l l
r3H
Figure 4.15
Comparison of Heat Transfer Results between LHS and RHS for Zero Gap
TABLE 4.2
CORRELATIONS FOR LHS TEST SPACE
EMPIRICAL CONSTANTS
EQUATION
FORM
ci
C2
AVERAGE %
DEVIATION
MAXIMUM %
DEVIATION
1.09
.92
3.28
3.05
1.91
1.72
3.52
2.93
1.64
1.42
4.91
4.04
C4
=3
Zero an gap
4.1
4.2
.9043
.1518
.2075
' .2483
.
.1689
0.635 an (0.25 in.) gap
Ov
4.1
4.2
1.2538
.0528
.1928
.2686
.2848
ON
2.54 an (I in.) gap
4.1
4.2
2.4135
0.0330
.1647
.2354
.2542
I
•
All
4.1
. 4.2
4.3
4.4
1.3657
.0232
1.395
.0419 .
.1894 .
.2859
.1877 '
.2709
.3704
-.2078
.3183
. -,1720
2.37
1.89
2.11
• 1.77
8.26
6.29
6.18
4.82
■
TABLE 4.3
CORRELATIONS FOR RHS TEST SPACE
EMPIRICAL CONSTANTS
EQUATION
FORM
AVERAGE %
DEVIATION
MAXIMUM %
DEVIATION
Zero cm gap
.1088
.2522
1.37
0.635 cm (0.25 in.) gap
4.1 .
1.2560
.0265
.1871
1.61
2.54 cm (I in.) gap
3.6106
1.5734
.00793
1.6107
.01340
.2481
.3832
.1778
.3032
.1760
-.2159
6.67 .
-.1726
68
The correlation which provides the best fit of all combined
data is
N u h = 0.0419 RaH °-270 9(Pr)0 ^3183(HiZ H ) e172
with
an
average
percent
deviation
percent deviation of 4.82.
of
1.77
and
(4.11)
maximum
The equation with one parameter
using all of the data c o m b i n e d in. the RHS test space is
N u h = 1.5734 Ray0 -1778
(4.12)
with a 3.09% average deviation and 8.65% maximum deviation.
The best co rre lat ion for the RHS test space of all three
inner body gap sizes combined is
N u h = 0 . 0 1 3 4 RaH0 '2897(Pr)0 *4 3 5 (Hi/H)“0 *1726
(4.13)
with 2.37% average deviation and 6,45 % maximum deviation.
Co rr ela tio ns for all three inner body gaps only dealt
wi t h the overall heat transfer data,
4.1),
from
the
inner
body
presented in Table 4.4.
to the
(as shown in Figure
outer
body.
These
are
The correlation which provides the
bdst fit of all combined data with three parameters is
N u h = 0 . 0 3 2 9 RaH0 -2719(Pr)0 ^3496(HiZH)-0 -1543
(4.14)
with a 1.98% average deviation and 5.07% maximum deviation.
Table 4.4 also presents
the cal cu late d constants for the
data of all gap sizes using one parameter.
N u h = 1.5079 Rail0 *1823
The equation is
(4.15)
with 2.65% average deviation and 8.11% m a x i m u m deviation.
TABLE 4.4
OVERALL HEAT TRANSFER CORRELATIONS
EMPIRICAL CONSTANTS
EQUATION
FORM
4.1
4.2
ci
C2
.9131
.1616
.2047
.2442
. ____
Zero cm gap
AVERAGE %
DEVIATION
MAXIMUM %
DEVIATION
C4
1.16
1.13
.1641
4.54
4.31
0.635 cm (0.25 in.) gap
o\
to
2.54 cm (I in.) gap
4.1
4.2
3.0013
.1132
.1522
.2321
.2878
1.66
1.23.
4.55
3.49
2.60
1.98
2.45
1.98
8.11
5.07 .
6.17
5.07
.
All
4.1
4.2
.4.3
I ^
1.5079
.01949
1.5395
.0329
.1823
.2853
.1807
.2719.
.3958
-.1940
.3496
-.1543
70
The best correlation for overall heat transfer data combined
including geometric effects is
N u h = 1.5395 RaH 0 *1807(HiZH)"0 '194
with
an
average
percent
deviati on
of
(4.16)
2.45 and
maximum
percent deviation of 6.17.
All of the above correlating parameters are only based
on the height of the enclosure,
H.
In order to obtain the
empirical equation based on the length of the enclosure, a
transformation was derived for the living space
Nu
= C1 (Ra
L2
)c2 (H/Lo)3c 2“1 Prc3 Ac4 ,
l2
(4.17)
and for the Trombe wall space
Nu
= C1 (Ra
lI
)c2 (HZL1 )3 c 2-1 Prc3 Ac4
lI
where HZL1 = 12 and HZL 2 = 0.75.
not
change
the
average
and
(4.18)
These transformations will
maximum
deviations
of
the
correlations.
The flow regime in. the living space may be similar to a
vertical plate in the infinite medium since the aspect ratio
of 0.75 could be consid ere d small.
found
the
transition
(to
Eckert and Jackson [4]
turbulent)
regime
was
in
the
between
1 08 to I O 10 for the ir
i so th er ma l vertical plate study.
This may imply that the
Rayleigh
number
range
flow in the living space seen in the present study is in the
71
transition regime,
since Rafi is in that range.
The flow in
the Trombe wall space may be expected to resemble flow in a
rectangular enclosure with adiabat ic horizontal.surfaces.
As Elder [12] mentioned, for RalllA3Z P r V S greater than IO9 a
turbulent behavior may appear.
A 3) / P f V 2
ig
in
the
range
of
In the present S t u d y X R a lll
4.IxlO^
to
9.8xl03.
This
suggests that the flow in the Tro m b e wall space is also in
the transition regime.
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
This inv est igation has present ed a study of natural
convection heat transfer from an isothermal vertical heated
wall with the same size gap at the top and the bottom to a
rectangular enclosure.
gap,
Three different gap sizes, 2.54 cm
0.635 cm gap, and zero gap, were investigated.
Several
conclusions can be drawn:
1)
The
2.54 cm vented
inner
body,
shows a higher
heat
transfer behavior than the 0.635 cm vented and unvented
i n ner
body
especially
2)
within
in
the
the
rectangular
lower
Rayleigh
space,
number
ranges.
As the gap size of the inner body increased,
thermal
stratification within the living
3)
test
The d i m e ns ion le ss
space also increased.
tempe ra tu re profile increased with
increasing vertical distance from the enclosure bottom.
An
exception
occurs
very
near
the
top
where
the
dimensionless temperature approaches zero.
4)
The tem pe ra tu re profile is nearly linear near the top
of
the
T r om be
wall
space.
T e m p e rat ur e
constant in the bottom of the test space.
is
nearly
Similarities
of the temperature profiles to Warrington's are a steep
drop at the inner body,
an
inner transition region,
a
nearly constant t em perat ur e zone (not in Trom be wall
73
s p a c e ) , an outer
transition zone, and a fairly steep
drop at the cold wall.
5)
The flow pattern at the upper portion of the living
space
for
all
clockwise
of
the
inner
circulation
body
cel l
ci rcu lat ion cell enclosing
gap
instead
sizes
of
shows
a
a
single
the entire region of the
living space as seen in other studies.
This is due to
a n .asymmetry of horizontal surface boundary conditions.
6)
For
the
numbers,
higher
aspect
ratios
and
higher
Rayleigh
heat transfer in the cavity is independent of
the horizontal boundary conditions.
Since the results of this inve stiga tion are the first
reported concerning the natural convection heat transfer
from a vertical heat ed wa ll wi th various vent sizes to a
rectangular enclosure, no direct com pa r i s o n with previous
results can be made.
The present exp er i m e n t is a 1 / 1 8 th
scale model study of the natural convection process in a two
story high room in passive solar heating with a Trombe wall
geometry.
However,
steady convection heat transfer data for
the full-scale structure has not been published to date, nor
co u l d
it be
ca lcu lat ed
from
the
past
investigations
of
similar Trombe wall houses. Direct comparison of the present
results with full-scale structures is truly difficult at the
74
present time.
This study, however,
has provided not only a
method to tackle the problem of natural convection within a
rectangular
enclosure
relevant
passive
to
but
solar
also
a met h o d
hea tin g
wit h
to
a
obtain
Tro mbe
data
wall
geometry.
The present t em per at ur e profile results show a large
thermal st rati fic ati on in the living space because of the
existence of a circulation cell at the upper portion of the
living space.
This stratification could possibly be reduced
by two methods.
could be m o v e d
posi ti on
One m e t h o d is the top gap of the inner body
downward
of the inner
to s o m e w h e r e
body.
above
the
central
The other is that the upper
horizontal surface could be insulated as well as the bottom
surface to create a single circul ati on cell enclosing the
entire region of the living space according to the previous
rectangular
enclosure
studies
with
adiabatic horizontal
surface boundary conditions.
Several correlations, in terms of one, two, or three
correlating parameters,
transfer
were developed to predict
in the living space.
the heat
The r e c o m m e n d e d empirical
equations are
N u h = 1.5734 Rali0 *1778
(5.1)
N u h = 1.6107 RaH0 *1760 (Hj/H)™ ° •2159
(5.2)
75
and
N u h = 0.0134 Rali0e2897Pr0e423l5 (HiZH) “ °*1726
(5.3)
for
6 .2 x 108 < R a 0 < I.Sx l O 1 0 ? 124.7 < Pr < 276.9;
0.833
< (HiZH) < 1 . 0 which have average percent deviation of 3.50f
3.01 and 2.37,
respectively.
This study could be extended to determine the extent of
the th re e- d i m e n s i o n a l
effects by mak ing
add ition planar
obse rvati ons and flow velocit y measurements.
from the tracer particles that
Reflections
have settled onto the upper
vent surface of the inner body and the bot tom face of the
4.
'
outer body caused visual and photographic problems that need
to be eliminated. Future work with the apparatus is planned.
Other test fluids are to be used to extend the Prandtl and
Rayleigh number ranges to,detect the transition from laminar
to turbulent flow.
APPENDICES
APPENDIX I
H E A T T RA N S F E R D A T A R E D U C T I O N PR O G R AM
The
computes
All
of
following
and
the
is
correlates
variables,
a
data
all
of
reduction
the
s ub r ou ti ne s
are de fi ne d w i t h i n the program.
program
dimensionless
and
fu nction
which
gr o u p s.
su b pr o g r am s
78
C**** HEAT TRANSFER DATA REDUCTION AND CORRELATION PROGRAM
C
DIMENSION T<B >» DTK(3)»TEMP(300)> HLOSS(3)»TK(10>> TM(10)
DIMENSION TH<3>,TC(O)
DIMENSION ORD(B), P R O ) ,GR(O),RA(O),ZZNUS(O),OTOTAL(O),A(B)
DIMENSION AL(Z),BL(O),OCONV(O),TAVOI(O),TAVOO(O),OHM(IO),TB(B)
C***« ENCLOSURE LENGTH
GL=IB.O
102 FORMAT(1X,F4.2>
66 FORMAT(' TROMBE WALL POSITION AT ',FO.I, ' IN')
READ(IO S ,SB > AL(I)
OUTPUT ' '
READ(103,102) DG
88 F0RMAT(25X,F3.I )
C
AL(I):
TROMBE WALL SPACE
WRITE(108,66) AL(I)
WRITE(108.100) DG
103 FORMAT( '
** TROMBE WALL DUCT GAP EQUALS ',1X.F4.2,' IN')
AL(Z)=GL-ALU >»2.0
CALL TAV(TEMP,TB,TM>
TM(S)=O.O; TM(IO)=O.O
GTOT=O.O
C
CALCULATE HEAT FLUX
OUTPUT '
HEAT TAPE VOLTAGE CURRENT(A)
TEMP(F)
TEMP(K) '
DO 43 NI=I,10
TK(NI) = (TM(NI>+459.63)/I.B
TK(S)=O.O: TK(IO)=O.O
READ(105,333)EMF,VOL
333 FORMAT(IX,F5.I,1X,F6.4>
O H M (N I )=I.O
CURNT =VOL/OHM(NI)
111 F0RMAT(7X,I2,7X,F5.I,4X,FB.4,4X,'* ',F7.3.F9.0)
POWER=EMF*CURNT
WRITE!108,111) NI.EMF.CURNT,TM(NI),TK(NI)
GTOT=QTOT+POWER
IF(N I .E G .5) THEN
C#***#LHS: LEFT HAND SIDE HEAT FLUX INPUT
GTOTAL(I)»GT0T*3.413
GTOT=O.O
ELSE
END IF
43 CONTINUE
C*****RHS: RIGHT HAND SIDE HEAT FLUX INPUT
GTOTAL(2)=QT0T*3.413
79
QT0TAL(3>"QT0TAL(1)+0T0TAL(2)
QCONV( D - Q T O T A L d ) iQCONV (2 )-QTOTAL (2) ! QCONV (3) -QTOTAL (3)
OUTPUT ' '
OUTPUT ' ** AVERAGE TROMBE WALL TEMPERATURE (R) **'
OUTPUT '
LEFT-HAND SIDE
RIGHT-HAND SIDE
OVERALL'
TAVGI <D -TB (I );TAVGO( D-TB(S)
TAVGI(2)-TB(Z):TAVGO(Z)-(TB(3)+TB(4)+TB(6)+TB(7))/4.0
TAVGI(3)-(TAVGI(I )+TAVGI(Z))/2.0
TAVGO(3)-(TAVGO(I)+4.0*TAVGO(2> >/5.0
DO 888 MM-I,3
TH(MM)-TAVGI(MM)/I.8
TC(MM)-TAVGO(MM)/I.8
888 CONTINUE
WRITE*108,222) <T H (I ),I-1 ,3)
OUTPUT ' ** ITS AVERGE SURROUNDING COOLED WALL TEMPERATURE (K) **'
WRITE*108,222) (T C (I >,I■ I ,3)
OUTPUT ' '
222 F0RMAT(5X,3(F7.3,11X))
BL(I)»12.0
J J-2
OUTPUT 'I
BL
DT (K)
NUSSULT NO.
PRANDTL NO. GROSHOF NO.
* RALEIGH NO. '
DO 7 M-I ,2
IF(M .E Q .2) THEN
OUTPUT ' LHS CAVITY TEMP BASED ON AVERGING COLD WALLS TEMP'
TAVGO(I)-(TB(3)+TB(4)+TB(5)+TB(7)>/4.0
ELSE
END IF
DO 5 J-I,2
C
HAI
HEAT TRANSFER SURFACE AREA
HA-1.0
DO 6 1-1,3
DT=TAVGI<I )-TAVGO(I>
DO 38 11=1,3
DTT-TAVGI(II)-TAVGO(ID
DTKtII>=DTT/1.B
38 CONTINUE
IF(D G .E Q .0.00) THEN
HLOSS(I)-(EXP(ALOG(.257443)+1.048533*LOG(DTK(I))))*3.413
HLOSS(Z)-(EXP*ALOG(.248315) + !.Ol 1473*L0G(DTK(2>)> >«3.413
ELSE
HLOSS(I)=.1B7955*(DTK(1)**1.070327)*3.413
HLOSS(2)-.182527*(DTK(2)**1.030395>*3.413
END IF
HLOSS(3)-HLOSS(I)+HLOSS(2>
C
80
233
6
3
7
939
o no
8
232
HLOSS <3 >"HLOSS <I >+ML0SS<2>
OCONVt I )-0T0TAL(I >-HLOSS(I )
IFt I .EG.I ) BL(Z)=AL(I)
IF(I .E G .2) BL(Z)=AL(Z)
IFd.EQ.3) BL(Z)=ALt I ) ;HA»2.0*HA
H=GCONV(I)/(DT*HA)
TAVG= (TAVGI (D T T A V G O d ))/Z.O
VIS=U(TAVG,JJ)
SH=CP(TAVGrJJ)
COND=CON(TAVGeJJ)
DEN=RHO(TAVGrJJ)
BET=BETA(TAVGrJJ)
PR(I)=VIS*SH/COND
G R (I)=32.174*B.ET#DT*((B L (J )/IZ .O )* * 3 )*DEN*DEN*3G00*3B00/(VIS*VIS)
Z Z N U S d )=H#BL( J )/<COND* 12.0)
R A (I)=GR(I >*PR(I>
U R ITE(108 r233) I,BL(J),DTK(I),ZZNUS(I),P R (I),G R (I),RA(I)
F O R M A T d X , I l .2X.F4.1,2XrF7.3,2X,F10.3,3X,F10.3,4Xr2(E10.3r4X)>
CONTINUE
CONTINUE
CONTINUE
TC(I)=TAVGO(I)/!.8
URITEt108,833) TC(I)
FORMATdX, '**LHS AVERGING COLD WALLS TEMP: ',F7.3. 'K ')
OUTPUT ' I
QTOTAL
QCONV
HLOSS (WATT)* PER CENT LOSS '
DO 8 1=1,3
Q T O T A L d )=QTOTALd )/3.413: QCONVt I)=GCONVt I >/3.413
HLOSS(I)=HLOSS(I)/3.413
HPC =H L O S S d )/ Q T OTALd )*100
URITEt 108,232) D Q T O T A L d ) ,QCONVt I ) ,HLOSSd ) ,HPC
CONTINUE
FORMATtZX,Il,3<2X,F8.3),10X.F3.2)
END
C***» TEMP(R) IS A TEMPERATURE(R ) CONVERTED FROM VOLTAGE READINQ(MV)
C*»** TB IS AVERAGE TEMPERATURE
C***« IL-IW-H IS NUMBER OF DATA POINTS OF THE TEST FLUID
SUBROUTINE TAV(TEMP,T B ,T M )
DIMENSION TEMP(300),TB(B),T M (8),T T (300)
81
88
100
99
17
11
101
1
2
3
4
5
B
7
8
15
12
13
READ(105.88) LW
FORMAT(IX »13)
WRITEdOB.lOO)
F0RMAT(28X, 'DATA POINT',' READING(MV)','
TEMP(R)
NC =O
FORMAT!IX,F5.3)
READ(105,99) VOLT
NC=NC+!
IF(N C .E Q .I) OUTPUT ' LEFT HAND SIDE TROMBE WALL '
IF(NC.EQ.11)0UTPUT ' RIGHT HAND SIDE TROMBE WALL'
IF(NC.EQ.21) THEN
OUTPUT ' ** ENCLOSURE SURFACE **'
OUTPUT '
FRONT'
ELSE
END IF
IF(N C .E Q .2 B ) OUTPUT
REAR'
IF (N C .E Q .31) OUTPUT •
LEFT'
IF(NC.EQ.3 B ) OUTPUT •
RIGHT'
IF(N C .E Q .41) OUTPUT •
TOP'
IF(N C .E Q .4 G ) OUTPUT •
BOTTOM
TEM=T(VOLT)
TEMP(NC)=TEM
TT(NC)=TEMP(NC)-459. 69
W R IT E (108,101) NC,VOLT,TEMP(NC),TT(NC)
F0RMATO2X, I3,7X,F6.3,6X,F8.2,3X,F6.2)
DO 13 N = I ,8
GO TO 11,2,3,4,5,6,7,8) N
I-DL=IO
GO TO 15
I=Il?L=20
GO TO 15
1*21iL=25
GO TO 15
I=26;L=30
GO TO 15
1=31:L=35
GO TO 15
I=3B;L=40
GO TO 15
1=41;l =45
GO TO 15
I=46:L=47
TA=O.O
DO 12 J=I,L
TA=TA+TEMP(J)
TB(N)=TA/(L-I+l)
TM(N)=TB(N)-459.B9
CONTINUE
IF(N C .N E .L W ) GO TO 17
RETURN
'
TEMP(F)')
82
END
FUNCTION T(E)
DIMENSION C O )
C d >-491.96562
C(2>«46.381884
C(3)=-l.3918864
C<4)«.15260798
C(5)--.020201612
C O ) - . 0016456956
C (7)--6.6287090/(10.**5)
C O ) - I .0241343/(10.**6)
T0T-0.
DO 10 1=1,8
10 T0T=T0T+C(I)*(£**(I-I))
T-TOT
RETURN
60 END
C#*** CORRELATE THE D IMENSIONLES5 RESULTS
DIMENSION X O , 400) , C O ) ,8(10,10) ,ANP(IO) ,RNRL(400> ,RNRLS(400>
DIMENSION R A (100),SH(IOO).XNU(IOO),PR(IOO)
READ(105,102) DG
102 FORMAT(IX,F4.2)
H= 12.0
WRITE!108,103) DG
103 FORMAT('
** TROMBE WALL DUCT GAP EQUALS
,1X,F4.2,' IN')
C*#*# DETERMINE THE CONSTANTS USING THE LEAST SQUARE AND
C##** ERROR CURVEFITTING SUBROUTINES
READ!105,100) NV.NOB
100 FORMAT(IX,II,IX,12)
DO 10 N - I ,NOB
READ!105,99) RA(N),XNU(N),PR(N),SH(N)
99 FORMAT!1X,E10.5,2(1X.F7.3),1X,F4.I)
10
CONTINUE
DO 3 1-1,3
DO I NV-2.4
DO 11 N - I,NOB
X ( I ,N )» I .O
IF (N V .E Q .2) GO TO 55
IF(N V .E Q .3) GO TO 66
IF(N V .E Q .4) GO TO 77
55 X(2,N)=L0G(RA(N))
IF(I.E Q .2> X(2,N)»L0G(PR(N)/(.2+P R (N))*RA(N ))
IF(I.E Q .3) X(2,N)-LOG(RA(N)/(HZSH(N)))
GO TO 2
66 X (2,N )=LOGtRA(N) )
X(3,N)=L0G(PR(N)>
IF(I.E G .2) X (2,N )-LOG(SH(N)ZH) I X (3,N )-LOG(PR<N )/(.2+PR(N > >#RA(N>)
IF (I.E Q .3) X (2,N )-LOG (RA(N))J X (3,N )-LOG(H/SH(N I>
GO TO 2
77 X(2,N >=LOG(RA(N)>
X(3,N)=L0G(SH(N)ZH)
83
2
11
105
5
I
3
X(4,N )=LOGtPR(N)>
IF(I.GT.l) GO TO 3
M=NV+I
X(MfN)=LOG(XNU(N)>
CONTINUE
CALL CURFT(NV,N08,X,C)
CALL ERROR(N V ,N O B ,X ,C ,S S O )
C(I)=EXP(Cd))
DO 5 J=1,NV
WRITE(10B,105) J,C(J>
FORMATdX, 'C ', 11 , '= ',F 9 .6>
CONTINUE
CONTINUE
CONTINUE
END
U U
UUU
SUBROUTINE CURFT(NV,N O B ,X ,C )
DIMENSION X(6.400) fC(B) r S d O f 10)
DOUBLE PRECISION S.D
LEAST SQUARES - NV INDEPENDENT VARIABLES
Y = X (NV+1,I ) - NOB- NO. OF OBSERVATIONS
C(I)=COEFFICIENT OF X(I)
M=NV+I
MP=M+I
DO I I=1,M
DO I J=1,MP
1 S(IfJ)=O.DO
DO 2 I=1,N0B
DO 2 J = I fM
DO 2 K=1,M
2 S(J,K)=S(J,K)+ X (J,I)«X(K.I)
S(1,MP)=1.DO
IF (NV-I) 397,997,998
997 S(1,1)=S(1,2)/S(1,1)
GO TO 999
998 DO IS LK=1,NV
11 IF ( S d f D ) 13,12, 13
12 WRITE (108,20)
20 FORMAT (' EQUATIONS IN SUBROUTINE CURFT ARE DEPENDENT - IGNORE FOL
!LOWING ERROR ANALYSIS')
DO 30 III=IfNV
30 C(III)=O.
GO TO 31
13 DO 14 J=1,M
14 S(M,J)=S(1,J+1)/S(1,1>
DO 15 1=2,NV
D =St I,I)
DO 15 J=1,M
15 S d - l , J )=S (I,J+1)-D*S (M,J)
DO 16 J=1,M
84
UUU
999 DO 3 1=1,NV
C(I)=Sd,!)
3 WRITE (108,4) I,C d )
4 FORMAT ( ' ',!OX, 'COEFFICIENT OF X' ,12, '= ',E13.9)
31 RETURN
END
SUBROUTINE ERROR(N V ,N O B ,X ,C ,SSO)
DIMENSION X (6,400),C(6),5(10,10),ANP(10),RNRL(400),RNRLS(400)
DOUBLE PRECISION YC,TS,TE
ERROR ANALYSIS - NOB OBSERVATIONS
Y=X(NV+1,I ) - SSQ=STANDARD DEVIATION
C d ) - CONSTANTS
M=NV+I
I
TS=O.DO
TE=O.DO
EMX=O.
DO 5 1=1,5
5 ANP(I)=O.
WRITE (108,1)
1 FORMAT ( 'O',2 X , 'Y EXPERIMENTAL',4 X , ' Y CALCULATED',2X,
I
'NUMERICAL ERROR',3X,'PER CENT ERROR')
DO 2 I=I,NOB
YE=X(M,I)
C
ADD EQUATION FOR YC AT THIS POINT
YC=O-DO
DO 20 IJ=I,NV
20
Y C - Y C + C d J)»X(I J, I >
C
YC =C d )*<X<NV, I ) )**C(2)
YC=EXP(YC)
YE=EXP(YE)
E=YC-YE
EP=IOO.*E/YE
EPA=ABS(EP)
IF (ERA-EMX) 6.6,7
7 EMX=EPA
6 DO 8 J=I,5
ACD = J*5
IF (EPA-ACD) 9,9,8
9 ANP(J)=ANP(J)+!.
8 CONTINUE
TS= TS +E+E
TE=TE+ABS(E P )
2 WRITE (108,3) YE,YC,E,EP,I
3 FORMAT (4E17.8,15)
A =NOB
SSQ=(TSZA)**.5
TE=TEZA
WRITE (108,4) S S Q ,TE
4 FORMAT ('O',IX,'STANDARD DEVIATION-',E15.8,3X,'AVERAGE PER CENT D
85
IEVI AXIOM*',E13.8)
WRITE (108,10) EMX
10 FORMAT ('0',20X, 'MAXIMUM PER CENT DEVIATION-',E15.8)
DO 11 1-1,3
XNF-ANRf I )
XNP=IOO.*XNP/A
ACD= I»5
WRITE (108,12) X N P ,ACD
11
CONTINUE
12 FORMAT (1X,E1S.8,2X,'PER CENT OF DATA WITHIN',2 X ,E 1 3 .8,2 X ,
I
'PER CENT OF EQUATION')
RETURN
END
FUNCTION U (T ,IDF)
CO TO (1,2) IDF
C**** ABSOLUTE VISCOSITY OF AIR
1 CO-134.375
Cl-6.0133834
C2-1.8432299
C3-1.3347050
U»T**C2/(EXP(C1)*(T+C0)**C3)
GO TO 10
C**** ABSOLUTE VISCOSITY OF 20CS DOW 200 SILICONE
2 V = .03875*(4.G*10**5)/(T-359.89)**1.912
Cl=52.754884
C2=.045437533
C3-5.1832336/<10.**5)
RHO=Cl+(C2-C3*T)*T
U=RHO*.943*V
10 RETURN
END
FUNCTION CP(T,IDF)
GO TO (1,2) IDF
C»*«* SPECIFIC HEAT OF AIR
1 C0=2.236775/(10.**5)
C l = .22797749
CP=Cl+CO*T
GO TO 10
C**** SPECIFIC HEAT OF 20CS DOW 200 SILICONE
2 TP=5.*(T-491.69)/9.
Cl=.3448334
C2=7.7499/(10.**5)
C3=4.167/(10.**8)
CP=Cl+(C2+C3»TP)*TP
10 RETURN
END
FUNCTION CON(T.IDF)
GO TO (1,2) IDF
86
C**** THERMAL CONDUCTIVITY OF AIR
1 XP=.I
CO=-B.59G49G5
Cl=34490.89
C2=8G8.23837
C3=B056583.8
7 X =XP
F=C0+C1*X+C2*X*X+C3*X*X*X-T
FP=Cl+2.*C2*X+3.*X*X#C3
XP=X-FZFP
IF (ABS((XP-X)ZX)-.0001) 6,6,7
8 CON=XP
GO TO 10
C**** THERMAL CONDUCTIVITY OF 20CS DOW 200 SILICONE
2 CON=.00034Z0.004134
10 RETURN
END
FUNCTION RHO(T ,IDF)
GO TO (1,2) IDF
C**** DENSITY OF AIR AT ATMOSPHERIC PRESSURE <LBMZFT*#3>'
1 A= 12.5
RH0=A*144.Z<33.36*T)
GO TO 10
C**** DENSITY OF 20CS DOW 200 SILICONE
2 01=52.754884
C2=.045437533
C3=5.I832336Z<10.**5>
RHO=.949*(C1+(C2-C3*T)#T)
10 RETURN
END
FUNCTION BETA(T ,IDF)
GO TO (1,2) IDF
C***« THERMAL EXPANSION COEFFICIENT OF AIR
1 BETA=I.ZT
GO TO 10
C**** THERMAL EXPANSION COEFFICIENT OF 20CS DOW 200 SILICONE
2 BETA=.00107Z1.8
10 RETURN
END
APPENDIX II
PARTIALLY REDUCED DATA
GAP(cm)
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
. FACE
Th (0K)
Tc (0K)
LHS
355.150
LHS
3.22104
LHS
309.861
LHS
297.537
LHS
327.492
LHS
315.115
LHS
338.784
LHS
304.145
LHS
345.507
LHS
291.968
LHS
291.093
LHS . 325.508
LHS . 345.939
RHS
355.755
RHS
322.172
RHS
310.038
RHS
297.681
RHS
327.467
RHS
315.221
RHS
339.101
RHS
304.391
RHS
346.146
RHS
292.074
RHS
291.138
RHS
325.162
RHS
346.540
LHS
291.113
LHS
320.432
LHS
356.951
LHS
309.759
LHS
325.946
LHS
297.742
LHS
335.191
LHS
315.569
LHS
346.538
LHS
306.291
LHS
323.216
LHS
356.268
RHS
291.130
RHS
320.841
RHS
358.209
297.941
291.529
289.001
286.420
. 292.603
289.178
295.077
287.979
295.224
285.433
284.283
293.793
297.232
298.472
291.904
289.204
286.572
292.835
290.113
295.415
288.275
295.472
285.576
284.376
293.467
297.417
284.765
291.236
299.576
289.076
292.494
285.769
294.837
289.678
298.030
289.126
293.269
302.538
284.335
290.739
298.371
Qconv W
268.850
119.283
73.825
33.295
143.873
93.520
190.323
52.345
225.737
17.252
. 17.749
132.556
217.239
242.598
107.201
66.323
30.007
130.118
84;639
173.759
' 47.607
205.656
15.608
16.131
120.909
196.888
18.226
119.202
269.688
76.474
137.393
38.897
174.546
96.900
221.630
56.106
122.016
255.117
16.511
108.495
245.018
QcondW
17.818
9.218
6.199
3.197
10.632
. 7.480
13.459
4.717
15.632
1.830
1.923
9.760
15.137
14.900
6.800
5.357
2.836
8.957
6.464
11.328
4.132
13.163
1.649
1.716
8.189
. 12.755
1.438
7.408
14.567
4.858
8.073
2.674
9.809
6.100
11.968
3.887
7.355
13.948
1.315
6.093
12.368
88
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
2.540
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
LHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
RHS
309.854
326.369
297.821
335.601
315.645
347.113
306.335
323.646
357.475
303.857
314.548
326.726
296.375
306.967
338.505
353.007
361.467
321.237
289.768
364.610
308.082
354.623
319.072
304.166
315.181
327.730
288.818
292.329
285.831
294.961
289.799
298.087
289.448
292.197
299.491
286.375
287.511
289.726
284.544
286.755
291.865
295.696
297.662
290.036
283.236
298.694
287.935
297.712
289.623
287.006
287.511
296.532
289.726
284.896
307.200
339.326
354.547
363.030
321.724
289.703
. 365.816
308.329
355.712
319.455
287.388
293.000
296.982
299.090
290.784
283.290
299.818
288.514
298.960
290.375
68.463
122.733
32.220
155.602
86.391
197.200
49.903
111.857
234.431
. 61.023
105.971
156.401
37.122
75.865
201.110
262.701
301.894
123.208
18.513
315.441
74.951
259.877
116,902
57.842
100.604
148.246
35.421
69.002
181.305
235.031
269.875
110.314
16.614
282.484
66.620
229.195
103.141
4.212
6.916
2.360
8.302
5.208
10.072
3.359
6.375
11.974
3.896
6.449
9.022
2.583
4.570
11.251
14.049
15.761
7.324
1.403
16.390
4.565
13.937
6.868
3.415
5.587
7.748
2.289
3.960
9.501
11.885
13.243
6.268
1.239
13.682
3.960
11.718
5.880
i
APPENDIX III
EFFECTIVE HEAT TRANSFER AREA
The
inner
body
was
not perfectly
present experiments because
the entire inner surface.
the heater
Thus,
isothermal
in the
tapes did not cover
the effective heat transfer
area on each face of the inner body had to be obtained to
calculate the heat transfer coefficient.
HNVENTED INNER BODY
The following steady-state heat conduction problem was
solved for the unvented inner body (refer to Figure A3.I).
The problem was initially set forth in cartesian coordinates
as
32T
3 X2
3 2T
I
3 Y2
3 2T
2
3Z^
- n
U
(A3.I)
with boundary conditions
3T (L ,Y ,Z)
3T(0,Y,Z)
0,
'9
3X
3X3T(X,0,Z)
n
^?
3Y
3 T (X,Y,0)
3Z
and
- q ”/ k
0
3T (X, L, Z)
■=' o,
3Y
for b < X <
otherwise
9T(X,Y,&)
(h/k) ^ T o - T ( X , Y , m
By applying the method of separation of variables, a Fourier
90
Y
Figure A3.I
Unvented Inner Body
91
series solution was obtained:
T(X,Y,Z)
- 2 lI“ Z cnincos (XnX) cos (UmY) {sinh [(Xn2+um2)1/ 2 Z]
n ym=0
(n+m40)
" Dnmcosh I (
1/2zl I
V+
+ q"(L-2a)(L-2b) [ 1 + h / k U where
— n^/L
(n — Oy
Iy
Z)]/(kL2 )
miT /L
2 y «,»,o»»««<,y) and
(m = Oy Iy 2y ..........y) are eigenvalueSy
and
(Xn2+ Um2)lZ2 + (h/k) tanh[(Xn2+ Um2JVZi]
■’nm
(Xn2+ ym 2)V 2 tanh[ (xn2+Ul[2) 1/2& ]+ (h/k)
The Fourier series constants w e r e obtained by orthogonality
“2.q" (L-2b)
{sin[um (L-a)] - sin(vm a ) }
^Om
k Kl2 TT2L 2
for n = 0 and Ki = I,f 2
JU p
-2qn .(L-2a)
'
p e e e e e e f
{sin[xn (L-b)] - sin(Xnb)}
k
n2,2L2
for n = I, 2, .......y and m = Oy
-4q" {sin[xn (L-b)I-Sin(X^b)H sin Eym (L-S)]-sin(ym a)}
^nm
k nm(n2+m2 )V 2 K^ZL
for n = ly2y ....... y and m = Iy 2y .....
The effective heat transfer area was obtained:
Aeff = A Tz=£ / T(L/2y L/2yi )
where A = L 2 and T z_ ^ = Q/h.
Figures
A3.2 and A3.3 show
plots of Q versus A eff for the left and right faces of the
inner
bodyy
respectively.
These
figures
were
used
to
x 929 (an )
Figure A3.2
The Effect Hear Transfer Area for the Left Face of the Unvented
Inner Body
x 929 (cm )
Figure A3.3
The Effect Heat Transfer Area for the Right Face of the Hnvented
Inner Body
94
correct the heat transfer data only when making a comparison
with Raithby1S predictions.
BIBLIOGRAPHY
BIBLIOGRAPHY
Mussulman, R. L. and Warrington R. O., An Investigation
of Natural Convective Heat Transfer in the Trombe Wall
Geometry, A Proposal to MONTS. 1982.
Jakob M., Heat Transfer,
Inc., N e w York, 1949.
Vol.l,
John W i l e y and Sons
McAdams, E. and Davis, A. H., Heat Transmission, Second
Edition, McGraw-Hill Book Company, New York, 1942.
E c k e r t , E. R. G. and J a c k s o n , T. W., A n a l y s i s of
Turbulent Free-Convection Boundary Layer on Flat Plate,
NACA TB. 1015„ pp. 255-261, 1950.
.
5.
Bayley, F. J., An Analysis of Turbulent Free-Convection
Heat Transfer,
Proceedings of the Institution of
Jflg o h a n i c a l E n g i n e e r s , Vol. 169, No. 20, pp. 361-368,
1955.
W a r n e r , C. Y. a n d A r p a c i V. S., An e x p e r i m e n t a l
Investigation of Turbulent Natural Convection in Air at
L o w P r e s s u r e ;a l o n g a V e r t i c a l H e a t e d P l a t e ,
11, pp. 397-406, 1969.
7.
Vliet G. C. and Liu C. K., An e x p erimental
Turbulent
Natural
Convection
Boundary
fil ASJflJLe. J-Qiirnal a f Heat
91, pp. 517-531, 1969.
8.
Churchill, S. W. and Chu, H. S.,Correlating Equations
for Laminaf and Turbulent Free Convection from a
Vertical Plate, International Journal of Heat of M ass
T r a n s f e r , Vol. 18, pp. 1323-1329, 1975.
9.
Batchelor, G. K., Heat Transfer by Free Convection
across a Closed Cavity between Vertical Boundaries at
Different Temperatures,
Stisrhsrly
Asslisil
Mathematics. Vol. 12, pp. 209-233, 1954.
10.
Eckert, E. R. G. and Carlson, W. 0., Natural Convection
in an Air Layer Enclosed B e t w e e n Two Vert i c a l Plates
with Different Temperatures, International Journal of
Heat and Mass Transfer. Vol. 2, pp. 106-120, 1961.
11.
Elder,
J,
Laminar
Free Convection
Study of
Laye r s ,
Vol.
in a Vertical.
97
Slot, J o u r n a l o f Z l n l d M e c h a n i c s , Vol. 23, pp„ 77-98,
1965.
12.
Elder, J. W., Turbulent Free Convection in a Vertical
slot, Journal o f Flui d M e c h a n i c s , Vol. 23, pp. 99-111,
1965.
13.
MacGregor, R. K. and Emery, A. F., Free Convection
Through Vertical Plane Layers-Moderate and High Prandtl
Numb e r Fluids, Transactions of the ASMS. Journal of
Heat Transfer. V o l . 91, pp. 392-401, 1969.
14.
Ostrach, S. and R a g h a v a n , C., Effect of Stablizing
Thermal Gradients on Natural Convection in Rectangular
Enclosures Transactions o f th e A S M Ej,. Journal o f Heat
Transfer, Vol. 101, pp. 238-243, 1979.
15.
R a i t h b y , G. D., H o l l a n d s , K. G. T. a n d U n n y T. E.,
Analysis of Heat Transfer by Natural Convection Across
Vertical Fluid Layers,
Transactions of the A S M E .
Journal of Heat Transfer. Vol, 99, pp. 287-293, 1977.
16.
Berkovsky, B. M. and Polevikov, V. K., N u m e r i c a l Study
of P r oblems on High-I n t e n s i v e Free Convection, Heat
Transfer and Turbulent Buoyant Convection, Vol. II,
Hemisphere Publishing Corporation. Washington, 1977.
17.
C a t t o n , I., A y y a s w a m y , P. S., an d C l e v e r , R. M.,
Natural Convection Flow in a Finite, Rectangular Slot
Arbitrarily Oriented wi t h respect to the Gravity,
17, pp. 173-184, 1974.
18.
E l s h e r b i n y S. M., R a i t h b y , G. D., an d K. G. T.
Hollands, Heat Transfer by Natural C onvection Across
Vertical and Inclined Air Layers, T r ansactions of the
A S M E, Journal of Heat T r a n s f e r . Vol. 104, pp. 96-120,
1982.
19.
Bauman, F., G a d g i l , A., R. Kammerud, and G r e i f , R.,
Buoyancy-Driven Convection in Rectangular Enclosures:
Experimental Results and Numerical Calculations, A S M E .
80-HT-66. 1980.
20.
Nansteel, M. W., and G r e i f , R., Natural Convection in
Undivided and Partially Divided Rectangular Enclosures,
98
T r a n sa c t i o n s a f the ASMEp Journal of Heat T r a n s f e r f
V o l . 1 0 3 , pp» 623-629, 1981.
21.
Powe, R. E., C a r l e y , C. T., and Bishop, E. H., Free
C o n v e c t i o n F l o w P a t t e r n s in C y l i n d r i c a l A n n u l i ,
Tra n a a G h i o n a a f
M Heat T&ansfer.
Vol..-91, p p . 310-314, 1969.
22.
Yin, S. H., Powe, R. E., Scanlan, J. A., and Bishop, E.
H., Natural Convection Flow Patterns in Spherical
A n n u l i , i n t e r n a t i o n a l Jfinrnal A t H a a t ansi M a a a
Transfer. V o l . 16, pp. 1785-1795, 1973.
23.
Powe, R. E. W a r r i n g t o n , R. 0. an d S c a n l a n , J. A.,
N a t u r a l C o n v e c t i v e F l o w B e t w e e n a B o d y and its
Spherical Enclosure, International Journal of Heat and
Mass Transfer. V o l . 23, pp. 1337-1350, 1975.
24.
Warrington, R. 0., N a t u r a l Convection Heat Transfer
Between Bodies and Their Enclosure. Ph.D. Dissertation,
Montana State University, 1975.
25.
Stotts R. E., Warrington, R. 0., Mussulman, R. L.,
S i m u l a t i o n and a Pr e l i m i n a r y C o m p a r i s o n of Passive
Solar Heating Systems, ASM E r 8 O-HT-17 f 1980.
26.
Trombe, F., Robert, J. F., Cabenat, M., and Sesolis,
B., Concrete Walls to Collect and Hold Heat, Solar A g e ,
pp. 13-35, August 1977.
27.
Balcomb, J. D., Hedstrom, J. C., and McFarland, R. D.
Simulation Analysis of Passive Solar Heated BuildingsP r e l i m i n a r y Results, Solar E n e r g y . Vol. 19, pp. 277282, 1977.
28.
Balcomb, J. D., McFarland, R. D., and Moore, S. W.,
Passove Testing at Los Alamos, P r o c e e d i n g s of the
Second National Passive Solar C o n f e r e n c e . Vol., 2, pp.
602-609, 1978.
29.
Wary, W. 0., and Balcomb, J. D., Trom b e W a l l vs Direct
Gain:
A Comparative Analysis of Passive Solar Heating
Systems, Third National Passive Solar Conference. 1979.
30.
Akbari, A. and Borgers, T. R., Free Convective Laminar
Flow within the Trombe Wall Channel, Solar Energy. Vol.
99
22, 1979.
31.
Casper son, R. L.., and H o c e v a r , C. J., Experimental
Investigation■of the Trombe Wall Passive Solar. Energy
System, Third N ationa l P a a a l v e Solar C o n f e r e n c e / pp.
231-235, 1979.
32.
P a l m i t e r , L., T. Wheeling, and B. Corbett, "Performace
of Passive Test Units in Butte, M o n t a n a , " National
Center for Appropriate Techonology (NCAT), Butte,
Montana, 1978.
:
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