Relative size limitations for natural convection heat transfer within an enclosure
by Stephen Alan Smith
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Stephen Alan Smith (1977)
Abstract:
Natural convection heat transfer from .isothermal cubes and cylinders concentrically .located within an
isothermal cubical enclosure was investigated. Four fluids (air, water, 20 cs silicone oil, and glycerin)
were used in the test space in conjunction with seven inner bodies. Heat transfer data and temperature'
distributions were obtained.
The independent correlating parameters used in this study were the Rayleigh and modified Rayleigh
numbers (both based on the hypothetical gap width, L, the boundary layer length, b, and the diameter,
D), and the hypothetical gap width ratio. The ranges for these parameters were: [Formulas not captured
by OCR] Temperature profiles were taken at five angular positions in two vertical planes. The fluid
temperature dropped gradually directly above the body, but all other angular positions demonstrated a
sharp decline in temperature to very near the wall temperature within a short distance of the inner body,
indicating the small effect of the enclosure on the heat transfer.
The heat transfer data were compared to the equations for natural convection within an enclosure and
for natural convection in an infinite atmosphere. The best equations for the two regions were: Nub =
.585 Rab^*.236 (enclosure [10]) 25 Nub = .52 Rab^.25 (infinite [1]) For the range of L/R. considered,
these two equations differed by a maximum of fifteen percent in their prediction of the heat transfer.
Correlations were improved by using the equation: (L/R)i = 1.26(Rab)^.0593 i as the criterion for
switching from the' enclosure equation to the infinite atmosphere equation. STATEMENT OF PERMISSION TO COPY
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Date
RELATIVE SIZE LIMITATIONS FOR NATURAL CONVECTION
HEAT TRANSFER WITHIN AN ENCLOSURE
by
■
STEPHEN ALAN SMITH
A thesis submitted in partial fulfillment
of the requirements for the degree
of
. MASTER OF SCIENCE
in
Mechanical Engineering
Approved:
Chairperson
lraduate .Committee
Head, Major Department
Graduate l5Dean
MONTANA STATE UNIVERSITY
..Bozeman,.Montana
'October, 1977
ill
ACKNOWLEDGMENT
The author would like to express his gratitude to those who pro­
vided assistance and advice during.the course of this investigation.
Special thanks go to Dr. R. 0. Warrington, his thesis advisor, and to
the other members of his committee. Dr. H. W. Townes, and Dr. N. A.
Shyne.
Thanks also go to Gordon Williamson and Vincent Ko e t t e r , who.?
assisted in building and maintaining the apparatus.
The w o r k in this investigation was supported by the Mechanical
Engineering Department at Montana State University.
TABLE OF CONTENTS
Page
V I T A ..................................................................... ii
ACKNOWLEDGMENT’ ....................................................
iii
LIST OF T A B L E S .......................'............................
v
LIST OF F I G U R E S ..................................................
vi
NOMENCLATURE . . '.............
vii
A B S T R A C T ..................................... ......... '.......... -
ix
I.
INTRODUCTION .................... •................ ..
II.
LITERATURE R E V I E W ........... . V ............................
3
EXPERIMENTAL APPARATUS AND PROCEDURE '....................
11
DISCUSSION OF R E S U L T S ......................................
24
C O N C L U S I O N S ..................................................
51
III.
IV.
V.
'I
B I B L I O G R A P H Y ............................................. ..
APPENDIX I:
APPENDIX II:
APPENDIX III:
APPENDIX IV:
HEAT TRANSFER SIMULATION PROGRAM
. . .
54
. ...............
55
PARTIALLY REDUCED HEAT TRANSFER DATA. .............
61
HEAT TRANSFER DATA REDUCTION PROGRAM . . . . . .
.66
TEMPERATURE PROFILE DATA REDUCTION PROGRAM . . .
76
V
LIST OF TABLES
Table
Page
3.1
Inner Bodies and Cartridge Heaters
.........................
4.1
Inner Bodies Used in 10.5 Inch Cubical.Outer Body and
Parameter Ranges for Each
. . . ......... .
. . . . . . '.
28
4.2
Correlations by Inner Body T y p e ...........................
30
4.3
Correlations by Fluid Type . . ......................... ..
31
4.4
Correlations for All Transition Region Data
32
4.5
Correlation Comparison Between Infinite Atmosphere V...
Equation and Enclosure E q u a t i o n .........................-
36
.
13
vi
LIST OF FIGURES
Figure
.
3.1
Heat Transfer Apparatus and Supporting Instrumentation . . .
3.2
Heat Loss from Cubes by Radiation and Stem Conduction . .
3.3
Heat Loss from Cylinders by Radiation and Stem Conduction
4.1
Comparison of the Infinite Atmosphere and Enclosure
Equations
......................... ■................... .
Page
12
. 20
.
21
26
4.2
Correlation of All Transition Region Data with Enclosure
and Infinite Atmosphere E q u a t i o n s . . ■............................ 33
4.3
Correlation of All Transition Region Data with Enclosure
E q u a t i o n .................................. .. . ............ ..
4.4
.
Comparison Between Infinite Atmosphere Equation and
Enclosure C o r r e l a t i o n s .....................................
35
37
4.5
Comparison Between Infinite Atmosphere and Enclosure
E q u a t i o n s .........................................................38
4.6
Temperature Profile in the Perpendicular Pl a n e , 1.5 x 4.0
Inch Cylinder in Air . . . . •..................................41
4.7
Temperature Profile in the Diagonal Plane, 1.5 x 4.0 Inch
Cylinder in A i r ......... ‘ .............. .. ■.................. 42
4.8. Temperature Profile in the Perpendicular Plane,.1.5 x 4.0
Inch. Cylinder in 20 csSilicone
4.9
43
.Temperature Profile in the Perpendicular Plane, 2.625 Inch
Cube in A i r .......... ......................................... ■ 44
4.10 Temperature Profile in the.Perpendicular, Plane, 2.625 Inch
Cube in 20 cs S i l i c o n e ............................. .. . . .. .
45
4.11 Temperature Profiles.for All,of the.Cubical Inner.Bodies ■
and 20 cs,- Perpendicular Plane . ...
. . . . ... ... .
47
4.12 Variation of the Nusselt N u m b e r 'with Gap Width Ratio.for ■
Convection from a .Cube in A i r .........................
. .
49
vii '
NOMENCLATURE •'
Symbol
•' Description
a
Any characteristic length
A.
Su/tface area of the inner body
b
distance traveled by the boundary layer around the inner
body
- o ..'
Empirically determined constants
C
1-3
C
Specific heat at constant pressure
P
D
Diameter for a sphere or cylinder,
f
Denotes function
g
Acceleration of gravity, 32.174 ft/sec
Gr
Grashof number, p
h
Average heat transfer coefficient, q
k
Thermal conductivity
£
Height of a vertical plate or cylinder
L
Gap width or hypothetical gap width, R^ - R^
Nu
Nusselt number, h a/k
cond
conv
Pr
2
side length for a cube
2
3 2
gg(T^ - T^)a /p
conv
/ (A.AT)
i
Heat transfer by conduction
Heat transfer by convection
Prandtl number C y/k
pRadial position measured from the center of the test
space
r - Pi(B)
r
Dimensionless radius, -- t- t— =-=--- r-nr
^o(G) " ^i(B)
viii
Nomenclature (continued)
Symbol
Description
1^ ( 6)
Distance from the center of the inner body to the surface
of the inner body
V 9)
Distance from the center of the outer body to the surface
of the outer body
R i ’R o
Ra
a
*
Raa
Rg
Rayleigh number,
2
p gg(T^ - T q)a
3
C^/pk
Modified Rayleigh n u m b e r , Ra- (L/R^)
Ratio of characteristic dimensions
■Local temperature
T
T
T
Hypothetical r a d i us, defined as the radius of a sphere
having the same volume as the body in question (inner and
outer body, respectively)
am
T - T
Dimensionless temperature, — ---- —
i
1O
Arithmetic mean temperature, (T^ + T )/2
T.
i
Inner body temperature
T
Outer body temperature
O
AT
Temperature difference, T^ - T^
3
Thermal expansion coefficient
y
Dynamic viscosity. '
* .■
Ratio of circumference of circle to diameter, 3.14159
0
Temperature probe angle
p
Density .
ix
ABSTRACT
Natural convection heat transfer from .isothermal cubes and cylin­
ders concentrically .located within an isothermal cubical enclosure was
investigated.
Four fluids (air, water, 20 cs silicone oil, and gly­
cerin) were used in the test space in conjunction with seven inner
bodies.
Heat transfer data and temperature' distributions were obtained.
The independent correlating parameters used in this study were the
Rayleigh and modified Rayleigh numbers (both based on the hypothetical
gap width, L, the boundary layer length, b , and the diameter, D ) , and
the hypothetical gap width ratio.
The ranges for these parameters w e r e :
2.26 • IO6 < Ra* < 4.35
—
L —
io10
4.77
• IO6 < Ra* < 4.03
—
D
—
O
I-1
O
—I
4.14 • IO5 < Ra, < 1.42
b
• IO9
I
9.91': IO5 < 'Ra < 5.98
L
.■IO10
%
< 9.02
O
r—{
CO
2.92 • IOI
4 <
*
2.28 < L/R. < 5.89
—
i —
Temperature profiles were taken at five angular positions in two
vertical planes.
The fluid temperature dropped gradually directly above
the body, but all other angular positions .demonstrated a sharp decline
in temperature to very near the w a l l ‘temperature within a short distance
of the inner body, indicating the small effect of the enclosure on the
heat transfer.
The heat transfer data were compared to the equations for natural
convection w ithin an enclosure and for natural convection in an infinite
atmosphere.
The best equations for the two regions were:
Nu^
F=
.585 R
a
Nu^
?=
25
.52 R a ^ "
^
(enclosure
[10])
(infinite Jl])
For the range of L/R^ considered, these two equations differed by
a maximum of fifteen percent in their prediction o% the heat .transfer.
Correlations were improved by using the equation:
I
= I.2 6 (Rab ) '0593
i
as the criterion for switching from the'enclosure equation to .the infi- •
nite atmosphere equation.
■ CHAPTER I
INTRODUCTION
;
: '
Many studies have been done in ,the past concerning natural convec­
tion heat transfer into a fluid medium of infinite e x t e n t .
More recent­
ly several studies have examined natural convection heat transfer to an
enclosed fluid, where the convective motion is limited.
Useful equa­
tions have been developed for each case.
One difficulty often encountered in using these empirical equations
is determining the range of gap width ratios over which the equations
for convection within an enclosure are. applicable.. Another difficulty
is determining how large an enclosure has to.be relative to the enclosed
body in order for the equations for heat transfer into an infinite
atmosphere to apply.
The object of this study was to examine heat
transfer within an enclosure, increasing the gap width ratio over that
studied previously to determine the bounds within which each set of
equations is applicable.
First, the existing equations were analyzed to determine the range
of gap width ratios .which would most likely form the bounds for the two
sets of equations.
Bodies of varying sizes were built to cover the
range of gap width ratios required by the analysis.
The analysis showed that for Ra^ ranging from IO^ to I O ^ , the
transition from the enclosure equations.to the infinite atmosphere
equations would.occur within a range.of gap width ratios between 2.5
and
5.0-
The inner bodies built for use in the 10.5 inch cubical outer
body included cubes (2.28 <_ L/R^.
4..25) and cylinders
(2.84 <_ L/R^ <_
5.89).
Four fluids were used, yielding Prandtl numbers from .704 to
10189.
These included air (.704
Pr
.712), water
(3.17
Pr <_ 10.16)
20 cs silicone oil (94.8 <_ Pr <_ 269) and 96 percent aqueous glycerin ■
(250 <_ Pr <_ 10189).
The 20 cs silicone oil is a Dow-Corning 200 sili­
cone fluid with a kinematic viscosity of 20 centistokes at 25p Celcius.
The data from this experimental work was compared with previous
studies for both free convection within an enclosure and free convection
into an infinite atmosphere.
Empirical equations were derived from the
data obtained in this investigation.
More important, however, was the
analysis of how well this data fit the equations derived in previous
studies and the development of criteria for determining whether a system
should be considered enclosed or infinite.
This information will
increase the acceptability of extrapolating the existing enclosure
equations and infinite atmosphere equations for use in the transition
region where the gap width ratio is large.
CHAPTER TI
LITERATURE REVIEW
.Studies have been m a d e .concerning natural convection heat transfer
from three-dimensional bodies,.both within enclosures, where..the convec­
tive motion is limited, and in an infinite .atmosphere, where there are •
no external c o n s t r a i n t s .on t h e .fluid movement.
Extensive research has .been done on the infinite .atmosphere c a s e .
Correlation of experimental data shows that the Nusselt number is a
function of the Grashof and Prandtl numbers.[I]; that is:
Nu
where:
Nu
=
a
-
ha
-—
'k
f(Gra , Pr)
(2.1)
(Nusselt number)
2 3
gp a gAT
2
(Grashof number)
.
V
Pr
-
ticP
(Prandtl number)
and a is some characteristic length dimension.
If the inertia of the fluid is small, .the dependence pf the Nusselt
number on the Grashof and Prandtl.numbers is such that the Nusselt n um­
ber correlates well w i t h the product of the two :
Nu
a
=
f(Cr
a
-Pr)
By defining the Rayleigh .number as .the product of the Grashof and Prandtl
numbers, the equation b e c o m e s :
■4
Nu
where:
Ra
a
=
Gr
a
-
a
Pr
(2.2)
f (Ra )
x a
=
. C.3p 2a3gAT
— ^— ;-----ky
Correlation of data taken from a variety of shapes by Nusselt,
M c A d a m s , and King [1] shows that, for Ra>10
4
, equation (2.2) can be
. approximated by:
' Nu^
^
.SZ(R u d)-25
(2.3)
■
where the characteristic dimension, D, is the diameter for cylinders and
spheres, and the side length for cubes.
In a study restricted to horizontal pipes in air and w a t e r , McAdams
.[2] recommended the dimensionless equation:
Nu d
3
be used for Ra ranging from 10
-
.53 (RaD)'25
9
(2.4)
w
to 10 .
.
McAdams also gives the results
of an analytical study by H e r m a n n , which yielded.the equation:
Nu d
*=
.40(RaD )-25
(2.5)
L i e n h a r d .[3 ] did both theoretical and experimental, work .on. external
free convection using a variety of shapes.
He equated the drag force of
the body on.the boundary layer with the buoyant force pf .the boundary
layer on.the body and.derived the-equation:
5
Nub
=
.52(Rab )'25
(2.6)
He experimented with vertical p l ates, horizontal cylinders, and spheres
and found that the equation accurately predicted the heat transfer.
He
also recommends the use of the length of travel by the boundary layer as
the characteristic length dimension when calculating the Nusselt and
Rayleigh numbers.
Amato and Tien [4] experimented with isothermal spheres in water
and obtained the relation:
Nu d
=
2 +
.SO(Ra b)'25 '
(2.7)
for 3 •10~’< R a < 8 -IO^ and 10<Nu <90.
D
JJ
Yuge .[5] also worked with spheres and recommended the equation:
Nu d
=
2 +
.392 (Grjy)-25
5
for l<GrD <10 ".
'
■
Holman [6] extended Y u g e 's work to include Prandtl number effects,
yielding
Nu d
?=.. 2 + ■.43 (RaD ) ‘2^
•.
(2.8)
Kreith [7] derived,the equation for the average Nusselt number to
be:
.25
Nu£
=
-
555CrV
(2.9).
•
6
9
for 10<Ra^<10 , for vertical plates and cylinders.
The length dimension,
i, is the vertical length of the plate or cylinder.
A few studies have b e e n .done on natural convection within a n .enclo­
sure.
These show that the same dimensionless parameters, that were used
to correlate convection to an infinite atmosphere (the N usselt, Grashof,
P r a h d t l , and Rayleigh numbers) also work well in correlating the results
of experiments on convection within an enclosure.
In addition, some
dimensionless ratio of characteristic dimensions is n e e d e d .
The general
form of the equation is the same as equation (2 .1), with, the ratio of
characteristic dimensions
(Rg) added as an additional parameter.
convection within an enclosure,
Nua
For
the general form is:
F= f(Gr ,Pr,Rg)
(2.10)
Except for when the fluid inertia is high (as in liquid metals),
.*
the Rayleigh number can again .be substituted for the product of the
Grashof and Prandtl numbers, changing the general .form to:
Nua
-
f (Ra^ 5Rg)
(2.11)
Scanlan, Bishop, .and .Powe .'J8] .correlated free convection between
two concentric spheres with the equation:
NUp
F= ...0874(Rap)'279(.]. + L/R^(L/R.)-'°°8
where L is the difference .between .the' two radii, R q - R ^ ,
(2.1%)
7
Weber
[9] and Jakob [1] summarized natural convection within hori­
zontal and vertical enclosures, and.Weber included a summary of natural
convection with i n concentric spherical and cylindrical annuli.
■In
addition, he presented a study of free convection between eccentricallylocated s p h e r e s .
The interested reader is referred to these works.
Warrington [10] extended the data to include convection from cubes,
cylinders, and s p h e r e s , to a cubical enclosure.
By defining the hypo­
thetical radius as the.radius of a spherd of. the same volume as the body
in question, he was able to ■correlate, all available enclosure heat
transfer data using the hypothetical gap width ratio,
the geometric parameter.
(R -R^)/R^, as
He correlated the available data with the
equation:
(2.13)
x
with an average percent deviation of 14.54 percent.
.Because the expo­
nents on the Rayleigh number and hypothetical gap width ratio are nearly
the same, the two parameters can be combined with only a small loss in
accuracy.
This simplifies equation .(2.12) to:
Nu^
*..236
?= ..585 (Ra^)
V.:
where;
i
This equation had an average.percent.deviation of 14.75 percent.
(2.14)
8
Warrington also derived the equation:
Nu^
=
.954 Ra b '208
for free convection within an enclosure.
parameter,
(2.15)
This equation has no geometric,
so the correlation was less accu r a t e , with an average percent
deviation of 18.51 percent.
On examination of Warrington's equations
(2.13) and
(2.14), it can.
be observed that for large values of L / R z the Nusselt number becomes'
unbounded.
Rowe
[11] made this same observation of Scanlan's equation
(2.12), and noted that the Nusselt number should be bounded by the
Nusselt number yielded by the equations for convection into an infinite
atm o s p h e r e .
Using equation (2.12) for enclosures and equation (2.7) for
an infinite atmosphere, Powe noted that the enclosure equation could
be used until the gap width ratio was between 1.3 and 2.2 for air and
between 2.0 and 4.0 for water.
The exact value of L/R^ at which.he
recommended switching to the infinite atmosphere equation was a function
of the Rayleigh number.
Powe also noted that for small values of L/R^ the heat- transfer
would be primarily due to conduction rather than convection.
He recom­
mended that the enclosure equation be used until the gap width ratio was
reduced to where the equation for.pure conduction predicted a higher
heat transfer, at which.point t h e .conduction equation should.he imple­
mented .
9
Warrington [10] presented solutions for the conduction from bodies
of various shapes to their enclosure.
to this source.
The interested reader is referred
By evaluating conduction solutions and comparing them
to the equation for convection into an enclosed fluid, it can be deter­
mined whether convection or conduction dominate as the means of heat
transfer.
Powe recommends using whichever predicts the.higher value for
heat transfer.
,
.
Some of the studies on convection within an enclosure include a
study of the temperature distribution within the enclosure [8,9,10].
It
was found in general .that the temperature profiles could be divided into
five regions:
the inner body,
I) a region with a steep temperature gradient close to
2) a transition region where the temperature gradient
becomes less severe, 3) a region with a small temperature gradient, 4) a
transition region
where.the temperature gradient gradually increases,
.
and 5) a region w i t h a. sharp temperature gradient close to the enclosure
walls.
The large temperature gradients are due to the high fluid velo­
cities near the body surfaces.
It was also noted that the vertical plane in which the profiles
were taken had little effect- on the profiles.
An increasing Prandtl
number tended to increase the.thickness of the transition .regions and
generally increase the fluid.temperature.
.
Inversions were sometimes noted in. the temperature .distribution.
'
These were.attributed to a high rate,of convection,of energy parallel to
10
the fluid flow relative to the transport of heat across the flow.
Holman [6] shows the results of a study done by E. E. Soehngen, who
used an interferometer to photograph the temperature distribution in a
fluid involved in free convection around horizontal cylinders in an
infinite atmosphere.
The photographs show a very small temperature
gradient directly above the body.
The gradient increased as the angular
position increased from the upward vertical.
At $0° from the vertical
position, the temperature gradient was large and the fluid was at the
ambient temperature a short distance from the b o d y .
As the angular
.position increased further from the upward v e r t i c a l , t h e r e was little
change in .the profile.
Very little data are available for free convection within enclo­
sures for large values of L/R_ j' where the equations for enclosures and
those for an infinite atmosphere predict the same value for the Nusselt
number.
The.purpose of this study is to investigate that area.
CHAPTER III
EXPERIMENTAL APPARATUS AND PROCEDURE
EXPERIMENTAL APPARATUS
The data in this investigation were obtained by placing seven
different inner bodies inside a 10.5 inch cubical test space.
A ■ .....
detailed description of the apparatus comprising the test spacfe is given
by Warrington [10].
A condensed description is given here for. reference
Figure 3.1 is a schematic of the apparatus.
test space was 15.0 inches on a side.
The box housing the
Inside the outer walls there was
a water j a c k e t , consisting of six separate channels, each 1.25 inches
wide.
Each channel had four inlet and outlet ports equally spaced
across the ends of the channel.to ensure uniform flow through each
chan n e l .
The top and bottom channels were fed by separate manifolds,
while the sides were fed by two manifolds,
adjacent sides.
The cooling water flow rate to each manifold could be
adjusted independently,
isothermal.
each supplying water to two
ensuring that the test space walls could be made
.The cooling water"was circulated .in a closed system from
the water jacket through a pump, a water chiller, an i n s ulated.storage
tank, and back to the jacket.
The.outside walls of the water jacket and
the walls enclosing the test space were fabricated from 0.5. inch type
6061 aluminum.
The seven inner bodies .were fabricated from .solid 6061 T 6 aluminum.
Three cubes were used, with side.lengths' of 3.2 inches, 2.625 inches,
and 2.0 inc h e s .
The other four were cylinders with hemispherical e n d s .
12
Figure 3.1
Heat Transfer Apparatus and Supporting Instrumentation
13
They will be referred to throughout this work.by their diameter and
overall length.
The cylinders used were 2.5 inches by 5.0 inches, 2.5
inches by 3.5 i n c h e s , 1.5 inches by 4.0 inches, and I .5 inches by 2.5
in c h e s .
The inner bodies were all heated by cylindrical Watlow car­
tridge h e a t e r s .
The size and power rating for each heater ajre shown in
,
Table 3.1.
TABLE 3.I
Shape
Size .
(inches)
Cartridge Size
Length
Diameter
'Heat Power
(at 120 volts)
3.2
.371"
2.5"
500 watts
2.625
.371"
1.5"
250 w
2.0 .
.246"
1.5"
200 w
2.5.x . 5.0
.371"
2.5"
500 w
2.5 x 3.0
.371"
2.5"
500 w
1.5 x 4.0
.246"
3.0"
300 w
1.5 x 3.5
.246"
1.5"
200 w
Cubes
.
Cylinders
Inner Bodies and Cartridge Heaters
The 3.2 inch cube had four thermocouples, one in the lower corner
of a side, one in the diagonally opposed corner on the top face,, one in
the center, of a side face, and one in the center of the top f a c e .
junction was epoxied in place within 0.125 inch of the surface.
Each
These
thermocouples were used check the.ispthermality of.the inner body and
14
verify the results of the computer program .shown in Appendix I. - This
program was designed to simulate the conduction through the solid cube
from the cartridge heater to the convective medium,
to check the iso-
thermality of the surface of the cube.
O n c e •isothermality was established,
the other cubes were built with
only one thermocouple in the center of a side face.
The cylinders all
had one thermocouple in the center of the straight portion within 0.125
inch of the surface.
Each inner body was supported within the test space by a support
stem.
The 3.2 inch cube, the 2.625 inch cube, and the 2.5 inch diameter
cylinders were supported by 0.5 inch diameter, 0.038 inch thick steel
tubes, which fit through holes in the test space wall and the outer
water jacket wall.
The other three bodies were supported by 0.25 inch
diameter, 0.038 inch thick steel tubes.
Each of these tubes fit
concentrically inside a ' 0.5 inch diameter, tube, .with steel bushings
holding the small.tube inside the large one.
The bushings were sealed
to the tubes, one by a silicone gasket sealer and the other two by .
silver solder.
The 0.25 inch stem extended past the end of the 0.5 inch
tube so that when e a c h body was centered in the test space, the .top of
the 0.5 inch tube was finch with the test space wall,
inch stem extended into the test space.
so only.the 0.25
The 0.5 inch tube extended
through the holes in the test space and.outer water jacket walls.
.15
>
Each tube was enclosed in heat shrinkable tubing to insulate
against radial heat conduction through the stem.
There was an O-ring
seal around the stem, sealing the test space from the water j a c k e t , and
another sealing the water jacket from the outside room.
In addition, a
0.5 inch Conax packing gland was placed on the support stem to seal/
against leaks between the steal shaft and the shrink tube.
T T h e heater leads and thermocouple leads were inside the support
stem.
The power to the heater .was supplied by a Sorenson variable DC
,power.supply (150 volts,. 15 amp maximum).
The current was measured by
measuring the voltage drop across a Leeds and Northrup shunt box.
The
heater voltage, shunt voltage, and thermocouple voltage were read by
three Digitec digital v o l t m e t e r s .
Each face of the cube enclosing the test space had several thermo­
couples spaced in the face, 0.125 inches from the inner surface.
thermocouples were wired in parallel,
each face.
The
giving an average temperature for
Each face was measured separately so that the overall fso-
thermality. of the enclosing cube could be. monitored.
Access to the test space and inner bodies was made, through a remov­
able 15.0 inch square coyer on the water jacket and a
circular, coyer on the enclosing cube.
10.0 inch diameter
The rectangular coyer was fitted
with a rubber gasket and'..sealed■with a silicone gasket.sealant.
The
circular ■coyer was. flanged and sealed with an 0-rrlng .coated with Dow
Corning high vacuum grease.
16
The temperature profiles .were measured using thermocouple probe's
consisting of copper-constantan thermocouples epoxied inside 0.0625 inch
diameter stainless steel tubes.
Each tube was located concentrically
inside a 0.40 inch diameter steel.tube and held in place by a 0.07 inch
hole in the end of the.tube extending into the wall, and a Conax fitting
on the.exterior end.
The tubes screwed into the test .space wall and
were sealed with pipe sealant.
jacket wall, where an
They each passed through the outer water
0-ring seal was used to prevent water leaks.
The probes were positioned a t •0°
(measured from the upward verti­
cal), 34°, 80°, 120°, and 16.0° (Figure 3.1).
­
One set of probes was on a
vertical plane passing through t h e .center of the test space and the
center of the front wall of the test space.
The second set was on a
vertical plane passing diagonally through the center of the test space.
The probes were positioned inside the test space using a modified vernier
caliper.
Heat transfer data were taken with different fluids in-the test
space, including a vacuum, pumped by a Duo-Seal vacuum pump,, atmospheric
air, 96 percent aqueous glycerin, water, and 20 cs Dow Corning 200
fluid, a silicone oil having a .kinematic viscosity .of
25° Celc i u s .
20 centistokes at
The fluids were.put into and removed from .the ..test space
through a fill stem in the bottom.of the apparatus. . The fill stem con­
sisted of a 0.40 inch d i a m e t e r .steel tube that.passed.through a hole in
the' outer wall.of the water jacket a n d .screwed into a.threaded hole in
17
the test space wall.
The tube w a s .sealed with pipe sealant on the
threaded end and with an O-ring in the outer wall,.
EXPERIMENTAL PROCEDURE
The inner body was placed into the test space by sliding the stem
through the O-rings and centering the body inside the test space.
packing gland was placed on the stem and tightened.
The
The two covers were
placed and sealed, and the .top three thermocouple probes were put in
place.
The heater leads and thermocouple leads were connected,.comple­
ting the assembly.
The apparatus was activated by filling the water jacket and storage
tank with water, turning on the water pump and chiller, and turning on
the power supply for the cartridge heater.
Once thermal equilibrium
was achieved (approximately eight hours for the cooling system and four
hours for the inner body), the flow rates in the water jacket channels
were adjusted to make the enclosing cube isothermal.
Once equilibrium was again established, the following data were
recorded:
(1)
.heater current
(2)
heater voltage
(3)
inner body thermocouple reading(s)
(4)
outer body thermocouple readings
This procedure was repeated at four-to'ten power settings for each
18
inner body and each fluid.
The three liquids
(glycerin, water,, and 20 cs silicone oil) were
introduced into the system by placing reservoirs above the apparatus
and connecting a hose from the appropriate reservoir to the fill stem
in the bottom of the test space.
The top three thermocouple probes
were removed to allow the air to escape as gravity fed the liquid into
the test space.
When the test space was full, the probes were replaced.
The line from the reservoir to the fill stem was left open to allow the
fluid to expand and contract as ..the temperature changed.
the above readings,
After taking
the fluid was drained by lowering the reservoir
below the test apparatus and removing the top thermocouple probes.
In— or-de-r— t o ob t-ain— the—heat— t-r-ansf er -by .convection--alone— -all— o ther
forms of heat loss from the inner body must be calculated and subtracted
from t h e :total-heat transferred from .the body.
Since all of the fluids
used except air were opaque to .radiation,.the only non-convectfve heat
loss was conduction through the support stem.
..The stem was ,insulated
against radial conduction, so .the losses., could be computed from the
one-dimensional steady state conduction equation:
■klA l + k 2A 2 ^ k 3A 3
^cond
^
(T,
where the subscripts refer to the.steel tube walls, thermocouple leads,
and heater leads, and d i s .the distance between the.bottom of the inner
body and.the'bottom.of the enclosure.
19
When air was used as the test fluid, conduction and radiation both
contributed to the heat loss from the inner body.
convective heat loss,
was used.
To determine the non-
the same procedure that Warrington [10] .described
The test space was evacuated to a pressure under fifty m i ­
crons for the 3.2 inch cube, and under ten microns for the other inner
bodies.
This evacuation of the test space essentially eliminated con­
vection as a mode of heat transfer.
With the enclosure wall.temperature
remaining constant, the radiative and conductive heat losses were deter­
mined to be a function of the inner body temperature. . These data are
plotted in Figures 3.2 and 3.3.
Once the heat transfer by convection alone is known, the average
heat transfer coefficient, can be calculated u s i n g t h e equation:
T-
_' t^coriv____
" W
V
The existing -equations for natural convection both within an enclo­
sure and in an infinite atmosphere .were derived from data taken from
isothermal bodies and enclosures.
Therefore,
the temperature variation
over the surfaces of both the inner and outer bodies must.be.kept to a
minimum.
The-.percent temperature variation over either.the inner or
outer body was defined as:'
temperature variation
.T
local,max
■ i^lbcal ,min
•
'-T
o
I
100
50 O
3 .2" cube
A
2.625" cube
D
2.0" cube
40 -
Q
□
A
A
□
A
O
A
i
100
150
200
INNER BODY TEMPERATURE (0F)
Figure 3.2
Heat Loss from Cubes by Radiation and Stem Conduction
250
□
2.5" x 5" cylinder
O
2.5" x 3.5" cylinder
A
1.5" x 4" cylinder
V
1.5" x 2.5" cylinder
HEAT LOSS
(BTU/HR)
20 -
□
10 "
□
O
□
A
V
M
H
a?
□ O
A
100
150
200
INNER BODY TEMPERATURE (0F)
Figure 3.3
Heat Loss from Cylinders by Radiation and Stem Conduction
250
.22
where T"
' and T 1
1
.refer to the minimum and maximum temperalocal, m m
local,max
r
ture readings on the inner or outer body.
for the outer body was 2.25 percent.
The average percent variation
The average percent variation for
the 3.2 inch cubical inner body was 6.78 percent for all the data except
when water was in the test space.
transfer involved,
.Because of the high rate of heat .V- '..'.
the average percent deviation for the 3.2 inch cube
with water in the test space was 25.0 percent.
According to Warrington
[10], this temperature variation will have a negligible effect on the
heat transfer correlations.
Based on this information, the average inner body and outer body
temperatures were used.
The results of the computer program in Appendix
I and the readings on the 3.2 inch cube indicate that the center of a
side face of a cube will be at the mean temperature of the body.
There­
fore, the remaining cubes had only one thermocouple each,- installed in
the center of a side face, and the cylinders had one thermocouple
installed in the middle of the straight portion of the body.
The temperature profile data were obtained by inserting the probe
into the test space until it contacted.the inner.body.
It was then
withdrawn in small increments and the.temperature recorded at each
interval, providing data for the temperature as a function of radial
distance.
-The inner and outer body temperatures.were recorded in order
to evaluate the profiles in terms of .the dimensionless temperature
ratio:
23
T
T
T- =
o
as a function of the dimensionless radius ratio:
r
.r.L(Q) .
ro (0) “ r.(0)
The heat transfer data obtained in this investigation were reduced
with a Texas Instruments SR-50 calculator to the partially reduced form
given in Appendix II.
A data reduction program, written in Fortran IV,
was used to further manipulate the data.
The heat transfer parameters
and correlation coefficients for the empirical equations were calculated
by this program.
The fluid properties were evaluated at the arithmetic
mean temperature:
.Ti .+ T0 .
T
am
by subroutines written by Weber [9] and Warrington JlO]. . A complete
listing of this program is found in Appendix III.
The temperature pro­
file data, was reduced by another program, listed in Appendix IV.
CHAPTER IV
DISCUSSION OF RESULTS
HEAT TRANSFER RESULTS
This investigation was done in two p a r t s .
First the existing equa­
tions for free convection within enclosures and in an infinite atjnpsphere
were -analyzed to find the range of hypothetical gap width ratio (L/R/)
that would most likely compose the transition region between the enclo­
sure region" and the infinite atmosphere region.
Data were then taken to
verify the analysis and determine equations that would best predict the
heat transfer in this transition region.
Most of the existing equations for free convection into an infinite
atmosphere yield s i m i l a r ■results for the range of Rayleigh numbers of
concern in this s t u d y , .so one equation was chosen as a representative
standard for free convection into an infinite.atmosphere.
equation reported by Jakob
Since the
[1] represented the correlation of data from
several different, sha p e s , including those used in this investigation,
equation (2.3) was chosen.
It is repeated.here for reference:.
Nty
(_2.3)
=
Warrington .[10] developed-his empirical equation from a wide ran§e
of data from several sources,
so his equation was.used as the.standard
for natural convection within enclosures.
reference:
It is repeated.here for
.25
Nub
=
(.2 .14)
./585(Ra*)'236
To compare the two equations, equation (2.3) was rewritten using
the distance travelled by the boundary layer as the characteristic
dimension.
For convection from cubes, the infinite atmosphere equation
becomes:
Nub
=
.618(Rab )*25
Figure 4.1 compares equations
"
(4.1)
(2.14) and (4.1) by showing the
Nusselt number as a function of the .hypothetical gap width ratio (L/R^)
for the two equations.
The dashed line connects the points where both
equations predict the same heat transfer.
This is where Powe [11]
recommends making the transition from the enclosure equation to the
infinite-atmosphere equation.
As L/R^ increases,
the heat transfer
predicted by the enclosure equation increases without bound.
Since the
infinite atmosphere equation predicts the heat transfer for infinite
L/R^, Powe recommends that if should form the upper bound for the enclo­
sure equation.
As L/R^ decreases,
the infinite atmosphere equation
predicts a greater, heat transfer than data indicate [8-10]., so the
enclosure equations form the lower bound for the infinite atmosphere
equation.
The equation for this transition line is:
L_
Ri
l.-26.(Ra)
.0593
. (4 .2)
Enclosure Equation Na
* .236
Infinite Atmosphere Equation Nu
.618 (Ra, )
Transition Line L/R
Enclosure
Infinite
Z
---
Transition Line
Enclosure
Infinite
Enclosure
Infinite
Enclosure
Infinite
Figure 4.1
Comparison of the Infinite Atmosphere and Enclosure Equations
10
. 27
The values of L/R^ defined by this transition line range from 2.5 to 5.0,
which is higher than the range of 1.3 to 4.0 that Powe found.
It should
be noted that Powe based his calculations on data from spheres only.
The
Nusselt numbers for convection from spheres to an infinite atmosphere is
six percent lower than the values for cubes shown in Figure 4.1.
This
would cause the range of L/R^ to vary from 2.0 to 4.2, which is in better
agreement w i t h Powe's results.
Based on the results of this analysis,
seven inner bodies were built
to provide data for L/R^ ranging from 2.28 to 5.89.
Table 4.1 indicates
the inner bodies tested, the value .of L/R_ for each body, and the ranges
of Prandtl and Rayleigh numbers over which each body was tested.
The data-“taken "from ’these seven bodies were correlated in several
ways.
The constants for the following empirical equations were evalua­
t e d using a least-squares .curve fitting technique:
Nu
d
=
F=
Nu
D
(4.3)
C1 itaDc2
C1 Ka8 tV
c
^
■
^
(4.4)
Nub
F=
C1 CBa*)02
(4.5)
Nu
l
FF
C1 HatC a ^ C 3
(4.6)
Nu
l
F=
C1 CSa*)^
(.4.7)
TABLE 4.1
I I
INNER BODIES USED IN 10.5 INCH CUBICAL OUTER BODY AND PARAMETER RANGES FOR EACH
J
.i
Shape
Size
L/ R ± ,
3.2"
2.28
i
CUBES
2.8-106
3.00
2.0-IO6
i
2.0"
max
m m
5.8-109
RaD
Fr
max
min
max ■
3.5 H O 5
7 . 3 -IO8
.705
I. O-IO4
2.2-109 . 2.4-105
2 . 7 -IO8
.704
7 . 5 -IO8
1.3-109
1.4-105
I.6 -IO8
.704
6 . 9 -IO3
1.4-1010 3 . 6 -IO5
8.4-108
.705
6.O-IO3
3.O-IO5
9.O-IO8
.704
5 . 5 -IO3
! ",
-
2.625"
Ra
b
min ,i,
4.25
I
' I.I-IO6
2.5" x 5"
2.8 4 .
6.I-IO6
2.5" x. 3.5"
3.46
2.3-106 I 6.9-109
I
CYLINDERS
9
1.5" x 3.5"
4.72
9 . 9 -IO5
5.3-10
2.9-104
1.6-108
.705
4 . 8 -IO3
1.5" x 2.5"
5.89
4.I-IO5
I.6 -IO9
3 . 7 -IO4
1.4 -IO8
.705
7 . 2 -IO3
29
The following tables
(4.2-4.4) show the correlation constants for
each equation form, the average percent deviation, and the percentage of
the data within twenty percent of the equation.
results broken down by inner body type.
Table 4.2 shows the
Table 4.3 shows the results for
each fluid, and Table 4.4 shows the results for all of the data.
It should be noted in Table 4.4 that the exponents of the two terms
in equation form (4.4) are nearly equal and that no significant increase
in accuracy is obtained by separating the Ra^ term in equation form
(4.5) into its components as shown in equation (4.4).
It also should be
noted that excluding the geometric factor altogether had little effect
on the accuracy of the equations, as can be seen by comparing equation
form (4.3) w i t h forms
(4.4) and (4.5).
This indicates that for this
range of L/R^ the enclosure has little effect on the heat transfer.
Warrington [10] found that for small L/R^ the geometric parameter had a
much larger effect on the accuracy.
Excluding the geometric factor, the
best equation for an enclosure was:
Nub
P=
.954(Rab ) l2°8
(2.15)
This had an average percent deviation of 18.51 percent.
Figure 4.2 shows all of the transition region data compared with
the enclosure equation (2.15) and.the infinite.atmosphere equation
(2.3). ..It should be noted that this enclosure equation predicts a lower
Nusselt number than either the'infinite atmosphere equation or the data
TABLE 4.2 .
CORRELATIONS BY INNER BODY TYPE
INNER BODY
TYPE
PERCENT OF
DATE WITHIN ■
' +20% OF EQUATION
Cl
c,
c,
AVERAGE PERCENT
DEVIATION
4.4
.270
.281
.131
10.39
93.33
4.5
1218
.284
10.50
88.80
4.6
.225
.281
10.39
93.33
4.7
.226
.281
'
4.4
.525
.249
'
4.5
.441
.251
4.6 '
.383
.249
4.7
.447
EQUATION
■ FORM
EMPIRICAL CONSTANTS
CUBES
.287
10.38
.151
•
. 93.33
10.38
95.65
10.50
92.39
. 10.44
94.57
L
CYLINDERS
■
.252
.394
10.82 .
94.57
LO
O
TABLE 4.3
CORRELATIONS BY FLUID TYPE
FLUID
EQUATION
FORM
AIR
GLYCERIN
(96%)
WATER,
20 CS
SILICONE
Cl
c,
4.3
1.27
.170
4.4
1.67
.172
4.5
.
EMPIRICAL CONSTANTS
4.6
.917
2.77
c,
.068-
.200
.104
:
deviation
9.83
'
PERCENT OF DATA
WITHIN +20% OF
• EQUATION
AVERAGE PERCENT
10.36.
^„
'
88.89
'
82.22
10.48 '■
.517
8.95 ■
80.00
'
84.44
4.7
.53 .
.233
11.10
84.44
4.3
.555
.357
6.99
6.38
97.67
4.4
.244
.262
4.5
.366
.262
4.6
.300
.259
4.7
.323
4.3
.275
6.34
97.67
'
97.67
5.42
97.67
.269
7.22
95.35
.319
.268
6.68
93.94
4.4
.183
.231
6.22
96.97
4.5
.167
.288
.288
6.73 '
4.6
.168
.285
.335
6.09
96.97
96.97
4.7
. .156
.292
6.17
. 93.94
4.3
.731
.235
3.92
100..
■ 4.4
.466
.259
4.5
.314
.272
4.6
.439
.249
4.7
.355
.266
.463
.168
.366
3.82
.100.
4.17
100.
2.69
100.
4.36
100.
TABLE 4.4
CORRELATIONS FOR ALL TRANSITION REGION DATA
EQUATION
FORM
EMPIRICAL CONSTANTS
Cl
c,
c,
4.3
.443
' .257 '
4.4
.340
.265
4.5
.328
.265
4.6
..280
.263
,• .314
.267
4.7 '
.234
.414
AVERAGE PERCENT
DEVIATION
PERCENT OF DATA
. WITHIN +20%. OF
- EQUATION
11.90
86.23
11.26
87.43
10.94
86.75
11.13
89.82
11.53
88.62
O
E
Cubes
- Cylinders
Infinite
UJ
UJ
Figure 4.2
Correlation of All Transition Region Data with Enclosure and
Infinite Atmosphere Equations
. 34
indicate.
This is as expected since equation (2.15) was obtained from
data where the gap width ratio was small and the equation form does not
account for changes.in the gap width ratio.
The best enclosure equation (2.14) includes the geometric parameter
and Figure 4.3 shows that the correlation was greatly- improved by adding
this parameter.
Table 4.5 shows that this form of the equation has
nearly -the same accuracy as the infinite atmosphere equation.
Figure 4.4 shows the correlations from four different inner b o d i e s .
The three correlations with open symbols.were obtained from Warrington's
data [10].
They were taken from three cubes in a spherical enclosure,
with L/IL ranging from .602 to 2.15.
The fourth correlation was
obtained from the.heat transfer data for the 2.625 inch cube inside the
10.5 inch cubical enclosure (L/R^ ^ 3.00).
Figure 4.4 shows that as
L/R. increases, .the enclosure .heat transfer correlations approach the
I
infinite atmosphere equation.
Figure 4.5 shows that the best enclosure equation (2.14) accounts
for the effect of changing L/R^,.seen in Figure 4.4.
As L/R^ increases
into the transition .region, both .the enclosure equation .(2.14) and the
infinite.atmosphere equation (2.3) predict nearly the same.heat transfer
Either-equation could be used in t h i s . r e g i o n , b u t Table 4.5 shows that
the use of equation (4.2) as.the transition line improves t h e .correla­
tions . . By including only.those points indicated by the transition line,
the.average, percent .deviation.involved in using the infinite .atmosphere
1000
Figure U .3
O
Cubes
EB -
Cylinders
Correlation of All Transition Region Data With Enclosure Equation
TABLE 4.5
CORRELATION COMPARISON BETWEEN INFINITE ATMOSPHERE EQUATION AND ENCLOSURE EQUATION .
INFINITE ATMOSPHERE EQUATION
%
' -52saD'25
AVERAGE PERCENT
PERCENT OF DATA
DEVIATION | ■ WITHIN +20% OF
EQUATION
DATA INCLUDED
ALL DATA
13.23
ENCLOSURE EQUATION
Nub = .585(Ra*)'236
AVERAGE PERCENT
DEVIATION
PERCENT OF.DATA
WITHIN + 20% OF
EQUATION
80.84
12.71
84.94
■:
•
CUBES
14.65
74.32
15.06
78.38
CYLINDERS
11.20
86:96
10.83
90.22
' . 11.96
90.11
ALL DATA:
L/R. < 1.26Ra •059
I
D
: i
L/R. > 1.26Ra ’°59
1
, b .
12.17
84.00
Figure 4.4
Comparison Between Infinite Atmosphere Equation and Enclosure Correlations
SdlM
SdIM
Figure 4.5
Comparison Between Infinite Atmosphere and Enclosure Equations
39
equation is decreased from 13.23 percent to 12.17 percent, and the
enclosure equation deviation is reduced from 12.71 percent to 11.96
percent.
As L/R^ increases.beyond this point, it is reasonable to
assume that the enclosure equation would no longer be accurate and the
infinite atmosphere equation should be used.
;
TEMPERATURE PROFILE RESULTS
Temperature distributions within the gap were obtained at two verti
cal planes and at five angular positions in each plane,(Chapter III).
The vertical plane that passes diagonally through the test space will be
referred to as the diagonal plane.
The vertical plane that passes per­
pendicularly through the center of the test space wall" will be called
the perpendicular plane.
All temperatures plotted in this chapter were plotted in terms of
the dimensionless temperature ratio:
..T . - .T
T
^
(4-8)
i
o
The radius was plotted .in.terms of the dimensionless radius r a t i o :
"_
r
For each angular position,
..r .
^
. ri (6) •
r,(:6) - ^(9)'
0, the dimensionless radius ratio, r , ranged
from.zero at t h e .i n n e r .body to one at the outer b o d y .
The temperature
ratio, T, ranged from one to zero between the inner a n d .outer body walls
• 40
respectively.
Profiles were obtained for two inner b o d i e s , the 2.625 inch.cube
and the 1.5 x 4.0 inch cylinder.
The profiles for each body.were ob­
tained with air (Pr = .71) and 20 cs silicone (Pr - 170) in the test
space.
Figure 4.6 shows the profile in the perpendicular plane for the 1.5
x 4.0 inch cylinder in a i r .
the diagonal plane.
Figure 4.7 shows the same profile taken in
As Warrington
[10] indicated, the vertical plane in
which the profiles were taken had little effect on the temperature dis­
tribution.
This was true of all of the profiles taken.
Using equation (4.2) as the limit for use of the enclosure equation,
the 1.5 x 4.0 inch cylinder was within the infinite atmosphere range.
Figures 4.6,- 4.7, and 4.8 show that the temperature profiles for that
cylinder are similar to those presented by Holman [6] for free convection
to an infinite atmosphere .(Chapter II).
was a small temperature gradient.
Directly above the body there
The gradient increased as the angular
position increased from the.vertical.
At 90p from the vertical position,
the temperature, gradient was large and the fluid was at the a m b i e n t .tem­
perature a short distance from the body.
As the angular position .I .'
increased further from the upward vertical there was little.change in
■
.the profile.
figures 4.9 and 4.10 show the .temperature profiles for the 2.625
inch cube in air and silicone, respectively.
According to equation
(4.2),
41
ii
0.90 -
1.834 x
TEMPERATURE RATIO
0.75
0.45
_
0.30 _
120
_ 160
0.15 -
RADIUS RATIO
Figure 4.6
Temperature Profile in the Perpendicular Plane
1.5 x 4.0 Inch Cylinder in Air
42
0.90 -
1.834 x 10
TEMPERATURE RATIO
0.60
0.45 _
0.30 _
0.15 _
RADIUS RATIO
Figure 4.7
Temperature Profile in the Diagonal Plane
1.5 x 4.0 Inch Cylinder in Air
43
I\
5.94 x IO
TEMPERATURE RATIO
0.60
0.30
x.)
RADIUS RATIO
Figure 4.8
Temperature Profile in the Perpendicular Plane
1.5 x 4.0 Inch Cylinder in 20 cs Silicone
44
0.90 4
6.97 x 10
0.75 4»
TEMPERATURE RATIO
0.60 -
0.30 ~
□
- 160
0.15 _
0.00
RADIUS RATIO
Figure 4.9
Temperature Profile in the Perpendicular Plane
2.625 Inch Cube in Air
45
A
0.90 -
Pr
=
171.1
Rafc = 7.63 x IO8
AT
=
102°F
TEMPERATURE RATIO
0.75 -
Figure 4.10
Temperature Profile in the Perpendicular Plane
2.625 Inch Cube in 20 cs Silicone
46
the system shown in Figure 4.9 is very close to the transition line be­
tween the enclosure and infinite atmosphere regions, and the system
shown in Figure 4.10 is within the enclosure region.
Both distinctly
show the five regions discussed in Chapter II; the large temperature
gradients close to both walls,
the small gradient in the middle of the
gap, and the inner and outer transition regions.
Figure 4.10 has the same characteristics as Warrington [10]
.
obtained for cubes with 20 cs silicone in the test space. -,His plot of
the profiles is repeated in Figure 4.11 for reference.
He showed that
as the inner cube size was reduced, the shape of the profile did not
change, but .the magnitude of the temperature ratio in the middle region
of -.the— gap—was.; reduced-;—
The— Zv625— inch— cube—used— in _this investigation
yielded similar results;
the profile has the same shape, but -the m a g n i ­
tude of the temperature ratio is lower than that obtained with the 6.4,
5.0, and 4 . 0 -inch cubes Warrington investigated.
Figure 4.9 shows a profile similar to those obtained by Warrington
for cubes in air for the vertical (0°) and the 34" probes.
However, the
other probes show profiles that resemble that described by Holman for
convection to an infinite atmosphere.
Warrington indicated that .temperature inversions.were present .when
fluids with high Prandtl numbers were in the test space. . These inver­
sions .wefe more pronounced .when spheres and cylinders were in.the test
space than.when.cubes were being.used.
The transition range data shows
47
Open Symbols - 4.0" cube, Pr = 298
Solid Symbols - 5.0" cube, Pr = 257
Solid Line - 6.4" cube, Pr = 278
<P •
n<> I
Figure 4.11
Temperature Profiles for All of the Cubical
Inner Bodies and 20 cs, Perpendicular Plane
■ • 48
the same tendencies,
only less pronounced.
There is a d e f i n i t e .inver­
sion shown in Figure 4.8, with.the cylinder connecting to silicone.
cube in silicone (Figure 4.10) showed a very '.small inversion.
The
Neither
of the bodies showed an inversion with air in the test space.
BOUNDS FOR THE ENCLOSURE REGION
Powe
[11] suggested that the range of L/IL values over which the
enclosure equations are accurate should be bounded from above by the
value of L/IL at which the infinite atmosphere equations become appli-
'
cable, and bounded from below by the point where more heat is transferred
by conduction than by convection." No data are available for the region
involved in the lower limit of L/R^.
Powe hypothesized that the enclo­
sure equation could be extrapolated for small values of L/R^ until the
conduction became the dominant mode of heat transfer.
Based on this hypothesis and Warrington's solution for the conduc­
tion heat transfer between concentric cubes. Figure 4.12 shows the
variation of the.Nusselt number with the hypothetical gap width ratio
7
for concentric cubes in air with a Rayleigh number of 10 ... The same
figure would show the variation of the Nusselt number for cylinders,
except the conduction heat transfer would.be slightly higher than shown "
for cubes, depending on t h e .eccentricity of the cylinder.
It is the author's opinion.that.near the lower limit.for h/R_ the
heat transfer would be due to both.conduction a n d .convection.
Until
INFINITE
ATMOSPHERE
ioL
R
Figure 4.12
i
Variation of the Nusselt Number with Gap Width Ratio for
Convection from a Cube in Air.
Ra, = io'
D
• 50
data are available for this region,.however, Powe's hypothesis could be
used with reasonable accuracy.
CHAPTER V
CONCLUSIONS
.
This study has increased the amount of available heat transfer and
temperature profile data by extending the range of the hypothetical gap
width ratio beyond that studied previously.
This has increased the
acceptability of the existing equations for natural convection both
within .an enclosure and in an infinite atmosphere by defining the range
of hypothetical gap w i d t h ratios over which each are.valid.
The author recommends Warrington's equation [10] for convection
within enclosures:
-~
Nub
^
.585(Ra*)"236
and the equation recommended by Jakob [1] for convection to an infinite
atmosphere:
NuP
-
-52(Ra0 )
The author recommends use of the equation:
F=
I. 2 6 (Rab ) *0593
as the upper limit for use of the enclosure equation.
BIBLIOGRAPHY
BIBLIOGRAPHY
1.
J a k o b , Max, Heat T r a nsfer, V o l . I, John Wiley & S o n s , N ew York,
1949, p. 523-525.
2.
McAdams, William, Heat Transmission, McGraw-Hill Co., Inc., New
York, 1954, pp. 175-177.
3.
L i e n h a r d , J., "On the Commonality of Equations for Natural Convec­
tion from Immersed Bodies," International Journal of Heat and
Mass T r a n s f e r , V o l . 16, pp. 2121-2123, November, 1973.
4.
Amato, W. S., and Tien, C., "Free Convection Heat Transfer from
Isothermal Spheres in Water," International Journal of Heat
and Mass T r a n s f e r , V o l . 15, ,pp. 327-339, February, 1972.
5.
Y u g e , T., "Experiments on Heat Transfer from Spheres Including
Combined Natural and Forced Convection," Transactions of the
A S M E , Journal of Heat Transfer, V o l . 82, pp. 214-220, 1960.
6.
Holman, J . P., Heat Transfer, Third Edition, McGraw-Hill Book
Company, N e w York, pp. 205-229, 1972.
7.
Kre'ith, F., Principles of Heat Transfer, Third Edition, Intext
Educational Publishers, New York, pp. 383-407, 1973.
8.
Scanlan, J . A., Bishop, E. H., and P owe, R. E., "Natural Convection
Heat Transfer Between Concentric Spheres," International
Journal of Heat and Mass Transfer, V o l . 13, pp. 1857-1872,
1970.
9.
Weber, N.,' Natural Convection Heat Transfer Between a Body and its
Spherical.Enclosure, Ph.D. Dissertation, Montana State Univer­
sity., 1971.
10.
Warrington, R. 0., Natural Convection Heat Transfer Between Bodies
and Their Enc l o s u r e s , Ph.D. Dissertation, M ontana State
University, 1975.
11.
P o w e , R. E . , ."Bounding Effects on the Heat Loss by Free/Convection
from Spheres and Cylinders," Transactions of the.A S M E , Journal
' of H e a t ;Transfer, Vol. 99, pp. 558-560, 1974..
APPENDICES
APPENDIX I
HEAT TRANSFER SIMULATION'P ROGRAM'
O O O O O O O O O O O
5
10
C
T H ] ? , P R P r lR/)'I I S D F S T G N F O
TU
S Y N T H T S I / F THR
HFAT
TRANSFER
IN
A C U H c OF
I" S Ti) C S V-1U H
A ? ” LONG,
i f ? " * I/?" H F A T E R
CENTERED
IN
JT
AND
THE
SURFACE
EXPOSED
TC
A CONSTANT
T E M R E P ATtJFF
MEDIUM
T-DR C O N V E C T I O N .
TEMPERATURE
TS N O R M A L I Z E D
SO
THAT
AMBIENT
TEMP=O
INPUT
A INTERVALS,
III I IAL
TEMP,
HF ATLR
WATTS,
ClIPF C O N D U C T I V I T Y , 3 C O N V E C T I O N
COEFF TCTUJTS ,
AMO
THF
SOR
FACTOR.
R E Al. K
TNl E G F R
X , Y , 7.
COM "ON
T C - 1 : 1 8 , - 1 : L O , - I R I I P ) , Il I , 112 , H 3
DIMENSION
TtiINCO : 10)
RE A D C J O S v S )
I MV ,C T , W A T I S ,K ,H I ,H 2 ,H 3 , W
F O R M AT C T A t I F l P . ? )
NTNV=-INV
DO
10
Z = N T N V 1JNV
DD
10
Y=O ,IMV
DO
10
X = O 1 TNV
T ( X 1Y 1Z ) = C t
DEFINE
HEATER
BOUNDARIES
I
F = TMV/6
M A X H T = I N V * 2/3
C
HXl = M A X H T - I
M X ? = I-MAXHT
TRl=IR+!
IRP=TR-I
TNVl=TNV- I
NTHVl=I-TNV
C
C
C
C
60
C
C
OX=DY=DZ
D X = . 1 2 5 / 1 NV
D T = ] UV . 2 1 * D X * V . ' A T T S / K
NlOdPS=O
NTDP=SOO
BEGIN
MEAT
TRANSFER LOOP
NLDOP S-NLOOPS + l
ERR = O .
CALCUL
DO
75
DO
7 I)
T ( X , m
ATE
HEATER SIDE
WALL
Z = M X 2 ,HXl
X = O, IR 2
, Z ) =T(X,TR+l,Z)+DT
TEMPS
70
75
TCTR,X, Zi = T C T R H 1X , / ) 4 m
T C l R , I P , Z ) = ( T ( T R+ I , ] U , Z ) 4 r C I R , I R M , 7 ) ) / Z . H j T
C
calculate
C
Z=HAXHT
HEATER
end
temps
V »' '
80
83
DO
no
Y = O ,TRZ
T C X 1 Y 1Z H T ( X rY fZ H ) - H l T
tty
_ 7 I=
- TCX
T r Y ,V,-Z-i;+DT
v _ 7_ 1 ■
>±
T C X 1v
Y ,-Z)
I C X , J R , Z ) = C I C X , I R + I ,Z ) + I C X , I K t / + I ) ) / Z . + O T
T ( I R , X , Z ) = (TC Iiml ,X .Z ) 4 T ( TL , X , /41 ) ) / R . +1)1
T C X , Tl? ,-Z H C T CX , I R + l , - Z ) + T C X , T R 1- Z - I ) ) / £ . + DT
T c 1 1? , X , - Z ) = C T C I R 4 I , X , - Z ; » T c T R , X ,- 7 - I ) ) / z . + O T
T C I P 1 T R 1 Z I - C T C T R + 1 » T R 1Z ) + T ( T R , TR + 1 tZ)
1-4 ICIH1TRfZ + !))/]. +I) I
TC I R , I R , - Z ) = C T ( I R + 1 , T R 1- Z ) + T ( I R f TR + 1, - Z )
C
C
85
C
C
1-+ TC IR , IF ,-Z-I ))/] .+ OT
INSULATE
OO U 5 Z=
DO
B5 X=
T C - I 1X 1Z
T ( X 1- I 1 Z
2 SIDES:
M lN V ,JUV
O 1INV
) = ! (I , X 1 Z )
) = T C X 1 I 1 Z)
CONDUCTION
SOLUTION
U S nif, S U C C E S S I V E
O V E R R EL A X A T I U M
Oil (H) Z = N T M V l , I N V l
OU 90
X = O 1 TNVi
1)0 9 0 Y = O 1 T N V l
I F C J A B S ( Z ) . L F . M A X H T . a n d .Y.l F . I R . A M D . X . L E . T R)
GO
TO
R = T ( X + 1 , Y , / ) + T ( X - l ,Y 1 Z ) + T ( X , Y - I , Z ) + T C X , Y + l ,Z)
/ + T C X 1Y 1Z + ! ) + T C X , Y , Z - l ) - 6 . * T ( X , Y , Z )
TT = H X 1 Y . Z ) + W / ( S . * R
E R R = C R i m A p s (I CX ,Y ,Z ) - T I )
TCX . v , Z ) =TT
90
C
C
C
CO M T I N U E
C 0 MV E C T J V E
P O U N O A R IFS :
STOESOO
H O
Z = N I M V 1INV
00
IUO X = O 1TMV
T T = T C X , INV-.1 , Z) /C I . + H 2 * D X / K )
E R R = F R K + A P S ( T ( X 1 T N V 1Z ) - T T )
100
H O
C
T ( X 1I N V 1Z ) = T T
DU
1 .10 V = O 1 T N V
TI = ! C T N V - I 1 Y 1 Z l Z C I .+ H 2 * O X / K )
E R R = E R R + A d S C T C T M V , Y 1Z l - T T )
TC I N V 1 Y 1 Z ) = T I
90
C
OOOO
12 0
131
132
133
135
140
155
15 0
160
170
100
ENOS00 120 X = O f T M V
on 120 Y - O i INV
T T l = T C X ,Y , I N V - I ) / (I .-+Hl * O X / K )
T T 2 = T ( X , Y , I - INV ) / C I . H_1 *0 X/ K )
E r r = E r r ^ a b s c t c x , Y , i n V ) - t i d t A B S c T c x, Y 1- I N v j - T T 2 )
T ( X 1Y 1I N V ) = Tl I
TC X 1Y j- I N V ) = T T 2
END
Ul= H E A T
TRANSFER
UlOP
D E C I S I O N AND P R I N T S E C T I O N
I F C E R R . L T . . 0 1 ) 0 0 TO 133
J F C N L O U P S .L I . N T O P )GU IU 60
W R T T f C l O P 1 13 I ) NLO QP S
F U R M A T C / , ' D T O NJT C O N V E R G E I N ' , I ^ 1 ' I T E R A T I O N S " )
WR ITE Cl 0.8, I 3? H RR
F O R M A T (' E R R = ' , F 1
J .3)
C A L L Ii e A T ( D X 1 IIIV1K)
GO TO 140
W R I T E( I OS, I 3 5 ) NL Q CJP S
F O P N A T C / , / , ' S O L U T I O N C O N V E R G E D I N " ,15," I T E R A T I O N S ' )
O O 1 6 0 / = M I N V 1 INV
WR IT E C I 0 (i , I 5 5 )
FORKAT(Z)
DO 160 Y = O 1 INV
W R I T E C I O b . 1 5 0 ) ( TC X , Y , Z ) , X = O , I N V )
F O R M A T ( I O F o . I)
'CONTINUE
W R I T E 1 1 0 8 , 1 7 0)
FORRATC/,/,/)
T F f F R R . L T . . 0 1 ) GO T D IdO
IF ( M T O P .GT . I O 0 O ) GO T D 180
NTUP = NTD p +5 00
GO TO 60
END
C
C
C
S U B R O U T I N E H E A T ( O X 1T N V 1K)
C A L C U L A T E T H E H E A T C O N V E C T E O F R U M THE
O F THE C U B E E X P O S E D T O T H E M E D I U M
C O M M O N T ( - 1 ; I d i - I : 18 ,-ID: lo) ,H I ,112 ,M3
I M T E G F R X fY lZ
DEAL K
A=ox*nx
O= O .
Z=IhV
M I N V 1 = l-TNV
00 10 Y = I , INV
FOUR
FACES
10
15
17
20
30
C
C
no
10 X = I 1T N V
Q = 0 + A m 1 * ( 7 ( X , Y ,Z ) + T C X - I 1 Y l Z m c x
O = O - H U H 3* O ( X 1Y 1Z K K X - I 1Y 1 Z K T ( X
Y=TNV
DO
]5
Z = K T N V l 1T N V
HO
15
X = I 1INV
0 = 0 + A * H 2 * ( I C X 1Y 1 Z K T C X - I 1 Y 1 7 ) + T ( X
0 = 0 + A * H 2 * ( T C Y » X , Z )+-T C Y , X - I , Z ) + T C Y
W R l T E ( 1 0 8 t1 7 )
F O R M A K / . '
C O N V EC T I O M F R O M
SU R F AC
WRTTEC100,20)0
Q = Q * . 293
W R I T E ( I Otl , 3 0 ) 0
FORMAT ( ' O = K F ? . ? , '
HTUZHR')
FORMAT ( '
= ' , , = 7 . 2 , ' W A T T S ')
CONDUCTION
FROM
MAXHT=In v * 2/3+I
K X 2= I - M A X H T
l Y - I , Z) + K X - I
i Y - I 1 Z) + K X - 1
Y - I 1Z)
Y - I 1 Z)
/4 .
/4.
, Y 1 Z - I ) + T ( X - l tY , 7 - 1 ) ) / 4 .
, X ,Z - 1 J + T C Y , X - I 1Z - D T M .
T ')
CARTRIDGE
0= 0.
I R=IMV/ 6+I
DO 4 C X = I ,IR
DO
40
Y = I 1 TR
Z=KAXHl - I
T I = C T C X 1 Y f Z K K X - I 1Y 1Z ) + K X , Y - I 1 Z ) + K X - I 1 Y - I , Z ) ) / 4
Z=MAXHT + 1
T Z = I K X 1Y 1 Z K T C X - I 1 Y 1 Z K U X 1Y - I 1 Z M K X - I 1 Y -I , Z ) ) / 4
0=Q + K*l)X*CTl-T2)/2.
Z=I-MAXHT
T 3 = ( T ( X ,Y , Z D + T C X - I , Y , Z ) + T CX , Y - l . Z T + T C X - D Y - l , Z ) ) / 4
Z=-I-MAXH I
40
T 4 = a c x , Y . Z M T U - I , Y 1Z M K X M - I f Z K K X - l , Y - I , Z ))/4
0=Q+K*DX*CT3-T4)/2.
on
5 U Z = M X Z 1M A X h t
OO
50
X = I 1 TR
Y=IP-I
Tl = C T U 1Y 1Z M T C X Y = I R + I
I, Y 1 Z M K X
, Y 1Z - I M r C X - I
,Y , Z - I ) ) / 4
i l 5 « ^ ,5ill(T, 1
) l { ?S x 7 i . - Y ’ n 4 r a i ', ’ z - n 4 K X - 1 'v ' z - 1 ) ) / 4
Y=TR-I
T 3 = ( T ( X ,Y ,Z ) + T C X - M Y , Z ) + 7 ( X 1 Y 1 Z - I ) K
50
45
( X - l , Y , Z - I ) ) /4
IQ ^ I L cxD i i I i r 3 i U 5 7 1
2 : v ’ z > , I C X - Y > z - u , u x - 1 ' Y > z - U ) / '’ WRTTFC I08,45 )
F U R M A T C /,
CONDUCTION
FROM
C A R T R I D G E ')
WRIT L ( I O H 1Z O ) O
Ln
VD
Q=Q * .293
W R J T E C IOQ
WRJJEClUn
F O R M A K / )
RETURN
EMO
3 0) Q
55)
APPENDIX II
PARTIALLY REDUCED HEAT TRANSFER DATA
The data in this section is all of the heat transfer data taken in
this investigation.
The data has been partially reduced in that the
thermocouple millivolt readings have been converted to degrees Rarenheit
and the power losses have been subtracted from the total power to obtain
PCONV.
The column headings are:
IDB
Body Identification as defined on page 67
IDF
Fluid Identification, 1-air, 2- w a t e r , 3-20 cs, 4-350
cs, 5-glycerin
0. B.DIA
Outer body diameter or side length (inches)
1 . B IDPA-^-Inner- body- -diameter^"Or- side -length— (inches) — XXXX
Eccentricity for cylindrical inner b o d i e s ; defined
as (H-I.B i D I A)/(O.B.DIA - I.B.DIA)
TAVGI .
Inner body temperature (0F)
TAVGO
PCONV
PLOSS
. .Outer body temperature
(0F)
Heat transfer by.free convection (btu/hr)
. Heat loss due to conduction and radiation (btu/hr)
IDB
IDF
O.B.D IA
I.B .D IA
XXXX
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
I
I
I
I
I
I
I
I
5
5
5
5
5
5
5
2
2
2
2
3
3
3
3
3
3
3
3
3
I
I
I
I
I
I
I
5
5
5
5
5
5
2
2
2
2
10.5000
10 . 5 0 0 0
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
19.5000
10.5000
10.3000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0 . 000O
0.0000
3 .2000
0.0000
0.0000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3.2000
3 .2000
3.2000
3.2000
3.2000
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6259
2.6250
2.6250
2.6250
2.6250
2.6250
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0 . 0 0 O0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
TAVGI
81.9740
50.8080
132.7190
102.1040
217.2380
171.8830
155.5630
109.2440
67.2940
204.2160
226.4520
146.6900
258.7700
70.4130
108.1210
57.8040
101.9540
129.4560
136.5210
78.7290
145.7340
108.7170
229.6050
195.0740
278.9510
171.8530
204.6850
121.9440
04.5420
192.01OO
11 1 . 9 6 0 0
230.4540
135.9720
218.0670
158.4430
98.2810
124.0340
179.3510
143.2520
227.1690
7 I .0890
I 19.7860
74.4920
93.7510
79.9910
TAVGO
38.1210
38.3130
37.9910
36.1240
37.0920
36.0390
37.0390
36.8850
45.30 IO
51.7630
58.4300
42.0320
60.8300
37.7220
38.88O0
37.1770
52.1930
67.1240
71.3470
43.9690
48.3000
43.5270
6 I .4720
55.8800
71.4210
53.5640
55.0990
42.2460
45.5070
46.8130
47.2910
45.6660
43.6190
43.2000
42.6010
5 I .2720
48.3680
53.4590
52.4190
58.8340
43.9510
63.4320
52.9690
53.7450
52.2610
PCONV
12.4300
3.8800
32.8100
2 0 .0800
76.8100
52.6000
43.9300
61.5400
47.6700
1641.3000
2101.3000
748.8900
2900.1000
106.7000
361.6000
3 0 6 . I 100
1645.60O0
2612.5000
2097.0000
206.6000
784.3000
441.3000
1676.1000
1265.0000
2310.5000
997.3000
1375.9000
564.9000
7.4600
42.8000
14.0000
58.90OO
32.0000
54.6000
42.7000
155.4000
341.4000
047.8000
480.2000
1499.6000
43.0600
1490.8000
343.9000
049.4000
483.1000
FLOSS
7.0000
0.6000
16.8000
10.9000
33.5000
24.5000
21.3000
28.0000
I . 1320
7.8470
8.6470
5.3860
10.I860
1.6820
3.5630
1.0610
2.5600
3.2080
3.3540
1.7890
5.0140
3.3550
8.6520
7.1630
10.6800
6.0870
7.6980
4.1010
3.9000
14.5000
6.5000
18.2000
9.0000
17.0000
1 1 . 1000
I .7280
2.7820
4.6280
3.3390
6.1880
0.8140
2.0720
0.7910
1.4710
1.0190
IDB
IDF
O.B.DIA
I . B-DlA
XXXX
TAVGI
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2
3
3
3
3
3
3
I
I
I
I
I
I
5
5
5
5
5
5
2
2
2
2
3
3
3
3
3
3
I
I
I
I
I
I
5
5
5
5
5
5
2
2
2
2
10.5000
10.5000
10.5000
10.3000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.3000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
2.6250
0.0000
0.0000
0.0000
0.0000
2.0000
0.0000
0.0000
109.6650
05.8940
188.0080
108.0090
133.0940
214.7460
169.9330
209.0850
I 10.7280
186.3130
134.2270
149.9120
07.6790
134.8090
102.5760
167.3540
74.4920
193.5090
236.3500
131.5630
95.0050
100.6000
64.2430
170.2140
92.9300
209.5130
73.2120
142.3040
I 18.3100
94.5730
134.0190
101.6630
110.0400
210.9890
156.3670
123.5720
173.0670
7 9 . 0260
205.4650
104.5880
149.3790
182.4590
107.4070
126.2980
142.7580
2.0000
2.0000
2.0000
0.0000
0.0000
0.0000
0.0000
0.0000
2.0000
0.0000
2.0000
2.0000
0.0000
2.0000
2.0000
0.0000
O. 0 0 0 0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
2.0000
0.0000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
0.3125
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
0.0000
TAVGO
PCONV
58.59O0 1252.7000
48.4810
152.2000
55.8580
848.0000
53.2780
254.5000
52.0420
432.5000
56.0230 1084.6000
52.3060
709.0000
50.8340
20.3000
52.4570
7.5000
50.6300
23.3000
50.7740
12.20OO
48.9430
15.9000
52.O870
4.2000
52.5770
268.0200
52.2530
120.0700
54.7530
462.2900
48.7840
41.9300
55.1290
655.0400
5 9.0130 1045.5200
6 7 .0 7 2 0 I 163.7900
59.6120
470.0600
58.4450
771.0000
49.6250
I 17.0900
52.0800
456.6700
49.9500
I 15.5300
58.2200
655.5000
52.6300
42.2200
52.4870
313.9300
49.9580
214.3900
53.0300
9.6300
53.5720
20.5600
52.1550
36.3400
50.4700
13.3000
48.1560
49.2500
48.1030
28.9400
55.7230
303.8700
56.0980
764.7200
50.2900
75.5100
5 7 .6 6 6 0 I 169.0000
190.0000
54.6250
56.8640
523.8400
08.3610 2969.0000
72.4600
831.7300
7 2.8440 1423.0000
7 7.2390 1947.0000
FLOSS
1.8780
1.3750
4.8900
2.0150
3.0090
5.0350
4.3240
15.0900
5.2000
12.6000
7.5000
9.0000
2.9000
1.3710
0.8390
I .8770
0.4290
2.3070
2.9560
1.0620
0.5900
0.0360
0.2440
1.6960
0.7160
2.5220
0.3430
1.4970
I . 1390
2.7500
7.4300
13.1200
4.6000
16.2100
10.1000
3.6430
6.2800
1.5430
7.9370
2.6830
4.9680
5 . 1070
I .0770
2.0700
3.5180
IDB
IDF
O.B.DIA
I.B .D IA
XXXX
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2
3
3
3
3
3
3
I
I
I
I
I
I
5
5
5
5
5
5
2
2
2
2
2
3
3
3
3
3
3
I
I
I
I
I
I
5
5
5
5
5
5
2
2
2
10.5060
16.5000
10.5000
10.5000
10.5060
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
2.5000
2.5060
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5060
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
2.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
0.3125
0.3125
0.0125
0.3125
0.3125
0.0125
6.3125
0 . 1 25 0
O. 1 25 0
0.1250
0.1250
0.1250
0.1250
0.1250
0 . 1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1259
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.2778
0.2778
0.2778
0.2778
0.2770
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
TAVGl
163.0930
121.3440
104.5270
92.5400
149.9930
211.0660
69.8030
83.3190
151.3860
102.1900
209.5130
176.0330
127.8050
133.9360
181.6630
94.5290
156.2860
77.6640
200.9720
106.1940
131.0960
160.2310
86.5480
99.5710
127.6380
75.7270
191.6940
102.5330
219.0310
160.4750
96.8600
175.9530
63.2600
150.6490
124.9990
199.8360
132.1460
85.1530
154.2470
102.2760
199.9920
175.5930
114.0810
171.0190
84.4980
TAVGO
83.7630
56.5630
58.7440
55.9030
56.4360
59.6490
51.2650
54.3020
51.9510
50.0570
52.0870
54.2720
53.4060
54.0310
55.1970
54.4750
56.7730
55.2270
57.9130
8 4 . 6 5 10
75.1030
77.5900
64.7400
60.0230
50.2220
49.2240
53.1430
50.6150
54.2340
51.4760
49.1780
49.6930
5 I .8460
52.0640
50.6070
49.2310
60.9570
53.6400
53.6170
49.0570
53.0070
54.1670
60.0080
66.6340
57.3210
PCOKV
2 5 0 6 . 70O0
304.7300
766,0300
149.5000
510.4660
991.3100
57.8500
4.9800
10.9600
9.0600
34.0900
2 4 . IOOO
13.1900
304.0000
688.5000
107.2500
457.9700
47.5600
850.5700
2949.5000
I 172.4800
2O 51.47O 0
301.3500
673.7200
300.9600
75.3500
670.7400
176.9100
849.7300
475.6100
6.0900
19.0300
I .6400
13.7500
9.8400
23.5200
197.0400
48.9900
327.6900
I 10.0400
611.1200
454.6700
750.3800
1880.2300
278.4700
PLOSS
4.2760
3.4790
6.7550
1.9670
5.0240
8 . 1 31 0
0.9950
I .4000
9.5300
3.6500
16.4500
12.4500
6.7000
3.2360
5 . 12 2 0
1.6220
4.0300
0.9090
5.7940
4.1120
2.3000
3.3470
0.8830
1.6020
3.1350
I . 0730
5.6110
2.1020
6.6740
4.4140
2.0000
7.0300
0.0000
5.6100
3.8000
8.6000
I .5280
0.6770
2 . 16O0
I .1430
3 . 1560
2.6070
1.1610
2.2410
0.5830
IDB
IDF
O.B.DIA
I.B .D IA
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2
3
3
3
3
3
3
I
I
I
I
I
I
5
5
5
5
5
5
2
2
2
2
2
3
3
3
3
3
3
3
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5060
10.5000
10.5000
10.5000
10.5000
10.5000
10.5060
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5060
10.5000
10.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5009
1.5000
1.5060
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.3000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
1.5000
XXXX
TAVGI
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
0.2778
140.3630
121.6920
82.5310
153.2260
102.4900
202.6930
176.2730
127.6800
181.2650
155.7970
201.5590
105.3150
68.8720
124.7470
170.2330
69.2710
199.7580
102.7470
149.5840
1 1 3 . 1910
166.5060
70.5860
137.3410
9 I .3270
143.3760
77.5760
212.0360
96.6870
161.3670
110.3950
185.7970
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
0.1111
TAVGO
PCONV
62.0010 1245.2900
5 I .3330
107.7700
50.0940
60.2000
51.8010
317.0100
51.1670
126.0600
54.0990
538.5300
5 1.0980
420.7500
44.9020
10.0900
40.7690
14.4900
50.2140
10.6000
40.1710
17.2400
49.0720
5.0700
4 9 . 0 2 10
1.7900
51.3850
126.0200
52.4070
313.0900
52.6230
16.0400
54.5350
411.9600
53.8730
69.4600
52.7960
202.1600
60.2470
475.3400
6 5.4250 1213.7900
57.9950
124.3600
59.5520
013.2700
53.0900
276.5600
55.7600
176.0500
54.8660
30.7900
55.5650
306.5500
52.3130
69.9700
52.1400
234.0600
52.0720
120.9200
53.6550
306.0200
FLOSS
1.6820
1.4900
0.6960
2.1780
I .1020
3.1900
2.6070
1.6500
6.4100
5.2000
7.7500
2.4000
0.4000
1.2800
2.1930
0.2900
2.5330
0.8530
1.6080
0.9240
1.7630
0.3590
1.3570
0.6670
1.5280
0.3960
2.7290
0.7740
1.9050
I . 1570
2.3050
APPENDIX III
HEAT TRANSFER DATA REDUCTION PROGRAM
This program reduces the data from the form in Appendix II to the
Nusselt, Grashof, Prandtl, and Rayleigh numbers, calculates the geo- - .
metric parameters, and evaluates the constants for empirical equation
forms using the subroutine CURET.
O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OOOOOOO OO
HEAT
TRANSFER
WRITTEN
BY
RD
ADAPTED
BY SA
•N A T U R A L
CONVEC
DATA
REDUCTION
WARRINGTON
SMITH
TION
IN A N
ENCLOSURE
M
tI sIilysSyiIIiI-IrIIisIUS......
SA
IS
SURFACE
AREA
OF
INNER
BODY
V1«EAN ?EHP^E
» r5yGDlF^REScFT0RS
T
VlA V G
1 S AI Ea
S V? R n , 5 E
UI!N
R N^°L S ^
TAVGO
IS A V E R A G E
OUTER
BODY
^
IS
J J IS
12345 IDB
IS
THE
TOTAL
POWER
LOSS
FLUID
IDENTIFIER
ATR
H20
20
CS
SILICON
350
CS
SILICON
GLYCERIN
THE
BODY
IDENTIFIER
!-SPHERE
2-CYLINDER
J- CUBE
DIMENSION
DIMENSION
NNNN=I
PI=3.1415927
OO
1000
IJ K = I , N N N N
K IS C O U N T E R
K= O
READ
IN
TDR=O
-OR
LAST
10.5
" INU5 R a D I a T I 0 N
»N0
CRTU/HR)
9.B2B
4 - SPHERE
5-CYLINDER
6~ C U BE
7-CUBE-ROTATED
u"6
TEMPERATURE
P C O N O U C T I O ^ L O S S E S ^ C S T U / H R 8 ! 8110'
PL
“ VOLUMETRIC
IN.SPHERICAL
IN.
CUBICAL
OUTER
OUTER
BODY
BODY
c^iii§§)
,:Sp,R
6(
)ii^;:,)^ r i ^ ^ R1ST6Ra500) 9 Z Z N U S C 1 5 0 0 )
C
C
50
RE ADC 1 0 5 , 2 5 )
HT
RUN
I D Q 1 J J , X C U B E , S D f X X X , I A V G I , I AV G O » P , P L
FORMAT(2T5,7F10.4)
T F C IDB.EQ.O)
GT
TO
100
K=K+ I
I F C K . E O . I) O U T P U T
I D B ,J J ,S D ,XXX
GO 7 0 ( 1 , 2 , 3 , 4 , 5 , 6 , 6 )
IDB
SPHERE,
SPHERICAL
S A = P K - S D * SO
V l = I . ;V
OUTER
BOOT
2= I.
G A P = ( X C U B E - S D ) /2.
R I= S D / 2 .
BLL=P K S D / 2 .
GO
TO
11
CYLINDERS,
SPHERICAL
OUTER
BODY
XLEN=XXX*CXCUBE-SO)+SD
SA=PI* S D * CXLEN-SD)+PI*SD*SD
V l=I . ?V 2 = l .
Rl=CSD/4.)*(C12.*XLEN/S0)-4.)**(1./3.)
GAP = ( X C U B E / 2 . )-RI
BLL=PI*SD/2.+XLEN-SD
GO TO
11
CUBES,
SPHERICAL
OUTER
BODY
SA=6.*SD*SD
Vl=I. ;v2=i.
RI=CC3.*S0*SD*SD)/C4.*PI))**Cl./3.)
GAP=(XCUBE/2.)-RI
B L L = S D* 2.
GO
TO
11
SPHERES , CUBICAL
OUTER
BODY
SA=PI*SD*SO
v i = i . ; v 2= I .
R0=((3.*XCUBE**3)/(4.*PI))**(l./3.)
RI = S D /2 .
GAP=R0-S0/2.
B L L = P I * S O /2.
GO TO
11
CYLINDERS , CUBICAL
OUTER
BODY
XLEN=XXX*(XCUBE-SD)+SD
SA=PI*SD*XLEN
Vl=I.;
V2=l.
RO=XCUBE+C3./(4.*PI))**(!./3.)
RI=SD/4.*((12.*XLENZSD)-4.)**(l./3.)
GAP=RO-RI
BLL=PI*S0/2.+XLEN-SD
GO TO
11
CUBES,
CUBICAL
OUTER
BODY
SA=6.*SD*SO
VI=I.;
V2=1.
11
C
C
C
C
C
C
C
C
C
C
C
R0=XCU8E*(3./C't .*PT))**C1 ./3.)
R I = S D * ( 3./(<♦.«?[))*«■( I . /3.)
GAP=RO-Rl
OLL=SD*?.
CONTINUE
DT
IS T E M P E R A T U R E
DIFFERENCE
BETWEEN
INNER
AND OUTER
BODIES
DT=TAVGI-TAVGn
M .I S A V E R A G E
HEAT
TRANSFER
COEFFICIENT
CBTU/HR-FT**I-F)
H=Pfr1 4 4 . /CDTfrSA)
MEAN
ABSOLUTE
TEMPERATURE
CR)
T A V G = C T A V G O f r V l f T A V G I * V 2 ) / CV I * V 2 ) + 4 5 9 . 6 9
CALCULATE
FLUID
PROPERTIES
VISCOSITY
CLBM/HR-FT)
V I S = U C T A V G , JJ)
SPECIFIC
HEAT
CBTU/LBM-R)
SH=CPCTAVG,JJ)
THERMAL
CONDUCTIVITY
CBTU/FT*»2-R)
C O N D = C O N C T A V G , J J)
DENSITY
C L B M / F T*fr3)
O E N = R H O C T A V G , JJ)
THERMAL
EXPANSION
COEFFICIENT
CFTfr*3/R)
B E T = B E T A C T A V G tJ J )
PRANOTL
NUMBER
PR=VlSfrSHZCOND
XPRCK)=PR
GRASHOF
NUMBER
CEITHER
GAP
OR D L L
OR SD,
3RD
CARD)
CF)
GR-32.174*BET*DTfrCGAP*GAP*GAP/1728.)*DEN*DEN*3600.*3600.ZCVIS*VIS)
G R = 3 2 . 1 7 4 * 0 E T f r Q T f r C S D f r S O frSD/ 1 7 2 8 . ) f r D E N f r D E N f r 3 6 0 0 . * 3 6 0 0 . Z C V I S f r V I S )
J
C
^is3E
2L « ; 5 ^ i 3T^ ^ r ^ E
%Ls%%LL^ 7^ u ,5g%N;?E5 ; « p oo-,cvis‘vi»
ZZNUS(K)=HfrGAPZCC O H O * 12.)
Z Z N U S C K ) = H * SOZ(C O N O f r l 2 . )
Z Z N U S ( K ) = HfrBLLZCGONDfr12.)
RAYLEIGH
NUMBER
RA=GRfrPR
XRA(K)=RA
C
RASTAR
IS M O D I F I E D
RAYLEIGH
RASTARCK)=RA*GAPZRI
GEO(K)=GAPZRI
GO TO
50
....
100
CONTINUE
C******
NU = C.l*RASTARfr*C2
DO
115
N = I,K
XC1,N)=1.
X ( 2 , N) = A L O G l O C R AS T A R ( N ) )
115
X ( 3 , N ) = A L D G 1 0 ( Z Z N U S CN))
h v = 2 ; n o b = k
CALL
CURFTCNV,NOB,X,C)
C
NUMBER
(RAfrGAPZRI)
ON
VO
OOOO
1000
1
2
997
998
11
12
20
30
13
14
15
16
999
3
4
31
CALL
E R R O R C N V , N 0 3 , X 1C , S S Q )
C l = I O . * * C C l ) !OUTPUT
Cl
OUTPUT
0(2)
CONTINUE
END
SUBROUTINE
C U R F r ( N V 1N O B 7 X 1 C)
DIMENSION
X(6,1500),C(6),S(7,7)
DOUBLE
PRECISION
5,0
.
LEAST
SQUARES
- NV
INDEPENDENT
VARIABLES
WRITTEN
BT R . E . P O W E
- MECHANICAL
ENGINEERING
Y = X f N V + ! , I)
- NOB=
NO.
OF O B S E R V A T I O N S
C(I)=GOEFFICIENI
OF
X(I)
M=N V+ 1
MP=M+I
DO
I 1=1,M
DO
I J = I 1 MP
S ( I 1 J ) = O . DO
DO 2 I = I 1 NOB
DO 2 J = I 1M
DO
2 K = I 1M
S ( J 1 K ) = S ( J , K ) + X ( J , I ) * X ( K , I)
S ( I 1 M P ) = I . DO
IF ( N V - I )
997,997,998
S ( 1 , 1 ) = S ( 1 1 2 ) Z S ( 1 11)
GO TO
999
DO
16
L K = I 1 NV
IF
L S ( L 1D )
13,12,13
WRITE
(108,20)
FORMAT
('
EQUATIONS
IN S U B R O U T I N E
CURFT
ARE
DEPENDENT
!LOWING
ERROR
ANALYSIS')
DO
30
I I I = I 1 NV
C(III)=O.
GO
TO
31
DO
14
J = I 1M
S ( M 1 J ) = S ( I 1 J t l ) Z S ( I lI)
DO
15
1 = 2 , NV
D = S(I, I )
DO
15
J = I 1M
S ( I - I 1 J ) = S ( I 1 J f l ) - D t S ( M 1J)
DO
16 J = I 1M
S ( N V 1 J ) = S ( M 1 J)
DO 3 I = I 1NV
C ( I ) = S ( I 7 I)
WRITE
(108,4)
! , C d )
FORMAT
( ' D O X 1 COEFFICIENT
O F X ' 1 2, ' = ' E 1 5 . 8 )
RETURN
END
SUBROUTINE
E R R O R ( N V 1N O B 1 X 1C 1 S S Q )
.
-
..
IGNORE
FQL
OOOO
C
DIMENSION
X ( 6 , 1 5 0 0 ) , C ( G ) 1A N P ( T )
OOUDLE
PRECISION
YC , T S , T E
ERROR
ANALYSIS
- NOB
OBSERVATIONS
Y = X ( N V - H 1 T)
- S S O = S T A N d AR O D E V I A T I O N
C d )
- CONSTANTS
WRITTEN
BY
R » E. POWE
- MECHANICAL
M= N V + I
TS=O-DO
TE=O-OO
EMX=ODO
5 1=1,5
5 ANP(I)=O.
WRITE
(108,1)
I FORMAT
('O rS X 1 e Y E X P E R I M E N I A L ' 8 X ,
Y
1R"5X ,'PER
CENT
ERROR')
D]
2 I= I 1 NOB
Y E = X ( M d )
ADD EQUATION
FOR
YC A T
THIS
POINT
YC=O-DO
DO
20
IJ=I.NV
20 Y 0 = Y 0 + 0 ( T j ) t X ( I J , I )
Y C = I O .**Y C
Y E = I O d t Y E
E=YC-YE
E P = I O O d E Z Y E
EPA=ABS(EP)
IF ( E P A - E M X )
6,6,7
7 EMX=EPA
6 D O 8 J = 1 ,5
A C O = J* 5
IF
(EPA-ACD)
9,9,8
9 ANP(J)=ANP(J)+!*
8 CONTINUE
TS = T S + E*E
TE=TE+ABS(EP)
2 W R I T E ( 1 0 8 , 3 ) Y E 1 YC , E t E P
3 FORMAT
(4E20.8)
A = NOB
SSQ=(ISZA)**.5
. TE=TEZA
WRITE
(108,4)
S S Q 1TE
4 FORMAT
(' O d O X 1 'STANDARD
DE V I AT IO N = ' E
H A T I O N = ' E 15.8)
WRITE
( 108,10)
EMX
10
FORMAT
( ' O d O X d M A X I M U M
PER
CENT
DEVI
DO
11
1=1,5
X N P = A N P (I)
X N P = I O O d X N P Z A
ENGINEERING
CALCULATED'5X
NUMERICAL
ERRO
H
.......
I 5 . 8, 5 X , ' A V E R A G E
A T I O N = 'E
1 5 . 8)
PER
CENT
DEV
=:i * 5
WRITE
(108,12)
X N P fACD
FORMAT
(5X.E15.8,2X,'PER
IOh
EQUATION
)
RETURN
a c d
11
12
CENT
OF
DATA
WITH I N '2X ,E 15 . 8 , 2 X , 'PER
CENT
END
FUNCTION U(TiJJ)
GO TO ( I , 2 , 3 , 4 , 5 ) JJ
C . . . . A B S O L U T E V I S C O S I T Y OF A I R
1
CO=I34•375
01=6.0133834
C2=l.8432299
C3=l. 3347050
U=Tt*G2/(EXP(Cl)*(T+C0)**C3)
GO TO 50
C....
ABSOLUTE
VISCOSITY
OF W A T E R
2 T P = T - 5 9 3 . 33203
Cl=.0071695149
C2=.011751302
03=.0087791942
C 4 = . 81654704
VIS=01*TP+C2*(1.+C3*(TP**2))**.5+G4
U=1./VIS
GO
TO
50
C....
ABSOLUTE
VISCOSITY
OF 2 0 C S
OOW
200
SILICONE
3 V = . 0 3 8 7 5 * 0 4 . 6 * 1 0 * * 5 ) / ( T - 3 5 9 . 6 9 ) * * l . 912
01=52.754684
02=.045437533
0 3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . **5)
RH0=C1+(C2-C3*T)*T
U = R H O * . 94 9 * V
GO
TO 5 0
r
ABSOLUTE
VISCOSITY
OF
350CS
COW
200
SILICONE
V= O.03875*05.495*10**9)/(T-259.69)**2.943
01=52.754684
02=.045437533
0 3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . **5)
RH0=C1+<C2-C3*T)*T
U=RHO*.970*V
GO TO 50
C*****VISCOSITY
OF
96%
GLYCERIN
5 01=103.51199
02=-.54040736
C3=.11128595E-02
•
0 4 = - . I 0 5 2 6 0 6 4 E - 05
C5=.38202064E-09
U = C 1 + C 2 * T + 0 3 *T * T + C 4 * T * * 3 + C 5 * T * * 4
U=IO.**U
,
K>
\
50
RETURN
END
FUNCTION
CPCT.JJ)
GO
TO
(I,2,3,4,5)
JJ
C - ...
SPECIFIC
HEAT
OF
ATR
1 C0=2.2 3 6 7 7 5 / ( 1 0 . )
.
Cl-.22797749
CP = C H - C O * !
GO TO
10
C....
SPECIFIC
HEAT
DF
WATER
2 Cl=I.3757095
C2=.OOl2960965
0 3 = 1 . 1 1 1 0 5 3 3 / ( 1 0. * * 6 )
CP=C l-(C2-C3*T)*T
GO TO
10
C....
SPECIFIC
HEAT
OF
20CS
3 T P = 5 . * ( T - 4 9 1 . 693/9.
Cl=.3448334
C2=7.7499/(10.* * 5 )
0 3 = 4 . 1 6 7 / ( 1 0 . *»0)
CP=C1+(C2-C3*TP)*TP
GO TO
10
SPECIFIC
HEAT
DF
35 O C S
T P = 5. K T - 4 9 1 . 6 9 ) / 9 .
C =.3259583
C 2 = 2 . 4 2 5 / ( 1 0 . **4)
0 3 = 5 . 4 1 6 6 7 / ( 1 0 . **7D
CP = C l + ( C 2 - C 3 * T P 3 * TP
GO
TO
10
C****
SPECIFIC
HEAT
DF
100%
5 IF(T.LT.599.69)
50
TO
Cl=.22420651
0 2 = . 66666701E-03
CP = C I + 0 2 *T
GO TO 10
6 Cl=.15481514
C2=.78571402E-03
CP=01+C2*T
10 R E T U R N
END
FUNCTION
C O N ( T , JJ D
GO
TO
(1,2,3,4,5)
JJ
C....
THERMAL
CONDUCTIVITY
I XP = . I
C0 = - 8 . 5 9 6 4 9 6 5
01 = 344 9 0 . 8 9
02=868.23837
03=8056583.8
DOW
200
DOW
200
GLYCERIN
6
OF
AIR
SILICONE
SILICONE
7
6
C....
2
X=XP
F=C0+Cl*X+C2*XtX+C3*X+X*X-T
FP=C1+2.*C2*X+3.*X*X*C3
XP=X-FZFP
IF
( A B S ( C X P - X ) Z X ) - O . 0001)
6,6,7
C O N = X P •
GO TO
10
THERMAL
CONDUCTIVITY
OF
WATER
Cl=.23705417
C?=.0017156797
C 3 = l. 1 5 6 3 7 7 0 / ( 1 0 . * * 6 )
C 0 N = - C 1 + ( C 2 - C 3 * T ) *T
GO
TO
10
THERMAL
CONDUCTIVITY
OF
2 OCS OOW
200 SILICONE
C O N = . 00034/0.004134
GO
TO
10
C....
THERMAL
CONDUCTIVITY
OF
350CS
DOW
ZOO
SILICONE
4 CON=O.00038/0.004134
GO
TO
10
C*****THERMAL
CONDUCTIVITY
o f
96%
GLYCERIN
5 C l = . 14340127
C2=.47222275E-04
C0N=C1+C2*T
10 R E T U R N
END
FUNCTION
R H O ( T 1JJ)
GO
TO
(1,2,3,4,5)
JJ
C....
DENSITY
OF
AI R AT A T M O S P H E R I C
PRESS.
LBM/FT3
1 A = 12.5
R H O = A t 144./(53.36*T)
GO
TO
10
C....
DENSITY
OF W A T E R
2 01=52.754684
02=.045437533
C 3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . **5)
RH0=Cl+CC2-C3*r)*T
GO
TO
10
DENSITY
OF
20CS
DOW
200
SILICONE
C 1=52.754684
C 2=.045437533
0 3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . **5)
RH0=.949*(C1+(C2-C3*T)*T)
GO TO 10
C....
DENSITY
OF
3 5 OC S D O W
200
SILICONE
4 01=52.754684
02=.045437533
C 3 = 5. 1 8 3 2 3 3 6 / ( 1 0. * * 5 )
RHO=.97*(C1+(C2-03*T)*T)
Ul«
C....
3
GO
TO
10
C*****OENSITY
OF
96%
GLYCERIN
5 01=89.789932
C2=-.2221163E-01
RH O = C 1+C2*T
10 R E T U R N
END
f u n c t i o n
BETA
Cl,J J )
GO TO
Cl,2,3,6,5)
JJ
C....
THERMAL
EXPANSION
C O E F F . OF
1 B E T A = I . ZT
GO
TO
10
C....
THERMAL
EXPANSION
COE F F . OF
2 TP=TZlOO.
IF ( T - 5 4 9 . 5 9 )
6,6,7
6 01=603.11841
02=-353.03882
03=68.2970 I2
C4=-4.3611460
RP=01+(02+(C3+04*TP)*TP)*TP
GO
TO
8
7 C l = - 1 2 8 .44920
02=68.827927
0 3 = - 1 3 . 858489
04=1.2608585
05=-.042495236
8P=Cl+(G2+(C3+CC4+C5*TPj*TP)
8 BETA
= BP/(10.t*4)
GO
TO
10
O....
THERMAL
EXPANSION
C O E F F . OF
3 BETA=.00107/1.8
GO TO
10
C....
THERMAL
EXPANSION
COEFF.
OF
4 BET A = . 0 0 0 9 6 / 1 . O
GO
TO
10
C * * * * * T H E R M Al
EXPANSION
COEFFICIEN
5 01=89.789932
02=.2221163E-01
BETA=C2/CC1-C2*T)
10
RETURN
END
AIR
WATER
*TPJtTP
20CS
350CS
T
OF
DOW
DOW
96%
200
200
SILICONE
SILICONE
GLYCERIN
APPENDIX IV
TEMPERATURE PROFILE DATA REDUCTION PROGRAM
This program reduces .the millivolt readings and micrometer readings
to the dimensionless temperature ratios and dimensionless radius ratios
for plotting.
OOOOOO
TEMPERATU
NR
IS R U N
NP
IS P R O
TAVGJ
IS
TAVGO
IS
5
10
RE
PROFILE
NUMBER
BE
NUMBER
INNER
BODY
OUTER BODY
PROGRAM
TEMPERATURE
TEMPERATURE
f i ^ )
"ROL( 6 0 “
y K ^ 0 C 6 O ) INNER
0UTER
80DIES
R E A D f l O S , 1 0 ) N R , TI X f 0 ( 1 ) , I = i;6)
FO
M A T (7 1
'
-R
.......
F 62.,3 )
IF(NR.EQ.0)G0
0 •
TO
100
TO A =
DO
15
20
25
30
35
GO
40
C
C
45
47
50
60
80
1=1,6
TO
5
,TP
40
)
.
30
K=K-I
OLR = D I M E N S ION LESS
DISTANCE
O L T = O I M E N S I O N LESS
TEMPERATURE
W R I T E ( 1 0 8 , A 5 ) N R ,NP
FORMAT ( 'Ik UN
NUMBER ',I3,8X, '
W R I T E d 06,47)
NR, NP
F0RMAT(2I3)
WRXTE(108,50)
PROBE
N U M B E R ' , 13,/)
F O R M A T (' R A D I U S ' , 6 X , ' T E M P E R A T U R E ' )
DO
80
1=1,K
DLR=RD(T)ZRO(K)
O L T = ( T E M P R O d ) - T A V G O ) /I AV GD
W R I T E d O W , 6 0 ) D L R 1O L T
WRITE(106,60)
DLR ,OLT
F O R M A T ( F 6 . 3 , 9X, F 6 . 3 )
CONTINUE
WRITE( 106,60)
0,0
GO
100
15
T0A=T0A+T0(I)/6.
TAVGI=TEMP(TI)
TAVGO=TEMP(TOA)
TAVGD=TAVGI-TAVGO
READ(105,25)NP
F O R M A T (12)
IF(NP.EQ.O)GO
TO
K=I
READ(105,35 )RD(K)
FORMAT(2F6.3)
I F C T P .E O . 0)G0
TD
TEMPRO(K)=TEMP(IP
K=K+ I
TO
20
END
C
F U N C T I O N M T E M P ( E ) TS
70
DEGREES
CF:)
F0R
COPPER-CONST ANTAN
THERMOCOUPLE
01=491.96562
02=46.381884
C3=-l.3918864
04=0.15260790
05 = - 0 . 0 2 0 2 0 1612
0 6 = 0 . 00 1 6 4 5 6 9 5 6
07=-6.6287090E-5
08=1.02413436-5
........
TEMP=01+E*(02+E*C03+Et(C4+E*(05+E*(C6+E*(C7+E*08))))))-459.69
RETURN
END
I.
OO
<1
MONTANA STATE UNIVERSITY LIBRARIES
3
762 1001551
Sm66
cop.2
Smith, Stephen A
Relative size limita­
tions for natural convec­
tion heat transfer ...
DATE
ISSUED
TO
6
Z<oot-
2
WEEKS
we
a
/
353*.
t
I S K W jBSARr l o a -
^^7 £ £
1