Natural convection heat transfer between a fixed array of cylinders and its cubical enclosure
by Gordon Crupper
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Gordon Crupper (1977)
Abstract:
Natural convection heat transfer from a fixed array of four isothermal , heated cylinders to an
isothermal, cooled cubical enclosure was experimentally investigated for both a horizontal and vertical
position of the array. The cylinders were arranged in a square array and four test fluids (air, water, 99%
aqueous glycerin, and a silicone oil) yielding Prandtl numbers in the range of 0.7 to 3.1 x 10^4 were
used with each geometric position.
The heat transfer results are presented in correlation equations for each fluid and geometry as well as
an overall correlation utilizing all data. Several equation forms were used in which the Nusselt number
was correlated as a function of combinations of the Rayleigh, Prandtl, and Grashof numbers. The best
equation found to correlate all data with a single parameter was NuB = 0.286 RaB 0.275 using the
boundary layer length as a characteristic dimension. This equation provided results with a 9.30%
average deviation. The results were compared with equations developed for single bodies to cubical
and spherical enclosures for the same Prandtl number range. Correlations of the data are provided in
tabular form for all fluids and geometries.
Several geometric effects were observed. The vertical configuration convected less heat than the
horizontal while a rotation about the vertical axis for each of these configurations had negligible effect.
The resulting decrease in heat transfer for the vertical configuration was attributed to a complex
interaction between the boundary layer length, the flow patterns which resulted from the geometry, and
the cross sectional area exposed to the upward flow. As the Prandtl number of the fluid media
increased it tended to damp out the geometric effects. Flow visualization studies and temperature
profiles were used to aid in evaluating the geometric effects. STATEMENT.OF PERMISSION TQ COPY
In pre s en ti n g - t h i s t h e s i s in p a r t i a l f u l f i l l m e n t o f t h e re quir e m e nts
f o r an advanced degree a t Montana S t a t e U n i v e r s i t y , I a g re e t h a t th e
L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r i n s p e c t i o n .
I f u r t h e r a gre e
t h a t pe rm is si on f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y
purposes may be g r a n t e d by my major p r o f e s s o r , , o r , in h is a bsen ce, by
the Director of L ib ra rie s .
I f i s unde rs too d t h a t any copying or
p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l no t be allowed
w i t h o u t my w r i t t e n p e r m i s s i o n .
Signature
Date
7
/##4?
7 7
Z
/ f 7 7
NATURAL CONVECTION HEAT -TRANSFER BETWEEN A FIXED ARRAY
OF CYLINDERS AND ITS CUBICAL ENCLOSURE
■ by
GORDON CRUPPER, JR.
A t h e s i s s ubm itt e d in p a r t i a l f u l f i l l m e n t
o f th e requi re me nts f o r th e degree
of
MASTER OF SCIENCE
in
Mechani cal Engin e e r in g
Head, Major Department
Gra duate’ Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
May, 1977
■ m
ACKNOWLEDGMENT
The a u t h o r e x p r e s s e s h i s a p p r e c i a t i o n to a l l t h o s e who have ai ded
in h is work. Spec ial a p p r e c i a t i o n i s extended to Dr. Robert 0. W a r r in g to n 5
J r . f o r his guidance and c o u n se l d uri ng t h e p r e p a r a t i o n o f t h i s t h e s i s
and a l s o t o Dr. J . A. S c a n l a n , Dr. T. C. Reihman 5 and Dr. R. L. Mussulman
f o r t h e i r a dvice and i n s t r u c t i o n . The a u t h o r i s e s p e c i a l l y a p p r e c i a t i v e
o f th e p a t i e n c e and u n d e rs ta n d in g o f h is wife and c h i l d r e n .
The work in t h i s t h e s i s was s uppor te d by th e United S t a t e s Army
under t h e Advance Degree Program f o r ROTC I n s t r u c t o r Duty. .
• IV
' TABLE OF CONTENTS
Chapter
Page
V I T A ............................................................................................
ACKNOWLEDGMENT............................................................................................ . . . „
ii
iii
LIST OF TABLES . ..........................
LIST OF FIGURES ..........................................................................
ABSTRACT..........................
v
vi
viii
NOMENCLATURE.......................................
ix
I.
INTRODUCTION.................................................................. ..............................
]
II.
LITERATURE REVIEW
3
III.
EXPERIMENTAL APPARATUS AND PROCEDURE ........................................ .
IV.
DISCUSSION OF RESULTS................... . . ' ..............................................
24
V.
CONCLUSIONS AND RECOMMENDATIONS................................... . . . .
55
............................................ . . . . . . . . .
13
APPENDIX I ........................................................................................................................... 69
APPENDIX I I .......................................
7r
BIBLIOGRAPHY
80
V
LIST OF TABLES
Table
Page
4.1 1 Ranges o f t h e Dimensionless Parameters . . .
25
4.2
C o r r e l a t i o n Equations f o r Air A l l ...................... ■
48
4 .3
C o r r e l a t i o n s f o r G lyc erin Data . ' ......................
50
4 .4
C o r r e l a t i o n s f o r Water D&ta...................................
54
4.5
C o r r e l a t i o n s f o r S i l i c o n e Data . ......................
57
4.6
C o r r e l a t i o n s f o r Al I D a ta ........................................
59
4 .7
F i t o f Data to P r e v i o u s l y Developed Equations
64
vi
LIST OF FIGURES
Fi gure
Page
3.1 ' Heat T r a n s f e r A p p a r a t u s ...........................................................................
14
3.2
S in g l e Cy lin de r with Heat Tape I n s t a l l e d ...................................
17
3 .3
i n n e r Body P a r t i a l l y A s s e m b l e d ............................................
17
4.1
Nu|_ versu s Gr^ f o r Al I D a t a ...................................
4.2
Flow P a t t e r n f o r H ori zo nta l C yli nde rs with 20 cs F lu i d
AT = 1 0 . 7 1 K .....................................................................................
and
.. .
31
4 .3
Flow P a t t e r n Around Upper H oriz on ta l C yli nder with 20 cs
F lu i d and AT = 1 0 . 7 1 K .....................................................
.". . .
32
I
Flow P a t t e r n Around Lower H oriz on ta l Cy lin der with 20 cs
F lu i d and AT = 10.71K. . . ............................................ .... . . . .
.33
Flow P a t t e r n Around Center Su pport Sphere with 20 cs Fl u i d
and AT = 10.71K...........................................................................................
34
4.6
Flow P a t t e r n f o r H o ri z ont a l C ylin de rs with 20 cs F l u i d
and AT = 5 4 . 4 3 K ........................................................................................
35
4.7
Flow P a t t e r n f o r Ho riz onta l C yl in de rs with A i r and
AT = 7.67K . . .............................................................................................
36
4.8
Flow P a t t e r n f o r H ori zo nta l C yl in de rs with A ir and
AT = 2 1 . 6 1 K ...................................................................... ..........................
37
4 .9
Flow P a t t e r n f o r V e r t i c a l C yli nde rs w ith Air and
T = 1 7 . O l K .................................................................................................
38
4.10
Nu. ve rsus Gr, FOR O0 and 45° R o t a t i o n s with t h e Cylind ers
i n Lth e H or iz ont a l P o s i t i o n . ' .................................................................. 40
4.11
Nu, v e rs us Gr, FOR 0° and 45° R o ta tio ns with t h e Cylinders
in t h e V e r t i c a l P o s i t i o n ...........................................................................
41
Temperature P r o f i l e f o r R ota tio n About th e V e r t i c a l Axis
with S i l i c o n e - H o r i z o n t a l P o s i t i o n .................................................
42
4 .4
4.5
4.12
.. .
2a
vi i
Figure
•
Page
4.13 Temperature P r o f i l e f o r R ota tio n About th e V e r t i c a l Axis
I with S i l i c o n e - H o r i z o n t a l P o s i t i o n ................................
4. 14
43
Temperature P r o f i l e f o r R ot a tio n About th e V e r t i c a l Axis
with G l y c e r i n - V e r t i c a l P o s i t i o n ........................................' . . .
44
N u s s e l t Number vers us Rayleigh Number f o r A i r Data with'
C o r r e l a t i o n Equations Superimposed .......................... . . . . .
47
N u s s e l t Number ve rs us Rayleigh Number f o r Glyc erin Data
with C o r r e l a t i o n Equations Superimposed. . . . .
..................
52
N u s s e l t Number versus Rayleigh Number f o r Water Data with
C o r r e l a t i o n Equations S u p e r i m p o s e d ................................................
55
N u s s e lt Number versu s Rayleigh Number f o r S i l i c o n e Data
with C o r r e l a t i o n Equations Superimposed........................................
56
4.19, Nu 1 ve rsus Gr|_ f o r Al I Data with C o r r e l a t i o n Equations
Superimposed .................................................................................................
6]
Nug versu s Gr 6 f o r Al I Data with C o r r e l a t i o n Equations
Superimposed .................................................................................................
62
1.1
Flow P a t t e r n Between Cylinders in t h e V e r t i c a l P o s i t i o n .
.
71
1.2
S t e a d y - S t a t e Heat Balance f o r t h e Support Sphere ..................
73
4.15
4.16
4.1 7
4.1 8
4.20
viii
ABSTRACT
Natural conv ectio n h e a t t r a n s f e r from a f i x e d a r r a y o f f o u r i s o ­
t h e r m a l , h e a t e d ' c y l i n d e r s t o an i s o t h e r m a l , cooled c ub ic a l e n c l o s u r e
was e x p e r i m e n t a l l y i n v e s t i g a t e d f o r both a h o r i z o n t a l and v e r t i c a l po-.
s i t i o n of t h e a rr a y .- The c y l i n d e r s were arra nged in a sq uar e a r r a y and
f o u r t e s t f l u i d s ( a i r , w a t e r , 99% aqueous g l y c e r i n , and a s i l i c o n e o i l )
y i e l d i n g Pr a n d tl numbers in the range o f 0 . 7 to 3.1 x !Cr were used with
each ge om et ri c p o s i t i o n .
The h e a t t r a n s f e r r e s u l t s are p r e s e n t e d in c o r r e l a t i o n e q u a ti o n s
f o r each f l u i d and geometry as well as an o v e r a l l c o r r e l a t i o n u t i l i z i n g
a l l d a t a . Several e q u a ti o n forms were used in which t h e N u s s e l t number
was c o r r e l a t e d as a f u n c t i o n o f combinations of the R a y l e i g h , P r a n d t l ,
and Grashof numbers. The b e s t eq u at i o n found t o c o r r e l a t e a l l da ta
wit h a s i n g l e pa ra m et e r was
Nub = 0.286 RagO.275
us in g th e boundary l a y e r le n g th as a c h a r a c t e r i s t i c dimension. This
e q u a ti o n provided r e s u l t s with a 9.30% average d e v i a t i o n . The r e s u l t s
were compared with e q u a t i o n s developed f o r s i n g l e bodies to c ubic a l and
s p h e r i c a l e n c l o s u r e s f o r th e same P r a n d tl number r a n g e . C o r r e l a t i o n s
o f the d a ta are pro vid ed in t a b u l a r form f o r a l l f l u i d s and g e o m e t r i e s .
Several geo metric e f f e c t s were obs er ved . The v e r t i c a l c o n f i g u ­
r a t i o n convected l e s s h e a t than the h o r i z o n t a l w hile a r o t a t i o n about
the v e r t i c a l a xis f o r each o f t h e s e c o n f i g u r a t i o n s had n e g l i g i b l e
e f f e c t . The r e s u l t i n g d e cr ea se in h e a t t r a n s f e r f o r t h e v e r t i c a l con­
f i g u r a t i o n was a t t r i b u t e d t o a complex i n t e r a c t i o n between th e boundary
l a y e r l e n g t h , the flow p a t t e r n s which r e s u l t e d from the geometry, and
th e cr os s s e c t i o n a l a r e a exposed t o th e upward flow. As th e Pr and tl
number of th e f l u i d media i n c r e a s e d i t tended t o damp out t h e geometric
e f f e c t s . Flow v i s u a l i z a t i o n s t u d i e s and te m p e ra t u re p r o f i l e s were used
t o a i d in e v a l u a t i n g the geo me tr ic e f f e c t s .
,
ix
NOMENCLATURE
I Symbol
A
AC
B
______________________ D e s c r i p t i o n _____________________
Total h e a t exchanger a r e a
Total a r e a o f th e c y l i n d e r s
D is tan ce t r a v e l e d by t h e boundary l a y e r on th e
i n n e r body
Specific heat a t constant pressure
Denotes f u n c t i o n
g
A c c e l e r a t i o n o f g r a v i t y , 9.81 m/sec
p
Grashof number, d e fi n e d by e q u a ti o n ( 2 . 2 )
Grx
4
2
Local Grashof number, g3qx /kv
h
Average he a t t r a n s f e r c o e f f i c i e n t , Qqqnv^ c^
k
Thermal c o n d u c t i v i t y
L
Gap width o r h y p o t h e t i c a l gap w id th , R -R.
I
Length o f s t r a i g h t c y l i n d r i c a l s e c t i o n
Nux
N u s s e l t number, h x /k , where x i s any c h a r a c t e r i s t i c
dimension
Pr
P ra ndt l number, CpP/k
qCOND
qCONV
qradiation
r
Heat t r a n s f e r by co nduction
Heat t r a n s f e r by co nvec tio n
Heat t r a n s f e r by r a d i a t i o n
Local p o s i t i o n o f thermocouple probe
2
3
Rayleigh number, p g|3(T..-To )x c / y k , where x i s
any c h a r a c t e r i s t i c dimension
X
Symbol
Description
Modified Rayleigh' number, Ra (L/R..)
I n n e r ( o u t e r ) body h y p o t h e t i c a l ra d iu s equal t o the
r a d i u s o f a sphere having an equal volume
Ri ( o )
Dimensionless r a d i u s d e fi n e d as R/R(9)
RND
R(G)
Length o f probe f o r r a d i a l p o s i t i o n 9
T
Local te m pe ratu re
Film t e m p e r a t u r e , I f, = (Tw + T00)/ 2
Tf
Dimensionless te m p e ra t u re d e fi n e d a s , (T-Tq ) / (Tj- -To)
1ND
Temperature of t h e i n n e r ( o u t e r ) body
TI ( 0 )
Temperature a t the o u t e r s u r f a c e o f any body
Tw
t w l I(O)
Ts
■
Temperature o f th e i n n e r ( o u t e r ) body as measured
by th e thermocouple in m i l l i v o l t s
Temperature of the s u p p o r t sphere
Tco
Temperature o f th e amb ie nt, f a r from the i n n e r body
X
Any c h a r a c t e r i s t i c dimension
6 .
Thermal expansion c o e f f i c i e n t
U
Dynamic v i s c o s i t y
V
Kinematic v i s c o s i t y
n
R at io of ci rcum ference of a c i r c l e t o th e d ia m e te r ,
3.14159
0
Temperature probe a n g u l a r l o c a t i o n as d e f i n e d on page 15
p
Density
CHAPTER I
INTRODUCTION
I n v e s t i g a t i o n in the a re a of h e a t t r a n s f e r by n a t u r a l co nvection
w it h in e n c l o s u r e s has i n c r e a s e d d r a m a t i c a l l y in the l a s t decade.
This
has been in response t o advances in e l e c t r i c a l pa c k a g in g , s o l a r h e a t ­
ing t e c hn ol og y, and i n c r e a s e d demands f o r hand lin g n u c l e a r w a s t e .
Most
r e c e n t l y th e demands f o r energy c o n s e r v a ti o n have i n c r e a s e d the impor­
ta n ce f o r a b e t t e r u n d e rs ta n d in g o f n a t u r a l c onvec tio n w i t h i n e n c l o ­
sures.
The purposes o f t h i s stu dy a re t o determine th e h e a t t r a n s f e r r e d
between a s e t of f o u r i s o t h e r m a l , h e a te d c y l i n d e r s ( m u l t i p l e bodi es)
and an i s o t h e r m a l , cooled cu bical e n c l o s u r e , t o determine the e f f e c t
o f th e p o s i t i o n o f t h e tube s w i t h i n t h e e n c l o s u r e , and to compare th e
r e s u l t s with the f i n d i n g s o f pr e vio us s t u d i e s on h e a t t r a n s f e r from
s i n g l e bodies t o th e same form o f e n c l o s u r e .
Four f l u i d s and i n n e r
body p o s i t i o n s are u t i l i z e d in s tu d y i n g th e h e a t t r a n s f e r r e d .
The body
p o s i t i o n s in c l u d e th e s e t o f c y l i n d e r s in both a h o r i z o n t a l and v e r t i ­
cal p o s i t i o n and in c l u d e a 45 degree r o t a t i o n about th e v e r t i c a l ax is
f o r each p o s i t i o n .
The f l u i d s used are a i r , w a t e r , 99 p e r c e n t g l y c e r i n
and a Dow-Coming 20 cs f l u i d .
The f l u i d s provide a P r a n d t l number
range o f 0 . 7 t o 31,000.
Although th e h e a t t r a n s f e r problem i s coupled w it h a f l u i d - f l o w
problem, the i n t e n t o f t h i s s tu dy i s d i r e c t e d p r i m a r i l y toward the h e a t
t r a n s f e r problem.
Flow v i s u a l i z a t i o n and te m p e ra t u re p r o f i l e s w i t h i n
•/
I
- Z -
t h e e n c l o s u r e a re o b t a i n e d t o a i d i n e v a l u a t i n g the h e a t t r a n s f e r .
CHAPTER' II
LITERATURE REVIEW
Heat t r a n s f e r by n a t u r a l c onvec tio n i s a f i e l d i n which s e v e r a l
subdivisions are e v id en t.
The m a j o r i t y o f the i n i t i a l s t u d i e s d e a l t
s o l e l y with h e a t t r a n s f e r from common shapes ( e . g . p l a t e s , c y l i n d e r s ,
s p h e r e s , e t c . ) t o a s u f f i c i e n t l y l a r g e su rr o u n d in g f l u i d medium as t o be
termed an
infinite
atmosphere.
F u r t h e r developments found a need t o
st udy th e n a t u r a l c o n v e c ti v e h e a t t r a n s f e r from one body to' an e n c l o s i n g
body through a f l u i d medium t h a t could n o t be t r e a t e d as i n f i n i t e . Both
o f th e major s u b d i v i s i o n s mentioned above have e x p e r ie n c e d emphasis on
t h e s e p a r a t e a r e a s o f h e a t t r a n s f e r from s i n g l e b o d i e s , s e r i e s o f l i k e
shapes u s u a l l y s y m m e tr ic a ll y p l a c e d , and o t h e r geo me tr ic e f f e c t s as
well as s p e c i a l i z e d f l u i d p r o p e r t i e s .
cu ss io n w i l l be covered i n two p h a s e s .
For purposes o f c l a r i t y t h i s d i s ­
These a re ( I ) n a t u r a l conv ectio n
t o an i n f i n i t e at mo sp here, and ( 2 ) n a t u r a l c onvec tio n i n e n c l o s u r e s ,
NATURAL CONVECTION TO AN INFINITE ATMOSPHERE
There have been many major developments i n r e s e a r c h i n t o t h e h e a t
t r a n s f e r r e d from a s i n g l e geo me tr ic shape t o an i n f i n i t e atmosphere.
Seve ral t e x t s pro v id e an e x c e l l e n t summary of t h e s e advanc es.
McAdams [1] pro v id e s a comprehensive review o f e a r l i e r r e s u l t s of
i n v e s t i g a t i o n s i n t o n a t u r a l c onv ect ion h e a t t r a n s f e r from s i n g l e
■
h o r i z o n t a l c y l i n d e r s t o an i n f i n i t e atmosphere with Rayleigh numbers
ra nging from 10 ~4 t o 10^.
Both Gebhart [2] and Holman [ 3 ] pro vid e
e x c e l l e n t r e f e r e n c e s f o r the he a t t r a n s f e r from h o r i z o n t a l and
v e r t i c a l c y l i n d e r s w ith Rayleigh numbers from 10“^ t o IO 1^ 9 well i n t o
the tu rb u le n t region.
I
In n e a r l y a l l cases t h r e e dim e n s io n le s s pa ra m et e rs a r e u t i l i z e d
t o c o r r e l a t e the h e a t t r a n s f e r by n a t u r a l c o n v e c t i o n .
The r e s u l t i n g
e q u a ti o n i s o f t h e form
Nux = f ( Grx , Pr )
(2.1)
in which t h e dim e n s io n le s s pa rame te rs a r e d e fi n e d as
_
G rx
""X
8 e t V - J - 1 x3_
.
12.2)
V2
( 2 .3 )
= HT-
and
Pr , - cEDy-
where x i s a c h a r a c t e r i s t i c l e n g t h .
(2.4)
S im p lific a tio n of the c o rre la tin g
e q u a ti o n has been accomplished by combining the Grashof number (Grx )
and t h e Pr a n d tl number (Pr) t o form a s i n g l e dim e n s io n le s s p a r a m e te r ,
the Rayleigh number, as f o l l o w s ;
Ra
= G r - Pr
(2.5)
The e q u a t i o n form most commonly u t i l i z e d t o c o r r e l a t e h e a t t r a n s f e r
I
data is
C2
Nux = C-j (Grx Pr) c
( 2 .6 )
F u j i i ,and. F u j i i [ 4 ] , Roy [ 5 ] , and C h u rc h il l and Ozoe [ 6 ] p r e s e n t s t u d i e s
t o e x p re ss
e x p l i c i t l y as a f u n c t i o n o f t h e P r an dtl number in the
e q u a ti o n
Nux =
(2 .7)
(Gr* P r ) ! / ^
f o r la m in a r flow along v e r t i c a l s u r f a c e s with uniform h e a t f l u x .
The
r e s u l t s were s a t i s f a c t o r y , but are l i m i t e d in a p p l i c a t i o n to t h i s p a r ­
tic u la r situation,
Lienhard [7] p r e s e n t e d t h e o r e t i c a l arguments t h a t r e g a r d l e s s of
sh ape , th e o v e r a l l h e a t t r a n s f e r c o r r e l a t i o n in t h e l a m in a r range can
be e x p re ss e d as
(NuxZRax ) ^ * = 1/2
f o r wholly immersed is ot he rm a l b o d ie s .
(2.8)
He recommended t h a t t h i s r e l a ­
t i o n s h i p be used with th e l e n g th o f t r a v e l of the boundary l a y e r as th e
c h a r a c t e r i s t i c le ngt h (x) to achieve r e s u l t s varying only by a few p e r ­
cent.
Ecke rt and Soehngen [ 8J r e s e a r c h e d th e e f f e c t s o f a v e r t i c a l s e r ­
i e s o f h o r i z o n t a l c y l i n d e r s upon each o t h e r .
U t i l i z i n g 2.231 cm diame­
t e r , is ot he rm a l c y l i n d e r s 5 they observed t h a t when two c y l i n d e r s were
, I
- 6 -
u t i l i z e d , one d i r e c t l y over the o t h e r a t a d i s t a n c e o f f o u r d i a m e t e r s ,
t h e r e .was no change in t h e N u s s e lt number o f t h e lo w e st c y l i n d e r as
oppo'sed t o a s i n g l e c y l i n d e r .
However, t h e upper c y l i n d e r evidenced
a r e d u c t i o n in h e a t t r a n s f e r r e d .
The r e s u l t i n g N u s s e l t number o f th e
upper c y l i n d e r was 87 p e r c e n t o f th e lower c y l i n d e r .
Sim ilar re s u lts
were achieve d when the s e r i e s ‘was i n c r e a s e d to t h r e e c y l i n d e r s .
The
N u s s e lt numbers were 100, 83, and 65 p e r c e n t , r e s p e c t i v e l y , from bottom
to top c y l i n d e r s .
When the middle c y l i n d e r was moved l a t e r a l l y out o f
l i n e by one h a l f a d i a m e t e r , th e N u s s e l t number o f t h a t ' c y l i n d e r ro se '
t o 103 p e r c e n t o f t h e bottom c y l i n d e r w hil e t h a t of. the. top c y l i n d e r '
was 86 p e r c e n t .
. ,
Ec ke rt and Soehngen reasoned t h a t th e e f f e c t o f t h e warmer wake
around t h e upper tubes, reduced th e h e a t t r a n s f e r r e d s i n c e th e tempera­
t u r e d i f f e r e n t i a l had d e c r e a s e d .
Con ver sel y, the s t a g g e r e d tube was
no t in th e n a t u r a l c onvec tio n plume and an induced f l u i d movement of
c o o l e r a i r r e s u l t e d in a g r e a t e r c a p a c i t y t o t r a n s f e r h e a t .
Analysis
o f th e method o f h e a t t r a n s f e r from th e top c y l i n d e r i n t h e s t a g g e r e d
a r r a y was more complex as i t was e f f e c t e d by both th e induced wake and
the co n v ec ti v e plume.
Leiberman and Gebhart [ 9 ] , in 1968, i n v e s t i g a t e d t h e i n t e r a c t i o n s
between th e n a t u r a l co n v ec ti v e flows o f s e v e r a l c l o s e l y spaced s u r f a c e s
by usi ng lo n g , h o r i z o n t a l wire s in a p a r a l l e l a r r a y a t s e v e r a l s pac ing s
and i n c l i n a t i o n s .
Other i n v e s t i g a t i o n s which i n c l u d e te m p e r a t u r e and
- 7 -
’
v e l o c i t y measurements about a l i n e sou rce have been conducted by BrodojWicz and Kierkus [ 1 0 ] , and Forstrom and Sparrow [ 1 1 ] .
All o f the above
i n v e s t i g a t o r s found t h a t cool a i r i s induced i n t o th e plume from the
s i d e s and below the s o u rc e .
The i n f l u e n c e of tube spacing and a r r a y on n a t u r a l con vec tion h e a t
t r a n s f e r c o e f f i c i e n t s f o r h o r i z o n t a l tu be bundles has been determined
e x p e r i m e n t a l l y by Tillman [ 1 2 ] .
Five p i t c h to. d ia m e te r (P/D) r a t i o s
were s t u d i e d f o r a sq uar e a r r a y o f 16 t u b e s , and f o u r (P/D) r a t i o s were
s t u d i e d f o r a s t a g g e r e d a r r a y o f 14 t u b e s .
The f o ll o w i n g e quat io ns
were developed to c o r r e l a t e th e d a ta :
Nuf = 0.057 (Gr P r ) f
0.5
(2 .9 )
f o r square a r r a y s , and
Nuf = 0.067 (Gr P r ) f 0,5
( 2 . 10 )
for. s t a g g e r e d a r ra y s, where the thermal p r o p e r t i e s o f . t h e f l u i d were
e v a l u a t e d a t t h e f i l m tem pe rat ure
Tf
e x c e p t g which was e v a l u a t e d a t th e ambient te m p e r a t u r e .
(2.11
'The c h a r a c ­
t e r i s t i c dimension was d e fi n e d as
( 2 . 12)
- 8 -
i n which
*
i s the flow cross s e c t i o n a r e a , A i s t h e t o t a l h e a t ex-
, changer a r e a , and x i s th e h e i g h t o f th e tube bundle.
Conclusions
drawn by Tillman i n d i c a t e t h a t tube s pac in g has more e f f e c t on the h e a t
t r a n s f e r than t h e type o f a r r a y .
A ls o, an optimum s pa c in g f o r each
a r r a y was determined f o r s e p a r a t e te m p e ra t u re d i f f e r e n t i a l s .
Natural c onvec tio n from v e r t i c a l tube bundles has been re s e a r c h e d
by Davis and Perona [ 1 3 ] .
They u t i l i z e d 42 tubes having an o u t s i d e
d ia m e te r o f 1.58 cm and a le n g th o f 1.23 cm a rr a nged i n 7 s t a g g e r e d
rows o f 6 tubes p e r row.
Experimental r e s u l t s were compared with t h e o ­
r e t i c a l r e s u l t s o b ta in e d from a f i n i t e d i f f e r e n c e s o l u t i o n of t h e pro b­
lem.
The r e s u l t s o b t a i n e d from both exper imen ta l d a ta and t h e o r e t i c a l
c a l c u l a t i o n s c o r r e l a t e d e xtr em el y well with the e x c e p t i o n o f th e val ues
in th e re gi on where th e end s u p p o r t system had an a p p a r e n t i n f l u e n c e .
NATURAL CONVECTION IN ENCLOSURES
Natural c onvec tio n in e n c l o s u r e s , in t h i s d i s c u s s i o n , w i l l be
r e s t r i c t e d t o h e a t t r a n s f e r r e d from one body o r bodies completely en ­
c lo s e d by a second body.
With t h i s r e s t r i c t i o n imposed, one f i n d s an
a r e a of i n v e s t i g a t i o n t h a t has b a r e l y been i n i t i a t e d .
The primary em­
p h a s i s t o d a te has been with s i n g l e bodies g e o m e t r i c a l l y c e n t e r e d w i t h ­
i n a second e n c l o s i n g body.
As d i s c u s s e d p r e v i o u s l y , s e v e r a l dim e nsi onle ss p a ra m et e rs were
u t i l i z e d in c o r r e l a t i n g the h e a t t r a n s f e r d a t a f o r n a t u r a l convection
- 9 -
t o an i n f i n i t e atmosphere.
AT though t h e s e para mete rs remain v a l i d f o r
e n c l o s u r e s , a t h i r d elem ent was found h e l p f u l , i f n o t n e c e s s a r y , which
a l t e r s e q u a ti o n ( 2 . 1 ) t o the form
Nux = f(Grx , Pr , Nd )
(2 .13)
where Nd i s a r a t i o o f c h a r a c t e r i s t i c dimensions.
There a re s e v e r a l so urc es o f i n f o r m a t i o n on h e a t t r a n s f e r from a
s i n g l e body t o i t s e n c l o s u r e .
Most r e c e n t l y , Warrington [14] developed
e q u a ti o n s from exper imen ta l da ta which a re use fu l i n c o r r e l a t i n g h e a t
t r a n s f e r between i n n e r
bodies u t i l i z i n g s p h e r e s , c u b e s , and c y l i n d e r s
o f varying s i z e s t o both s p h e r i c a l and c ub ic a l e n c l o s u r e s using da ta
o b t a i n e d by Bishop [ 1 8 ] , Sca n!a n, e t aj_ [19] and o t h e r major i n v e s t i ­
g a t i o n s , as well as c o n s i d e r a b l e d a t a o f h is own.
The f o ll o w i n g c o r r e ­
lation:
Nul = 0.396RaL0,234
(_L_)0.496 p r 0.162
was found t o a d e q u a te ly d e s c r i b e th e h e a t t r a n s f e r r e d f o r 9 d i f f e r e n t ,
s i z e s o f s p h e r i c a l , 11 c y l i n d r i c a l , and 6 cu bic al i n n e r bodies when _
e nc lo s e d by a s p h e r i c a l o r c ubic a l o u t e r body.
Several d i f f e r e n t
f l u i d s were u t i l i z e d t o ext end th e Pr a n d tl number range from 0.706 t o
13,800.
The average p e r c e n t d e v i a t i o n was 13.50%.
C o r r e l a t i o n s are
a l s o given f o r s e p a r a t e combinations o f f l u i d s , i n n e r , and o u t e r b o d i e s .
Warrington Cl4] a l s o pro vi de s an e x h a u s t i v e review of t h e l i t e r a t u r e
■I
- 1 0 - '
p e r t a i n i n g t o h e a t t r a n s f e r from a s i n g l e ge om et ric shape t o i t s en. closure.
Kuehh and G ol ds te in [15] a n a l y t i c a l l y developed a c o r r e l a t i n g
equation fo r horizontal concentric c y lin d e rs.
U t i l i z i n g th e e x p e r i ­
mental r e s u l t s of e i g h t s t u d i e s by o t h e r a uth ors i n which th e r a t i o o f
o u t e r t o i n n e r c y l i n d r i c a l diame ters (DqZD1- ) was 2 and 3 , and a Pr and tl
number o f 0 . 7 , th e y found t h a t th e a n a l y t i c a l model f i t t h e d a t a w i t h i n
a few p e r c e n t .
Inc lu ded in t h i s s tu d y were two major e x t e n s i o n s t o t h e
c o n c e n t r i c c y l i n d e r problem.
F i r s t was an a n a l y t i c a l d e t e r m i n a t i o n of
t h e gap width a t which t h e s o l u t i o n was th e same as t h a t f o r a s i n g l e
cylinder.
I t was found t h a t when DQ/D^ > 360 a t R a ^ = IO7 and
DqZD1->
= 10""\ t h e h e a t t r a n s f e r c o e f f i c i e n t was w i t h i n 5% o f t h a t
i
for a single cylinder.
The second e x t e n s i o n of t h e i r a n a l y t i c a l model
700 a t Ran
was the s o l u t i o n t o th e problem of m u l t i p l e c y l i n d e r s c o n ta i n e d in a
sin g le c y lin d ric a l enclosure.
For t h i s case the c o r r e l a t i n g eq uat io n
became
2N
Nu =
(2. 15)
l n [ l+2ZNu1] - N ln[l -2Z Nu Q]
where th e N u s s e l t number o f th e i n n e r tube boundary c o n d i t i o n s i s
Nu1 = [( 0.518RaDV 4 [ l + (
( 2.1 5)
-T l
and th e o u t e r tube N u s s e l t number i s
Nu0 = ' { ( [ ( ------ ------:)5 /3 +(0.587Ra1 / 4 ) ] 3 / 5 ) 15+( 0.1Ra 1 /3 ) 15}1/15
T -0 .2 5
D
D
1-e
o
o
(2.17)
for a s e t of N inner c y lin d e rs.
f o r small numbers o f i n n e r t u b e s .
This e q u a ti o n i s e x p e c t e d . t o be v a l i d
Although th e a u th o rs are not s p e c i f i c
as t o th e term s m a l l , they le av e the im pre ssi on t h a t more than 4 o r 5
c y l i n d e r s would r e s u l t in e r r o r .
No o t h e r h e a t t r a n s f e r d a t a could be
found f o r t h i s c o n f i g u r a t i o n and, t h e r e f o r e , i t has n o t been t e s t e d f o r
val i di t y .
The s tu dy o f v e r t i c a l tube bundles i n e n c l o s u r e s has been mainly
w i t h i n th e low Pr a n d tl number range ( l i q u i d m e t a l s ) .
Dutton and
Welty [16] r e s e a r c h e d t h e e f f e c t s of c y l i n d e r s paci ng on t h e h e a t
t r a n s f e r r e d from a v e r t i c a l rod bundle in a v e r t i c a l c y l i n d r i c a l e n c l o ­
s u r e u t i l i z i n g l i q u i d mercury as t h e f l u i d medium, Pr = 0 . 0 2 3 , with a
uniform h e a t f l u x a p p l i e d t o the rod bundle .
t u r e d i s t r i b u t i o n s were a l s o measured.
Axial and r a d i a l tempera­
R es ults o f t h i s s tu d y showed
t h a t t h e r e i s a s t r o n g dependence on c y l i n d e r s paci ng and t h a t the
e q u a ti o n form f o r use in s t u d i e s a t low P ra nd tl numbers s hould in v o lv e
th e d im e ns io nle s s p a ra m e te r p ro du c t d e f i n e d as
Gr* • Pr = ( g p q x V k a 2 )
(2.18)
where q i s a uniform h e a t f l u x .
The stu dy o f n a t u r a l co nvection i n e n c l o s u r e s as evi den ce d by th e
.
- 12 -
l i t e r a t u r e has i n c r e a s e d in the l a s t decade.
That p o r t i o n which i n ­
cludes th e use o f . m u lt ip l e bodies s t i l l la cks a p p r e c i a b l e knowledge.
I
This s tu d y i s in te n d e d t o p ro vid e an e x t e n s i o n o f t h i s a r e a .
CHAPTER I II
EXPERIMENTAL APPARATUS AND PROCEDURE
|
EXPERIMENTAL APPARATUS
The a p p a r a t u s f o r t h i s i n v e s t i g a t i o n c o n s i s t e d o f a w a te r j a c k e t e d
c ub ic a l o u t e r body$ c o o li n g system, power s o u r c e , and a f o u r c y l i n d e r
i n n e r body with s u p p o r t i n g e l e m e n t s „
One o u t e r body system was used
t o o b t a i n h e a t t r a n s f e r d a t a and te m p e r a t u r e p r o f i l e s w hil e a s e p a r a t e
system was u t i l i z e d to photograph flow p a t t e r n s .
The assembled o u t e r "
body and p e r i p h e r a l components a re shown i n Figure 3 J *
The o u t e r body in
which t e s t s were conducted t o de ter mi ne the
h e a t t r a n s f e r was c o n s t r u c t e d from 1.27 cm t h i c k , t y p e 6061 aluminum
with an i n n e r 26.67 cm c ubi ca l chamber.
This was a j a c k e t e d design
c o n s i s t i n g o f a s e p a r a t e 3.175 cm wide r e c t a n g u l a r channel f o r each
f a c e o f th e cube.
Access to th e t e s t chamber was p r ov id e d through
a 25 .4 cm removable c i r c u l a r p l a t e on
p l e t e l y removable o u t e r f a c e .
the top i n n e r f a c e and a com­
A c lo s e d system c o n s i s t i n g o f a c h i l l e r ,
pump, and s t o r a g e r e s e r v o i r prov ide d w a t e r t o cool t h e o u t e r body.
Z
The flow r a t e o f c o o l i n g w a t e r through each channel was c o n t r o l l e d
by a valve which fe d f o u r i n l e t and o u t l e t p o r t s .
This arrangement
a s s u r e d uniform flow along the e n t i r e fa c e and a l s o allowed the temp era­
t u r e o f each face t o be c o n t r o l l e d i n d e p e n d e n tl y in o r d e r t o achieve an
is o th e rm a l o u t e r body.
The te m p e ra t u re of th e i n n e r wall was monitored
by 34 copper c o n s ta n ta n thermocouples epoxied 0.3175 cm from the i n n e r
- 14 -
Fig ur e 3.1
Heat T r a n s f e r Apparatus
. - 15 -
face.
The thermocouples f o r each f a c e o f t h e cube were connected in
p a r a l l e l which p r ovi de d an average te m p e ra t u re f o r each f a c e . .
By con­
t r o l l i n g the flow o f th e c o o li n g w a t e r , a l l fa c es could be main ta ine d
w i t h i n 2 K.
To o b t a i n te m p e r a t u r e p r o f i l e s w i t h i n th e t e s t chamber, th e o u t e r
body was designed with nine thermocouple p o r t s .
These c o n s i s t e d o f
one common and f o u r a d d i t i o n a l p o r t s on each o f two s e p a r a t e a x i s .
One a x i s was on a v e r t i c a l pla ne through t h e c e n t e r o f t h e cube while
t h e o t h e r was t h e v e r t i c a l pla ne through t h e edge o f t h e c ube .
The
f i v e p o r t s in each a x is were a t O05 3 4°, 8 0 ° , 120°, and 160° measured
downward from the top c e n t e r v e r t i c a l a x is of the body.
Each thermo­
couple p o r t had a c e n t e r tube which moved through a f i x e d p o r t t u b e .
The c e n t e r tube was a 0.1587 cm d ia m e te r s t a i n l e s s s t e e l tube which
c a r r i e d the copper-cons t a n t a n thermocouple epoxied t o t h e i n n e r end.
The outer, s l e e v e was a 1.016 cm d ia m e te r s t a i n l e s s s t e e l tube t h re a d e d
i n t o th e i n n e r j a c k e t w a l l , and s e a l e d in both j a c k e t w a l l s with r u b ­
ber 0 -rin g s.
The thermocouple l e a d tube was s e a l e d w i t h i n t h e s l e e v e
by a Conax f i t t i n g a t t a c h e d t o th e o u t e r end.
A v e r n i e r c a l i p e r was
m odif ied t o a t t a c h t o the Conax f i t t i n g w hil e th e thermocouple tube
was a f f i x e d t o t h e s l i d i n g s c a l e .
This p e r m i t t e d t h e l o c a t i o n o f the
thermocouple t o be e s t a b l i s h e d with 0.0025 cm.
The o u t e r body used f o r flow v i s u a l i z a t i o n was n e a r l y i d e n t i c a l
t o t h e one p r e v i o u s l y d e s c r i b e d .
The major d i f f e r e n c e was t h a t . t h e
body was c o n s t r u c t e d from a c l e a r p o ly v in y l and a l l p o r t i o n s were
p a i n t e d bla ck e x c e p t f o r a l i g h t so ur c e s l o t p la ce d v e r t i c a l l y on one
s i d e and one c l e a r f a c e which allowed photographs t o . b e ta ken o f the
pl a ne which was i l l u m i n a t e d by the l i g h t s o u r c e .
An i d e n t i c a l coo lin g
system was i n s t a l l e d and t h e m on ito ri ng and pow er.devi ces were s la v e d
from the h e a t t r a n s f e r system!
The i n n e r body de sign c o n s i s t e d o f f o u r i d e n t i c a l c y l i n d e r s and a
s u p p o r t i n g system which allowed i t to be s y m m e tr ic al l y c e n t e r e d in the
i n n e r body.
Figures 3.2 and 3 .3 show th e i n n e r body in d i f f e r e n t s t a g e s
o f c o n s t r u c t i o n -.
Copper tube (CDA #122) 17.78 cm long and 4.115 cm
o u t s i d e d ia m e te r Was used t o c o n s t r u c t each c y l i n d e r .
Each end o f th e
tube was t h r e a d e d and end caps were c o n s t r u c t e d from copper p l a t e
(CDA #110) t o c l o s e the en ds . ' I n o r d e r t o p ro vid e a s u p p o r t system,
rods' c o n s t r u c t e d from type 304 s t a i n l e s s s t e e l 4.191 cm long and
0.9525 cm.O.D,were t h r e a d e d i n t o each c y l i n d e r a t i t s mi dlength and
then connected t o a 5.1 cm d ia m e te r p o ly c a r b o n a te s p h e r e .
The r e s u l t i n g i n n e r body c o n f i g u r a t i o n was f o u r p a r a l l e l c y l i n d e r s
in a square a r r a y 12.192 cm c e n t e r t o c e n t e r .
The f i n a l assembly was
then p l a c e d sy m m e tr ic al l y c e n t e r e d in t h e o u t e r body s u p p o rt e d by a
main stem c o n s t r u c t e d from a 1.27 cm 0 . D..type 304 s t a i n l e s s s t e e l ro d .
This s u p p o rt stem was designe d to t h r e a d i n t o th e s u p p o r t sph ere so t h a t
th e e n t i r e tube system could be s u p p o rt e d e i t h e r on a h o r i z o n t a l o r v e r ­
t i c a l a x i s with t h e unused tap being plugged when n o t i n us e.
To
- 17
Figure 3 .2
S in g le Cylinder with Heat
Tape I n s t a lle d
Figure 3 .3
Inner Body P a r t i a l l y Assembled
. -
18 -
minimize c onvec tio n l o s s e s from a l l s t a i n l e s s s t e e l t u b i n g , th e tubes
were covered with a s h r i n k tube having a low thermal c o n d u c t i v i t y .
The
s u p p o r t stem pass ed through the j a c k e t w a l l s o f t h e bottom f a c e o f th e
o u t e r body and was s e a l e d with ru bbe r 0 - r i n g s .
i
A d i r e c t c u r r e n t power so urc e and p r e s s u r e s e n s i t i v e h e a t tape
pr ovi de d th e h e a t i n g f o r th e i n n e r body.
The he a t t a p e , 0,064 cm
t h i c k and 0.32 cm w i d e , allowed f o r a maximum o f 62 w a tt s p e r f o o t un­
de r o p e r a t i n g c o n d i t i o n s o f from 60 t o 533 degrees K.
The tape was
a p p l i e d t o th e e n t i r e i n n e r s u r f a c e o f each tube and on th e end caps
u t i l i z i n g two s t r i p s ap pr ox im at el y 2 . 4 meters long with each s t a r t i n g
a t t h e m idp oin t o f t h e tube and wrapped toward t h e end where i t t e r m i ­
n a t e d by c o n c e n t r i c wrapping on th e end cap.
Each end o f th e h e a t type
was connected t o power le ad s which were c a r r i e d i n s i d e th e s u p p o rt
tubing.
Since each h a l f tube had s e p a r a t e power s o u r c e s , i t was p o s s i ­
b l e t o keep each c y l i n d e r is o th e rm a l when i n t h e v e r t i c a l p o s i t i o n .
Als o, a high thermal c o n d u c t i v i t y copper m a t e r i a l f o r t h e tubes was
chosen t o f a c i l i t a t e i s o t h e r m a l i t y .
Once the h e a t t a p e s were in p l a c e ,
th e y were c oat e d with a s i l i c o n e caulk which ke pt them s e c u r e l y in
p la c e and pro vi de d an i n s u l a t e d backing and t h e rem ai nder o f t h e tube
opening was f i l l e d w ith f i b e r g l a s s to e nsu re t h a t c o n v e c t i v e c u r r e n t s
would n o t induce an uneven te m p e ra t u re d i s t r i b u t i o n on t h e t u b e s .
Each
of t h e e i g h t h e a t ta pe i n p u t le ad s was connected t o e i t h e r a 2 .0 or 0.5
amp ammeter (depending on the i n p u t power) then through v a r i a b l e power
- -19 - ■
r e s i s t o r s (35 ohm, 150 w a t t , 207 amp) and f i n a l l y to th e D.C. power
sources.
Voltages were re a d on a d i g i t a l v o l t m e t e r connected through
a m u l t i p l e p o s i t i o n swi tch t o the low s i d e o f each v a r i a b l e r e s i s t o r .
The te m p e ra t u re o f each h a l f tube was monitored by c o p p e r - c o n s t a n ta n thermocouples epox ied i n t o th e wall o f th e tube a pprox im at el y
0.025 cm from the o u t e r s u r f a c e s .
The two thermocouples p e r tube were
p la c e d on o p p o s i t e s i d e s o f th e c y l i n d e r a t th e o n e - q u a r t e r and t h r e e quarter lengths.
The e n t i r e system was assembled so t h a t no two a d j a ­
c e n t tu be s had thermocouples in th e same p o s i t i o n .
The- l e a d s t o th e
thermocouples were p a ss e d through th e s u p p o r t stem system and s e p a ­
r a t e l y connected t o a te m p e ra t u re r e f e r e n c e source and d i g i t a l miI l i voltmeter.
EXPERIMENTAL PROCEDURE
The f o ll o w i n g sequence o u t l i n e s t h e procedure f o r a c q u i s i t i o n o f
.
;
-'
h e a t t r a n s f e r d a t a , te m p e ra t u re p r o f i l e s , and v i s u a l i z a t i o n d a t a . In
each case th e i n n e r body was c e n t e r e d i n t h e t e s t chamber and th e p e r i ­
p h e ra l c o o li n g system was a c t i v a t e d .
The c o o li n g was mo nit or ed u n t i l
t h e o u t e r body became is o th e rm a l and e q u i l i b r i u m was e s t a b l i s h e d .
Al­
though e q u i l i b r i u m could be e s t a b l i s h e d w i t h i n 6 t o 8 h o u r s , a lon ge r
time was a l l o t t e d (24 hours) i n o r d e r . t o f u l l y t e s t and check th e s e a l s
on th e i n n e r chamber.
When i t was det ermi ne d t h a t t h e t e s t chamber was
o p e r a t i n g s a t i s f a c t o r i l y , i t was f i l l e d with a t e s t f l u i d , and the
- 20 -
d e s i r e d power (depending on f l u i d and d e s i r e d te m p e r a t u r e d i f f e r e n c e )
was a p p l i e d and a d j u s t e d to b r in g th e i n n e r body t o i s o th e r m a l e q u i ­
librium.
The h e a t t r a n s f e r d a t a p o i n t s r e q u i r e d the f o ll o w i n g measurements:
(1)
In p u t v o l t a g e
(2)
Inp ut amperage
(3)
In n e r body te m p e ra t u re
(4)
Outer body te mp erat ure
which were c o l l e c t e d f o r each power s e t t i n g .
U t i l i z i n g 16 d i f f e r e n t
f l u i d / g e o m e t r y 'co mbinations, 166 d a ta p o i n t s were o b t a i n e d ,
A listing
o f t h e f l u i d s , body p o s i t i o n s , and t h e p a r t i a l l y reduced d a t a i s p r o ­
vided in Appendix I I .
Temperature p r o f i l e s were always ta ken in con­
j u n c t i o n with s e l e c t e d d a ta r u n s .
Each o f t h e nine probes was p o s i ­
t i o n e d , in t u r n , a g a i n s t th e i n n e r body ( r e f e r e n c e p o s i t i o n ) and were
w ith d ra w n .i n incre men ts ra ngi ng from 0.013 t o 1.27 cm depending on the
p r e d i c t e d te m p e ra t u re g r a d i e n t .
P o s i t i o n and thermocouple re a d in g s
were re c o rd e d f o r 15 te m p e ra t u re p r o f i l e s e t s .
The above i n f o r m a t i o n
was reduced by a computer program which co nver te d thermocouple m i l l i ­
v o l t re a di ng s t o
K and c a l c u l a t e d power i n p u t in w a t t s .
s t u d i e s were conducted with a i r and t h e 20 cs f l u i d .
V isualization
Smoke, with a i r ,
and f l u o r e s c e n t p a i n t p a r t i c l e s mixed with t h e 20 cs f l u i d , pro vided
the t r a c e r p a r t i c l e s needed f o r t a k i n g t h e p h o to g r a p h ,
The same p r o ­
cedures f o r t h e v i s u a l i z a t i o n a p p a r a t u s were used as o u t l i n e d in the
21
heat t r a n s f e r data c o lle c tio n .
In o r d e r t o minimize e r r o r s i n d a t a , s e v e r a l p r e c a u t i o n s were
ob ser ved .
All i n s t r u m e n t a t i o n was c a l i b r a t e d twice d urin g t h e t e s t i n g
procedures.
Once p r i o r t o i n i t i a l t e s t i n g o f the body in th e h o r iz o n -
t a l c o n f i g u r a t i o n an d, f i n a l l y , p r i o r to t e s t s in the v e r t i c a l p o s i t i o n ,
Power and co ol in g w a t e r c o n t r o l s were a d j u s t e d f o r each' data run so
t h a t t h e v a r i a t i o n i n thermocouple r e a d in g s in m i l l i v o l t s (TMVL) f o r
t h e i n n e r ( I ) ( o u t e r ( 0 ).) body met t h e fo ll o w i n g r e q u ir e m e n ts :
^ (O linaX " tmvlU O W
TMVLt
- TMVLn
i Itrin
umax
=
(3.1)
10%
The average v a r i a t i o n i n te m p e ra t u re f o r e i t h e r the i n n e r or o u t e r body
was w i t h i n 4%, with a maximum v a r i a t i o n o f 15%.
The h e a t t r a n s f e r r e d by n a t u r a l c onvec tio n from th e tu be s was c a l X
c u l a t e d by
q CONV = qTOTAL - QRADIATI0N - QC0NDUCTED
.
(3 .2 )
Of th e f o u r t e s t f l u i d s w a t e r , 99% g l y c e r i n , and th e 20 cs f l u i d were
opaque t o r a d i a t i o n and t h e h e a t lo s s due t o r a d i a t i o n was needed only
for a ir .
The procedure used t o account f o r th e h e a t l o s s due to the
i n n e r body s u p p o r t system proved t o be complex and i s , t h e r e f o r e ,
covered in a s e p a r a t e d i s c u s s i o n provide d in Appendix I .
d i s c u s s i o n on r a d i a t i o n lo s s w i l l be p r e s e n t e d h e r e .
Only the
This procedure
a l s o accounts f o r conduction l o s s e s through the main s u p p o r t stem when
I
i
- 22 -
I
a i r i s used -as the working f l u i d . '
The h e a t t r a n s f e r r e d by r a d i a t i o n was determined e x p e r i m e n t a l l y
using th e fo ll o w i n g p ro c e dur e.
The complete i n n e r - o u t e r body system
was assembled and a l l thermocouple p o r t s were plugged with s o l i d w i r e s ,
A vacuum pump was a t t a c h e d t o th e access tube which had been, used to
f i l l the t e s t chamber with th e t e s t f l u i d s and t h e chamber was evacua­
t e d t o l e s s than 10 mic ron s.
A minimum of 15 d a t a p o i n t s were c o l ­
l e c t e d over th e range o f AT1S used in t h i s s t u d y .
These p o i n t s were
used in a l e a s t sq uar es curve f i t to develop an e q u a ti o n which could
be used t o c o r r e c t a l l a i r d a ta f o r r a d i a t i o n and conduction l o s s e s .
The d a t a p o i n t s f i t the e q u a ti o n with an average d e v i a t i o n o f 1.49%
and a maximum d e v i a t i o n o f 3.73%.
A second r a d i a t i o n c a l i b r a t i o n was
made f o r th e v e r t i c a l c o n f i g u r a t i o n .
Having determined t h e h e a t t r a n s f e r by n a t u r a l co nvec tio n from
th e f o u r c y l i n d e r s as i n d i c a t e d in e q u a ti o n ( 3 . 2 ) , a f i n a l da ta re d u c ­
t i o n f a c i l i t a t e d the development o f c o r r e l a t i n g e q u a t i o n s .
The h e a t ■
t r a n s f e r (Qco^y) allowed the computation o f th e average h e a t t r a n s f e r
c o e f f i c i e n t (h) d e f i n e d as f o l l o w s :
h
Qcqnv
Ac(Ti-To)
(3 .3 )
- 23 -
The f l u i d p r o p e r t i e s were c a l c u l a t e d using the a r i t h m e t i c mean temperature
TI + T0
(3.4)
Tavg
and th e computer f u n c t i o n s u b r o u t i n e s l i s t e d in Appendix I o f r e f e r e n c e
[14].
An e x c e p ti o n t o t h e above was th e c a l c u l a t i o n of th e p r o p e r t i e s
o f 99% g l y c e r i n e which were taken from c u r r e n t l i t e r a t u r e .
The s p e c i f i c
h e a t f o r 100% g l y c e r i n e was used as d a ta f o r 99% was n o t a v a i l a b l e .
observed by Warrington [ 1 4 ] , t h i s did n o t a f f e c t th e r e s u l t s .
As
CHAPTER IV
DISCUSSION OF RESULTS
Heat t r a n s f e r r e s u l t s a re d i s c u s s e d in terms o f t h e dimensionless,
pa rame te rs d e fi n e d in e q u a ti o n s (2 .2 ) through (2 .4 ) and c a l c u l a t e d from
th e d a t a a c q u i r e d from th e proce dures o u t l i n e d in Chapter I I I .
Further
c l a r i f i c a t i o n must be made here s i n c e the Grashof and N u s s e l t numbers
were c a l c u l a t e d using two d i f f e r e n t c h a r a c t e r i s t i c dim e ns io ns.
A hypo­
t h e t i c a l gap width which has been proven t o be a r e l i a b l e c h a r a c t e r i s t i c
dimension in p a s t i n v e s t i g a t i o n s [14] was used and i s d e f i n e d as
\
(4 .1 )
L = R0 -R1
where R0 (Ri) i s t h e r a d i u s o f a sp her e having th e same volume as the
o u t e r ( i n n e r ) body.
entations,
Since the i n n e r body was t e s t e d in two d i f f e r e n t o r i ­
a c h a r a c t e r i s t i c dimension based on the boundary l a y e r
le ng th was d e fi n e d f o r each.
For th e h o r i z o n t a l p o s i t i o n of the c y l i n ­
d e r s , th e boundary l a y e r l e n g t h was d e f i n e d as
B = JL (Diameter)
( 4 .2 )
2
t
and f o r th e v e r t i c a l p o s i t i o n
B = (Height + Diameter)
( 4 .3 )
with the assumption t h a t f o r the Rayleigh number range in v o lv e d in t h i s
s t u d y , t h e r e would be no s e p a r a t i o n o f the boundary l a y e r from the body.
The ranges o f t h e s e para mete rs are given in Table 4 . I .
Through flow
TABLE 4.1
RANGES OF THE DIMENSIONLESS PARAMETERS
FLUID
Nub
Rag
5.164X105
33.50
4.225X10?
4.831X1O6 5.516X10? 0.7121
7.889
6.353X105
• 10.57
2.766X1Q5
8.933X105 3.889X1O^ 0.7086
MAX
42.17
I .021X108
54.83
MIN
10.28
3.206X105
MAX
65.02
16.918X108
MIN
33.65
I 3.639X10?
■ MAX '
65.92
6 .887X108
MIN
37.50
13.013X10?
RANGE
Nul
MAX
21.18
MIN
RaL
Grl
Pr
6rB
AIR
3.748X1Q8 . I .936X105 3.419X105 3.13X104
GLYCERIN
6.353
114.8
7.566X104
I .449X1O1 3.427
I .62X1O3
4 . GllXlO 9 ' I .439X108 8 . 43OXI08 10.45
I WATER
8.586X1O^
3.828X106 9.053X1O^ 5.658
5.173X109
5.508X106
7 . 125X106
9.840X104 2.328X1O4 ' 187.3
,
X
O
CO
138.0
CD
LO
I
21.23
326.0
I SILICONE
23.23
/
,
- 26 -
v i s u a l i z a t i o n r e s u l t s t h i s assumption was found t o be a c c u r a t e and w i l l
be d i s c u s s e d l a t e r .
Reference to pa rame te rs which r e q u i r e a c h a r a c t e r ­
i s t i c dimension w i l l be s u b s c r i p t e d L o r B t o denote th e a p p r o p r i a t e
dimension.
Several e q u a ti o n forms were used t o c o r r e l a t e th e h e a t t r a n s f e r
data.
The f o ll o w i n g f o u r e q u a ti o n forms were found t o c o n s i s t e n t l y
p ro vid e t h e b e s t r e s u l t s and are used t o p r e s e n t th e r e s u l t s o f t h i s
st udy :
Nu% = Ci Ra%
(4.4)
Nu% = Ci Ra* %
( 4 .5 )
Nu% = Ci
( 4 .6 )
Nux = Ci Grx
( 4 .7 )
Pr
where t h e c h a r a c t e r i s t i c dimension (x) could be s u b s c r i p t e d e i t h e r L
or B.
Equation ( 4 . 5 ) i s in terms o f a modified Rayleigh number
'
( 4 .8 )
The c o n s t a n t s Ci through C3 were deter mine d by usi ng a s t a n d a r d l e a s t
sq uar es curve f i t .
Geometric e f f e c t s were-found' t o be s i g n i f i c a n t and were g e n e r a l l y
c o n s i s t e n t among th e d i f f e r e n t f l u i d s .
The remainder o f th e d i s c u s s i o n
1
on r e s u l t s w i l l be covered in th e o r d e r o f geometric e f f e c t s , h e a t
.
t r a n s f e r r e s u l t s f o r each f l u i d , and h e a t t r a n s f e r r e s u l t s , f o r a l l dat a
combined.
Temperature p r o f i l e s , flow v i s u a l i z a t i o n s , and comparisons
with s i n g l e body, s t u d i e s w i l l be i n c lu d e d in each s e c t i o n as they are
applicable.
GEOMETRIC EFFECTS ON HEAT TRANSFER
When the i n n e r body was changed from th e h o r i z o n t a l t o t h e v e r t i c a l
c o n f i g u r a t i o n t h e r e was a s i g n i f i c a n t d e cr ea se in the h e a t t r a n s f e r r e d .
This d e cr ea se was e x h i b i t e d f o r a l l f l u i d s , however, th e r e l a t i v e magni­
tude v a r i e d f o r each f l u i d .
The d e cr ea se in h e a t t r a n s f e r between th e
two geometries i s most pronounced in t h e low v i s c o s i t y range ( a i r ) and
becomes almost n e g l i g i b l e in th e high v i s c o s i t y - range ( g l y c e r i n e ) as
seen in Figure 4 . 1 .
The r e s u l t i n g d e c r e a s e was 28%, 8%, 8%, and 2% f o r
a i r , w a t e r , s i l i c o n e , and g l y c e r i n , r e s p e c t i v e l y , (in o r d e r o f i n c r e a s ­
ing Pr a n d tl number).
This s u g g e s ts t h a t the f l u i d v i s c o s i t y tends to
damp o u t geo metric e f f e c t s as was p o s t u l a t e d by Weber [17]..
However,
Warrington [14]. found no vis co us e f f e c t on changing the dia m e te r o f a
sp her e in the e n c l o s u r e .
This i s p o s s i b l y because h i s geometric change
was n o t as r a d i c a l as t h a t p r e s e n t e d h e r e .
The e f f e c t s o f t h e h e a t t r a n s f e r r e d by t h e lower c y l i n d e r on t h e .
1000
o - G ly cer in e
v - Air
& - Water
100
□ - Silicone
ff4.Q8c
c&m
Nul
o e°*
COe1
o y g°e
..
Vr f< X
t W^
e Q
Open Symbols - H ori zo nta l
Closed symbols - V e r t i c a l
........ i
Figure 4.1
I
I
I
I I I ! I !_ _ _ _ _ _ _ _ I
Nu, v e rs us Gr^ f o r All Data
I
I
I I I I I I_ _ _ _ _ _ _ _ I
I
I
I I I ! I !
I
I I I I I I
I
I
I
I I I I I I
- 29 -
upper c y l i n d e r in t h e h o r i z o n t a l p o s i t i o n a re s i m i l a r t o th o s e observed
by Ec ke rt and Soehngen [ 8] .
The lower c y l i n d e r p re h e a te d th e f l u i d r e ­
s u l t i n g in a reduced c a p a c i t y o f the f l u i d t o t r a n s f e r h e a t as the
v e l o c i t y f i e l d c a r r i e d i t around t h e upper c y l i n d e r .
The reduced ca pa c ­
i t y f o r h e a t t r a n s f e r in the f l u i d i s a d i r e c t r e s u l t o f th e lower AT
or driving force fo r heat t r a n s f e r .
When t h e average N u s s e l t number f o r
th e upper c y l i n d e r was compared t o t h e lo w e r, th e r e s u l t s were 84% o f
th e bottom c y l i n d e r f o r a i r , 70% f o r g l y c e r i n , 89% f o r w a t e r , and 75%
for silicone.
Eck ert and Soehngen [ 8] r e c ord ed 87% o f the bottom c y l i n ­
der usi ng a i r with t h e d i s t a n c e between c y l i n d e r s being f o u r d i a m e t e r s .
The r e s u l t s here are comparable and can be ex pec te d to show a s t r o n g e r
e f f e c t s i n c e a s e p a r a t i o n o f only t h r e e dia m e te rs was used.
Since each h a l f tube was s e p a r a t e l y h e a t e d , a s i m i l a r a n a l y s i s was
a v a i l a b l e f o r the v e r t i c a l p o s i t i o n .
The r e s u l t i n g average N u s s e l t num­
be r f o r . t h e top h a l f o f each c y l i n d e r was 81% f o r a i r , 58% f o r g l y c e r i n ,
64% f o r w a t e r , and 65% f o r s i l i c o n e when compared t o th e bottom h a l f .
These r e s u l t s were e xpect ed s i n c e th e le n g t h of th e high speed flow
along th e body i s long r e l a t i v e to t h e h o r i z o n t a l c y l i n d e r w hil e the
cross s e c t i o n a l flow a r e a i s much s m a l l e r .
Thus th e e f f e c t o f the
s m a l l e r d r i v i n g f o r c e (AT) f o r h e a t t r a n s f e r i s more pronounced in the
v e r t i c a l c o n f i g u r a t i o n than in the h o r i z o n t a l .
The ev id en c e o f a geo­
m e t r i c e f f e c t i s s upp ort e d by both an o v e r a l l d e cr ea se in h e a t t r a n s f e r
and a r e d u c t i o n in the average N u s s e l t number when upper and lower
- 30 -
hal ves o f th e i n n e r body are compared.
A h i g h e r h e a t t r a n s f e r from the h o r i z o n t a l c y l i n d e r s as compared
t o t h e . v e r t i c a l c y l i n d e r s r e s u l t s from the a b i l i t y o f th e c o o l e r medium
to mix with th e p r e h e a t e d f l u i d as i t r i s e s from the l o w e r •c y l i n d e r .
Figu res 4 .2 through 4 . 8 show t h i s e f f e c t .
The major eddy, as seen in
Figu res 4.2 through 4 . 5 , between th e two c y l i n d e r s , r e s u l t s from th e
mixing o f c o o l e r f l u i d i n t o th e flow .
Figures 4 .7 and 4 . 8 show t h a t
t h i s eddy system f o r a i r i s n o t as pronounced as f o r s i l i c o n e .
This
may be a p o s s i b l e e x p l a n a t i o n f o r t h e l a r g e r de cr ea se in t h e average
N u s s e lt number (28%) as compared t o the o t h e r f l u i d s s i n c e the medium's
c o o l i n g . e f f e c t caused by t h e . e d d y i s reduc ed.
As the te m pe ratu re o f
th e c y l i n d e r s i s i n c r e a s e d , a n o t h e r eddy i s formed a t t h e s i d e o f the
upper c y l i n d e r i n d i c a t i n g th e same mixing a c t i o n .
Figure 4 . 6 .
This i s e v i d e n t in
The i n t r o d u c t i o n o f c o o l e r f l u i d from t h e s i d e s and below
was observed in o t h e r s t u d i e s [ 9 - 1 1 ] .
The eddy a c t i o n as d e s c r i b e d
was n o t e v i d e n t in th e v e r t i c a l c o n f i g u r a t i o n , Figure 4 . 9 .
The flow
v i s u a l i z a t i o n s in Figures 4.2 through 4 .9 a l s o s u p p o r t th e assumption
t h a t t h e boundary l a y e r does n o t s e p a r a t e from th e c y l i n d e r with the
ex ce p ti o n o f t h e top cap o f the v e r t i c a l c y l i n d e r .
C o r r e c t i o n in t h e
boundary l a y e r le n g th f o r t h i s s e p a r a t i o n had no e f f e c t on th e d e v i a ­
t i o n in f i t f o r th e c o r r e l a t i n g e q u a t i o n s .
Two o t h e r geometries were s t u d i e d .
The i n n e r body in both the
v e r t i c a l and h o r i z o n t a l o r i e n t a t i o n s were r o t a t e d 45° about t h e i r
Figure 4 .2
Flow Pattern fo r Horizontal
Cylinders with 20 cs Fluid
and AT = 10.71 K.
- 32 -
Figure 4 . 3
Flow Pa tt er n Around Upper
Horizontal Cylinder with
20 cs Fluid and AT = 10.71 K.
- 33 -
Figure 4 . 4
Flow Patter n Around Lower
Horizontal Cylinder with
20 cs Fluid and AT = 10.71 K.
ii
- 34 -
Figure 4 . 5
Flow Patter n Around Center
Support Sphere with 20 cs
Fluid and AT = 10.71 K.
- 35 -
Figure 4 . 6
Flow Patter n f o r Horizontal
Cylinders with 20 cs Fluid
and AT = 5 4 . 4 3 K.
- 36 -
Figure 4 . 7
Flow Patter n f o r Horizontal
Cylinders wit h Air and
AT = 7 . 6 7 K.
- 37 -
Figure 4 . 8
Flow Patter n f o r Horizontal
Cylinders wit h Air and
AT = 21.61 K.
- 38 -
Figure 4 . 9
Flow Patter n f o r V e r t i c a l
Cylinders with Air and
AT = 17.01 K.
-
vertical axis.
fer.
39
This r o t a t i o n had n e g l i g i b l e e f f e c t oh th e h e a t t r a n s ­
U t i l i z i n g the graphs o f N u s s e l t number versus Grashof number f o r
a i r , w a t e r , and th e s i l i c o n e f l u i d as shown in Figures 4.10 and 4 . 1 1 ,
th e O0 ve rsus 45° r o t a t i o n d a t a p o i n t s c l e a r l y i n d i c a t e t h a t t h i s geo­
m e t r i c change had n e g l i g i b l e e f f e c t on t h e h e a t t r a n s f e r r e s u l t s .
Al­
though t h e g l y c e r i n d a t a a re not p r e s e n t e d , they e x h i b i t e d t h e same
p a t t e r n as the o t h e r f l u i d s .
The absence o f any r o t a t i o n a l e f f e c t was
a l s o e v i d e n t in the te m p e ra t u re p r o f i l e s .
Figures 4.12 through 4.14 show
show t h a t f o r t h e same r a d i a l p o s i t i o n the p r o f i l e s f o r t h e O0 and 45°
r o t a t i o n do not change.
These p l o t s use d im e ns io nle s s te m p e ra t u re and
r a d i u s r a t i o s and are d e f i n e d as
T
ND
(4.9)
where T i s t h e l o c a l te m p e ra t u re measured a t any r a d i a l p o s i t i o n and
\
d i s t a n c e from the s u p p o r t s p h e r e , and
where R i s the l o c a l d i s t a n c e from th e s u r f a c e o f th e s u p p o r t sphere
and R(Q) i s t h e d i s t a n c e from the s u p p o r t sphere to t h e o u t e r body de­
pending on r a d i a l p o s i t i o n .
As seen in Figure 4.12 f o r t h e r a d i a l p o s i ­
t i o n Q = 120°, a c y l i n d e r blocked th e t r a v e l of a probe from re a ch in g
th e s u p p o r t sp her e and, t h e r e f o r e , th e d im en si onles s d i s t a n c e could not
100.0
SILICONE
gpSSQ |@
50.0
O 3
O*
WATER
oo
©
s
B
O ®
#
Nul
AIR
I •
Q
Closed Symbols - O0
Open Symbols - 45°
10.0
! I ; I I
IO5
IO8
IO6
Gr
Fi gu re 4.10
L
NuLversu s Grl f o r O0 and 45° R ota ti o n s with the C yl in d e rs in
the Ho riz ont a l P o s i t i o n .
I
100.0
SILICONE
OS
50 .0
-
»
0
» ®
WATER
O GS
%
®o
ea
O©
I
Nu
AIR
A ^
Closed Symbols - O0
Open Symbols - 45°
A
10.0
J __ I
J_ _ _ _ _ I__ I__ L - L v L - L
IO5
Figure 4.11
IO6
1 : 1 :1
J ___ L ' I I I I
107
IO8
Nul ve rsus Gr^ f o r C0 and 45° R ota ti o n s with the C yli nde rs in
th e V e r t i c a l P o s i t i o n .
- 42 - •
I .0
o o
O0 R o t a t i o n , AT=36.24 K.
A A 45° R o t a t i o n , AT=37.18 K.
0.8
A.
e
0.6
o
A
'ND
9
9=120°
A
0 .4
Q=Oc
Z l
0
aOAOAOA
a
O T>A
4,
&
0.2
0.2
Figure 4.12
0.6
0.8
I .0
r ND
Temperature P r o f i l e f o r R ot a ti o n About the
V e r t i c a l Axis with S i l i c o n e - H o r i z o n t a l P o s i t i o n ,
0 .4
- 43 -
1 .0
o q
O0 R o t a t i o n , AT=3G.24 K.
a a
45° R o t a t i o n , AT=37.18 K.
0.8
0.6
t ND
0=34c
0.4
A.
2
^ A
Go ^
0 , 0 O^ O O O
O
0.2
-
O
O
o
^
^
V
r,
G=BQO
a
^
°
>€■
W
£) ®
A
A!
0.2
0.6
0 .4
0 .8
I .0
r ND
Figure 4.13
Temperature P r o f i l e f o r R ot a ti o n About the
V e r t i c a l Axis with S i l i c o n e - H o r i z o n t a l P o s i t i o n .
-
44 -
1.0 ro e
a
a
Ou R o t a t i o n , A1=27.78 K.
45O R o t a t i o n , AT=27.48 K.
>a 5
AA A
Ox oA oA Oa oa o a oa o a o a o a g a ®
9=120
9=160
Figure 4.14
Temperature P r o f i l e f o r R ot a tio n About the
V e r t i c a l Axis with G l y c e r i n - V e r t i c a l P o s i t i o n .
- 45 -
s t a r t a t th e z e r o r e f e r e n c e p o i n t .
ATI f l u i d s e x h i b i t e d th e same e f ­
f e c t as shown in Figures 4.12 through 4.1 4 .
I
HEAT TRANSFER RESULTS USING AIR
The d a t a usin g a i r as th e f l u i d medium were c o r r e l a t e d s e p a r a t e l y
f o r both th e h o r i z o n t a l and v e r t i c a l geometries and a l s o f o r a l l a i r
d a ta combined. . S e p a ra te c o r r e l a t i o n s f o r th e r o t a t i o n a l d a t a (0° and.
45°) a re not p r e s e n t e d s i n c e t h e r e was a n e g l i g i b l e e f f e c t as p r e v i o u s l y
described.
The n o t a t i o n th ro ugh out t h e remainder o f t h i s d i s c u s s i o n
w i l l c o n s i s t of p r e s e n t i n g a t w o - p a r t d e s i g n a t i o n i n which t h e f l u i d
w i l l be given f i r s t , foll owed .by t h e a x i a l geometry; i . e . , A ir- H ori zo ni,
t a l , A ir-V ertical.
A ll.
When a l l d a ta a re p r e s e n t e d th e l a s t term w i l l be
The s c a t t e r in th e d a ta f o r a i r , as seen in Figure 4 . 1 , demon­
s t r a t e s th e same p a t t e r n as d e s c r i b e d by Warrington [1 4 ] .
The e f f e c t
i s pronounced i n th e lower te m p e ra t u re range which seems to s u p p o rt
hi s co n cl u s io n t h a t conduction and r a d i a t i o n l o s s e s may become a domina nt f a c t o r as th e f l u i d and m o le c u la r motion are re d u c e d .
The b e s t o v e r a l l e q u a ti o n which c o r r e l a t e s th e h e a t t r a n s f e r f o r
t h e A i r - H o r i z o n t a l d a t a was
Nul = 0.899 Ral 0?20°
(4.11)
w it h an average p e r c e n t d e v i a t i o n o f 8.42 and maximum p e r c e n t d e v i a t i o n
of- 18.70.
C o r r e l a t i o n s usi ng e i t h e r t h e modif ied Rayleigh number
■
I
- 46 ~ •
o r the boundary le ngth pro vid ed the - same average d e v i a t i o n s due t o th e
c o n s t a n t geometry.
all flu id s.
The a i r d a ta had th e h i g h e s t average d e v i a t i o n of
This can be d i r e c t l y a t t r i b u t e d t o the h e a t l o s s by r a d i a ­
t i o n s i n c e a l l o f th e opaque f l u i d s had a r e s u l t i n g average d e v i a t i o n
approxim at el y one magnitude s m a l l e r , as w i l l be d i s c u s s e d l a t e r .
The
b e s t c o r r e l a t i o n f o r A i r - V e r t i c a l was
Nul = 0.293 Ra^ ' 260
(4.12)
with 4.32% average d e v i a t i o n and 19.79% maximum d e v i a t i o n .
The b e s t e q u a ti o n using a s i n g l e c o r r e l a t i o n p a ra m e te r f o r a l l a i r
d a ta was
Nu8 = 0.466 Ra60 -242
with 6.79% average d e v i a t i o n .
.(4.13)
C o r r e l a t i o n s using th e h y p o t h e t i c a l gap
width r e s u l t e d in an average d e v i a t i o n o f 16.48%.
Compared to o t h e r
f l u i d s , a i r was a f f e c t e d the most by geometric o r i e n t a t i o n as shown in
Figure 4.1 and i t s resp onse to a c o r r e l a t i o n using th e boundary l a y e r
l e n g t h f o r the c h a r a c t e r i s t i c dimension as shown in Fi gure 4.T5.
Table 4. 2 p r e s e n t s a l l c o r r e l a t i o n s used f o r a l l a i r d a t a combined.
Equation forms ( 4 .6 ) and ( 4 .7 ) are n o t p r e s e n t e d i n t h e t a b l e s i n c e t h e
s e t o f e q u a t i o n s g e n e r a t e d by a l e a s t sq uar es curve method form an
i l l - c o n d i t i o n e d s e t when the P r an dtl number i s used as an in de pen de nt
p a ra m e te r.
This i s a d i r e c t r e s u l t o f the very small v a r i a t i o n in
40
30
Nur
4^
Nu1
Figure 4 . 1 5
N u s s e l t Number versu s Rayleigh Number f o r Air Data with
C o r r e l a t io n Equations Superimposed.
TABLE 4 . 2
CORRELATION EQUATIONS FOR AIR-ALL
AVERAGE % DEVIATION
EQUATION
MAXIMUM % DEVIATION
'
Nul = 0.237 Ra^ - 285
16.487
36.212
Nul = 0.203 Ra* ° - 285
16.487
36.212
6.795
22.626
6.795
22.626
Nu
B
= 0.466 Ra ° - 242
B
Nud = 0.409 Ra* ° - 242
B
B
- 49 -
Pr a n d tl number f o r the a i r d a t a .
Al I a i r d a t a have’ been p l o t t e d in F i g ­
ure 4.15 in the form Nul versus RaL and Nub versus Ra6 with the b e s t f i t
e q u a ti o n s which used a s i n g l e c o r r e l a t i n g p a ra m et e r.
HEAT TRANSFER RESULTS USING' GLYCERIN
C o r r e l a t i o n s using g l y c e r i n d a t a pro vid e d very good . r e s u l t s with '
average p e r c e n t d e v i a t i o n s ranging from 0.9% t o 8.7% depending on th e
e q u a ti o n form used and t h e p a r t i c u l a r geometries i n v o l v e d .
Using the
gap width i n s t e a d o f boundary le n g th as a c o r r e l a t i n g p a ra m et e r a f f e c t s
only the c o e f f i c i e n t s and n o t th e average d e v i a t i o n when a p p l i e d t o
separate geometries.
T h e r e f o r e , only th e e q u a ti o n s usi ng th e gap width
w i l l be p r e s e n t e d in t h e t e x t .
The t a b l e s w i l l i n c l u d e the c o e f f i c i e n t s
f o r e q u a t i o n s using boundary l a y e r l e n g t h .
The b e s t f i t u t i l i z i n g a s i n g l e c o r r e l a t i n g param ete r f o r G ly c e r in H or iz ont a l i s
Nul = 0.547 RaL0 ‘236
( 4.1 4)
and f o r G l y c e r i n - V e r t i c a l
Nul = 0.711 RaL0 ’217
(4.15)
with r e s u l t i n g average d e v i a t i o n s o f 1.72% and 0.913%, r e s p e c t i v e l y .
C o r r e l a t i o n s using two para mete rs improved th e p e r c e n t d e v i a t i o n only
s l i g h t l y as shown in Table 4 . 3 .
TABLE 4 . 3
CORRELATIONS FOR GLYCERIN DATA '
G ly c er in -H ori z o n ta l
Equation
Form
C2
Cl
Max %
Devia t i on
Avg %
D e v ia ti o n
N
u
l -
C 1 R
0.548
2
s l
C3
0.236
C9
Nu l = C ^R a * -
0.482
0.236
1.722
Nu^= C ^ R a L - P r d
0.096
L .457
1.638
C2
0.476
C^ Rdg
1.722
Cp
0.419
Nug = C ^ R a*
0.236
I
Nu b =
Cg
C1Ra8 Pr
0,092
Nu l =
Co
C1Grl Pr
0.096
0.298
1.638
Cp
Nug =
C1Gr 8 Pr
Co
0.135
1.638
I 0.094
0.392
4.457
0.298
0.392
4.457
6.589
0.162
0,2 ' 7
6.589
j0.162
1.150
3.454
5.071 ‘
. 0.162
I .150
5.066
1.150
-0.086
0.076
5.070
0.162
0.076
5.071
7.246
0.230
2.624
1.511
7.246
0.197
2.540
0.287
24.88
0.287
24.89
8.714
Q.045
.0.321
1 .757
I . 5 11
1.757
0.139
11.67
0.197
2.539
0.045
-0.050
7.557
8.714
0.209
0.217
0.230
2.624
0.244
6.589
0.913
5.066
-0.086
5.070
0.913
Cl
C2
C3
Avg %
Max %
Devia t i on
D ev iatio n
0.525
0.21 7
1.150
0.818
■4 . 4 5 7
1.638
Cp
6.216 '
6.216
0 .2 98
3.454
Glycerin-A' I
0.595
6.589
0.913
0.918
0.236
I .722
C2
I 0.094
0.21 7
0.912
0.633
6.216
0.299
Cl
C2 '
c 3 .
Avg %
Max %
Devi a t i on
Devi a t i on
0.711
6.216
1 .722
G lycerin-Vertical
0.146
7.557
0.321
■
0.461
11.67
- 51
C o r r e l a t i o n s o f a l l g l y c e r i n da ta was b e s t accomplished using the
, h y p o t h e t i c a l gap width r a t h e r than th e boundary l a y e r when using one
parameter.
As d i s c u s s e d p r e v i o u s l y , t h e g l y c e r i n d a t a were a f f e c t e d
very l i t t l e by t h e geometry as a r e s u l t o f t h e dampening e f f e c t s of
viscosity.
The use of t h e boundary l a y e r le n g th as a c h a r a c t e r i s t i c
dimension o v e r - p r e d i c t s th e geometric e f f e c t s c au si ng th e c o r r e l a t i o n s
based on t h i s dimension t o y i e l d a h i g h e r average p e r c e n t d e v i a t i o n .
Figure 4.16 r e v e a l s t h i s o v e r - p r e d i c t i o n through a s h i f t in the v e r t i c a l
d a ta as seen in the graph of Nu^ ve rsus Ra^.
Both w a t e r and s i l i c o n e
were a f f e c t e d in th e same manner altho ugh t o a l e s s e r d e g r e e .
The c o r r e l a t i o n s which pro vid ed t h e b e s t r e s u l t s f o r a l l g l y c e r i n
d a t a were
Nul = 0.595 Rajl
0.230
(4. 16)
and
(4 .17)
wit h r e s p e c t i v e average d e v i a t i o n s o f 2.62% and 2.54%.
Figure 4.16 p r o ­
vi de s a p l o t of Nu versus Ra and the s i n g l e param ete r c o r r e l a t i o n equa­
tions.
HEAT TRANSFER DATA USING WATER AND SILICONE'
Water and s i l i c o n e e x h i b i t e d s i m i l a r h e a t t r a n s f e r r e s u l t s and
/
I . Nu
0.476 Ra
2. Nu
0.918. Ra,
3. Nu
0.244 Ra
0.236
0.217
2. V e r t i c a l
0.287
3. Al I Data
I . H ori zo nta l
I i i i l
I . Nu
0.548 Ra
2. Nu
0.711 Ra
3. Nu
0.595 Ra
0.236
□ Horizontal
o Vertical
I i i i i
I . Horizontal
0.217
2. V e r t i c a l
0.230
□ Horizontal
o Vertical
I
Figure 4 . 1 6
I i i f i
N u s s e l t Number v ers u s Rayleigh Number f o r Gl ycerin Data with
C o r r e l a t i o n Equations Superimposed.
- 53 -
are combined i n t o one s e c t i o n .
C o r r e l a t i o n s f o r both f l u i d s y i e l d e d
average d e v i a t i o n s which ranged from 1.26% to 6.16%.
These r e s u l t s
as ^ e l l as th o s e f o r g l y c e r i n a l l i n d i c a t e t h a t c o r r e l a t i o n s f o r a
s i n g l e f l u i d can be ex pec te d t o y i e l d s a t i s f a c t o r y r e s u l t s .
Table 4 .4 p r e s e n t s a l l e q u a ti o n s .and geometries f o r th e c o r r e ­
la tio n o f water data.
C o r r e l a t i o n s which prov ide d th e b e s t f i t of
d a t a using one and two pa rameters f o r a l l w a te r d a t a are
Nul = 0.888 Ral ° - 208
(4.18)
Nub = 0.05 3 Ra60 -310Prcu463
(4.19)
and
with average d e v i a t i o n s o f 4.40% and I .97%, r e s p e c t i v e l y .
Figure 4.17
p r e s e n t s a l l w a te r d a t a and t h e b e s t f i t eq u at i o n in g r a p h i c a l form.
The d a ta f o r s i l i c o n e i s p r e s e n t e d i n Figure 4 .1 8 and the r e s u l t ­
ing c o r r e l a t i o n s are p r e s e n t e d in Table 4 . 5 .
Best f i t e q u a t i o n s f o r th e
s i l i c o n e f l u i d using one and two pa rame te rs are
Nu
= 0.709 Ra.0-228
(4.20)
and
Nug = 0 .0 35 Rag0 -308Pr0 -284
with average d e v i a t i o n s o f 4.28% and 1.26%, r e s p e c t i v e l y .
(4.21)
TABLE 4 . 4
CORRELATIONS FOR WATER DATA
Water-Hori zo nta l
Equation
Form
cI
c2
Avg %
Deviation
C2
Nul = C^RaL ^
0.783
Max %
D e v ia ti o n
0.217
1.286
C2
Nul = C1Ra* ^
0.696
1.277
0.217
I
0.661
Nug= C1Rag 2
0.217
Cp Cg
Nub= C1Ra8 Pr
0.588
0.217
1.286
I
I .055
0.200
C2 Cg
Nul = C-|6rL Pr
I .277
0.200
Co
Nub= C1Gr8 Pr
Co
1.055
1 .240
5.069
-0.0 80
0.120
5.080
I .240
0.200
0.199
.563
0.199
.563
0.183
1 .580
I
0.540
0.120
5.080
0.183
0 !540
0.793
2.292
4.401
-0.076
2.291
0.183
-. 076
0.107
0.107
2.291
.
8.013
0.208
'
8.013.
0.244.
0.172
0.293
6.170
6.170
0.053
14.27
0.293
I 14.27
0.310'
1.970
0.318
1.970
0.463
4.973
0.244
4.277
0.053
'
8.133
4.277
0.154
2.291
0.183
0.318
0.180
2.291
0.540
2.226
'
2.292
0.540
1.580
4.401
0.199
0.563
2.226
2.292
2.292
.
0.208
0.888
0.199
0.563
1 .235
5.080
I .240
Cl
C2
C3
Avg %
Max %
D e v ia ti o n
D e v ia ti o n
I .379
5.0692
1.286
Cp
Nug= C1Ra* ^
-.080
5.080
.240 •
C2
Cl
C3
Avg % ■
Max %
Devi ati on
D e v ia ti on
0.917
5.069
0.200
Wa t e r - A l l
1 .020
5.069
1.286
G? C3
Nul = C1R a ^ P r d
C3
W a t e r - V e r ti c a l
0.4170
8.133
I 0.310
I.
0.773
4.973
I . Nu
0.661
Ra
2. V e r t i c a l
0.199
2. Nu
3. Nu
0.217
0.180 Ra
0.293
3. All Data
I . Horizontal
□ Ho rizo nta l
o Vertical
RaB
D Ho riz ont a l
o Vertical
I . H oriz on ta l
3. All Data
2. V e r t i c a l
Figure 4 . 1 7
0.217
I . Mu
0.783 Ra
2. Nu
I .020 Ra
3. Nu
0.888 Ra 0.208
N u s s e l t Number v ers u s Rayleigh Number f o r Water Data with
C o r r e l a t i o n Equations Superimposed.
0.199
I . Nu
0.678 Ra
2. Nu
0.984 Ra
0.198 Ra
3. Nu
0.224
2. V e r t i c a l
0.221
3. All Data
0.296
I . Horizontal
□ H orizo nta l
o V ertical
<— 3. Al I Data
□ H ori zo nta l
o Vertical
Figure 4 . 1 8
I . Ho rizon tal
2. V e r t i c a l
I . Nu
0.795 Ra
2. Nu
0.769 Ra
3. Nu
0.709 Ra
0.224
0.221
0.228
N u s s e l t Number v er s u s Rayleigh Number f o r S i l i c o n e Data with
C o r r e l a t io n Equations Superimposed.
TABLE 4 . 5
CORRELATIONS FOR SILICONE DATA
Si I i c o ne- H or iz ont ai
Equation
Form
cI
C2
C3
Max %
Avg %
Devi a t i on
Devia t i on
C2
Nul - C1Ral L
0.795
0.224
0.888
Nul = C1Ra* 2
Cg CNul = C1R a ^ P r 6
0.703
0.224
0.888
0.404
I
0.242
0.815
0.678_
Nug= C1Rag
.888
0.600
Co
Nug= C1Ra* c
j
C2 C2
0.354
Nug= C1Rag Pr
'
Co
Co
Nul = C1Grl Pr
.815
0.404
0.354
0.815
0.681
2.672
0.602
0.062
I 2.672
0.24 2
0.305
2.672
0.305
2.672
0.221
0.607
0.868
0.602
0.681
0.602
0.868
0.602
Cl
C2
C3
Avg %
Max %
D e vi at io n
Dev iatio n
0.709
I
0.228
2.867
0.221 ■
0.627
0.228
2.867
0.224
0.011
4.284
1.864
6 . 818
0.202
-0.090
4.196
0.198
7.013
2.948
0.221
2.867
0.607
0.873
2.985
0.242
0.607
0.984
0.24 4 .
0.815
C2 Cg
Nug= C1Grg Pr
2.985
0.062
2.985
0.242
0.607
0.683
0.224
.888
C2
Cl •
C3
Avg %
Max %
Devia t i on
Devia t i on
0.769
2.985
Si l ic o n e -A l I
Si I i c o n e - V e r t i c a l
0.221
2.867
4.284
5.912
6.818
0.296
15.14
0.167
0.296
5.912
0.114
0.0.35
15.14
0.308 I 0.284
2.948
0.225
0.236
I .256
5.892
1.864
2.948
0.225
0.236
4.196
0.035
0.225
2.948
1.256
0.202
0.113
7.013
0.308
0.591
5.892
-
58 -
R es ults co ncerning th e e f f e c t s o f geometry and f l u i d p r o p e r t i e s
have g e n e r a l l y the same c h a r a c t e r i s t i c s as th e r e s u l t s f o r g l y c e r i n .
These r e l a t i o n s h i p s between f l u i d s and g e o m e t r i e s ' a r e covered in the
next s e c tio n .
HEAT TRANSFER RESULTS' FOR ALL' FLUIDS
In a d d i t i o n t o th e c o r r e l a t i o n s a l r e a d y d i s c u s s e d , t h r e e o t h e r
combinations o f d a ta were c o r r e l a t e d .
Data f o r a l l f l u i d s were c o r r e ­
l a t e d a g a i n s t h o r i z o n t a l , v e r t i c a l , and both g e o m e t r i e s .
The r e s u l t i n g
d e cr ea se in t h e accuracy o f a l l e q u a t i o n forms, when compared t o the
s i n g l e f l u i d c o r r e l a t i o n s , can be d i r e c t l y a s s o c i a t e d with th e Pr andtl
number e f f e c t .
This i s e v i d e n t in Table 4.6 and Figure 4 . 1 .
Table 4.6 i n c l u d e s a l l c o r r e l a t i o n e quat io n forms usi ng d a t a com­
b i n a t i o n s d e s c r i b e d above.
The b e s t f i t f o r a l l d a t a , a l l v e r t i c a l
d a ta and a l l h o r i z o n t a l d a ta using th e gap width are
Nul = 0.358 RaL° - 257Pr 0 •014
. (4.22)
Nul = 0.262 Ral 0 l 268Pr 0 -028
(4 .23)
Nul = 0.498. Ral 0 e245P r "0 *002
(4 .24) .
and
wit h average d e v i a t i o n s o f 8.38%, 4.53%, and 6.92%,, and maximum d e v i a ­
t i o n s o f 40.45%, 18.48%, and 14.74%, r e s p e c t i v e l y .
TABLE 4. 6
CORRELATIONS FOR ALL DATA
A l l - H o r i zont.a!
v
Equation
Form
C2
Cl
C3
Avg %
Max %
D e v ia ti o n
Devi a t i on
C2
Nul - C^Rbl c
0.496
C2
Nul - C1Ra* ^
Nul = C1R a ^ P r J
Nug= C1Rag
Nug= C1Ra*
Co
0.24 5
6.947
0.435
15.27
0.245
6.947
0.498
6.916
0.437
6.947
0.383
15.27
0.2^ -5
0.438
0.245
0.241-5
15.27
0.24r5
C2 Cg
Nu8= C1Gr 8 Pr
0.498
0.2/ 15
6.916
0.438
6.916
0.264
9.340
0.274
0.365
9.500
0.259
0.227
0.274
0.317
0.259
0.262
9.340
0,259
0.268
0.262
30.58
0.268
0.268
0.302 ■ 0.268
.0 .3 58
0.028
18.48
0.297
18.48
0.297
18.48
47.71
0.257
8.384
9.305
0.277
0.275
28.63
0.275
28.63
0.274
8.437
0.358
8.384
0.277
8.437
0 J114__.
40.453
9.305
0.246
0.274
47.71
9.500
0.286
30.58
4.531
4.531
0.028
18i48
0.274
9.340
4.531
14.74
30.58
4.531
4.74
0.243
30.58
9.340
0.302
4.74
0.245
cI
C2
C3
Avg %
Max %
Devia t i on
Devia t i on
- 0.002
0.244
Al I Data
Cl
C2
C3
Avg %
Max %
D e v ia ti o n
Devia t i on
0.300
5.27
6.916
C2 C2
Nul = C1GrL Pr
- 0.002
14.74
6.947
Nu8= C1Rag2P r 03
r
A l l - V e r t i ca'
0.012
24.85
0.257
0.271
40.45
0.274
| 0.286
| 24.85
- 60 -
As mentioned p r e v i o u s l y , t h e c o r r e l a t i o n s f o r s i n g l e geometries
were no t a f f e c t e d by the change from one c h a r a c t e r i s t i c dimension t o
another.
C o r r e l a t i o n o f a l l da ta i n d i c a t e d an a p p r e c i a b l e improvement
when th e boundary l a y e r le n g t h was us ed , r e s u l t i n g in an 8 ;44% average
d e v i a t i o n and 24.85% maximum d e v i a t i o n versus 8.38% and 40.45%, r e s p e c ­
t i v e l y , f o r c o r r e l a t i o n s usi ng the gap w i d t h .
The b e s t e q u a t i o n f o r a l l
d a t a was
Nun = 0.277 Ra_0.274pr0.012
P
.
(4. 25)
D
The r e l a t i v e l y small valu e o f t h e exponent a s s o c i a t e d w it h t h e Pr and tl
number s u g g e s ts t h a t t h i s term can be dropped w i t h o u t a p p r e c i a b l e lo s s
in a c c u ra c y .
The r e s u l t i n g eq u at i o n
Nub = 0.286 Ra60 *275
(4.26)
has a 9.30% average d e v i a t i o n and. 28.63% maximum d e v i a t i o n .
Figures
4.19 and 4.20 prov ide a p l o t o f a l l d a t a a g a i n s t the b e s t f i t e q ua ti ons
usi ng a s i n g l e c o r r e l a t i n g p a r a m e te r .
Warrington [14] pro v id e s one of th e most e x h a u s t i v e s t u d i e s in
h e a t t r a n s f e r in e n c l o s u r e s t o d a t e .
In c o r r e l a t i n g s e v e r a l s i n g l e
body g e o m e t r i e s , f l u i d s and o u t e r body g e o m e t r ie s , he found t h a t the
b e s t s i n g l e pa ra m et e r eq u at i o n usi ng th e gap width was
Nul = 0.181 Ra*0 -282
(4.27)
1000
o - G ly cer in
v - A ir
A - Water
100
□ - Silicone
Nul
2. V e r t i c a l
I . H ori z o n ta l
3. All Data
L
...;
Open Symbols - H ori zo nta l
Closed Symbols - V e r t i c a l
i
i i I i i i i l ____ I
I I I I I 111
I
I I I IIIII
IOz
10
I
Nu^
2.
0.274
Nu^ = 0.264 RaL
3.
Nu,
I_ i I I
= 0.496 Ra^
0.245
1.
0.259
= 0.365 RaL
------- 1---- I— i
m .i
III
8
Ra,
Figure 4 . 1 9
Nul versu s RaL f o r All Data with C o rr el a tio n Equations Superimposed
- Glyc erin
- A ir
100
3. Al I Data
- Water
I . Horizontal
D - Silicone
Nub
•
10
I,
Open Symbols - H ori zo nta l
Closed Symbols - V e r t i c a l
IO5
IO6
IO7
IO8
I . Nu
0.437 Ra
2. Nu
0.300 Ra
3. Nu
0.286 Ra
IO9
0.245
0.274
0.275
IO10
Rafj
Figure 4 . 2 0
Nug v ers u s Rag f o r All Data with C o rr el a tio n Equations Superimposed
- 63 - .
with an 18.67 average p e r c e n t d e v i a t i o n , and the b e s t f i t using the
boundary l a y e r l e n g t h was
Nu
B
= 0.585 Ra*0 ' 236
P
with a 14.75% average d e v i a t i o n .
•
(4.28)
The d a ta encompassed by t h i s i n v e s t i ­
g a t i o n were c o r r e l a t e d with t h e above e qu a ti o n s and. r e s u l t e d in 13.04%
and 20.34% average d e v i a t i o n s , r e s p e c t i v e l y .
However, t h i s r e q u i r e d a
r e d e f i n i t i o n o f the boundary l a y e r le n g t h to i n c l u d e the t o t a l boundary
le n g t h f o r a l l f o u r c y l i n d e r s .
C o r r e l a t i o n s usin g t h e boundary l a y e r
le n g th f o r one c y l i n d e r as used in t h i s stu dy pro vid ed un a c c e p ta b le r e ­
s u l t s when a p p l i e d t o t h e e q u a ti o n s f o r s i n g l e bodies which have been
developed by Warrington.
These e q u a ti o n s with c u r r e n t d a t a f i t t e d t o
them a r e . p r o v i d e d in Table 4 . 7 .
Even with a r e d e f i n e d boundary l a y e r
l e n g t h , the r e l a t i v e l y l a r g e d e v i a t i o n from e q u a ti o n ( 4 .2 8 ) and th e
s l i g h t l y l a r g e r d i f f e r e n c e s as seen in Table 4 .7 f o r c o r r e l a t i o n s using
th e boundary l a y e r dimension, s u g g e s t t h a t the h e a t t r a n s f e r i s compli­
c a te d by t h e m u l t i p l e body geometry and th e tendency t o o v e r - p r e d i c t
t h e s e geometric e f f e c t s as p r e v i o u s l y d i s c u s s e d .
TABLE 4 . 7
FIT OF DATA TO PREVIOUSLY DEVELOPED EQUATIONS
The f o l l o w i n g e q u a t i o n s were e x t r a c t e d from r e f e r e n c e [14 ]:
AVG % DEV.
EQUATION
__________ CONDITIONS FOR USE
1.
Nu b = 0 . 5 8 5 R a * 0 *2 3 6
14.75
S in g l e i n n e r bodies (cube, s p h e r e , and c y l i n d e r ) with
s p h e r i c a l and c ub ic a l o u t e r bodies usi ng a l l f l u i d s .
2.
Nu b = 0 . 2 4 8 R a * 0 ' 2 4 8
15.80
Al I f l u i d s and i n n e r bodies usi ng a c ub ic a l o u t e r
body.
3.
Nu b = 0 . 5 7 8 R a * 0 ' 2 3 9
14.49
All f l u i d s and o u t e r bodie s usi ng a c y l i n d r i c a l in ne r
body.
4.
Nu l = 0 . 1 8 8
R a * 0 *2 8 2
18.67
Same as e q u a ti o n I .
5.
Nu l = 0 . 1 5 1
R a *0 "295
17.74
Same as e q u a ti o n 3.
R e s u l t s o f f i t to t h e s e e q u a t i o n s f o r c u r r e n t d a t a :
-- ----------- \-------------
EQUATION I
EQUATION 2
EQUATION 3
EQUATION 4
EQUATION 5
Avg %
Dev.
Max %
Dev.
Avg %
Dev.
Max %
Dev.
Avg %
Dev.
Max %
Dev.
Avg %
Dev.
ALL FLUIDS
HORIZONTAL
11.77
24.17
7.966
22.06
7.760
20.82
17.11
34.94
16.78
36.77
ALL FLUIDS
VERTICAL
28.91
41.81
22.27
36.04
24.76
38.14
9.785
30.42
9.310
32.81
20.34
41.81
15.12
■36.04
16.26
38.14
13.45
34.94
13.04
36.77
DATA
ALL DATA
Max
%
Dev.
Avg %
Dev.
Max
%
Dev.
CHAPTER V.
CONCLUSIONS AND RECOMMENDATIONS
This st udy was i n i t i a t e d p r i m a r i l y t o extend th e d a t a a v a i l a b l e
f o r h e a t t r a n s f e r in e n c l o s u r e s t o more complex i n n e r body ge om et rie s.
Previous i n v e s t i g a t i o n s by Bishop [ 1 8 ] , Scanlan [ 1 9 ] , Warrington [ 1 4 ] ,
and Weber [17] pro vid e a complete a n a l y s i s o f common s i n g l e body s h a p e s .
This i n v e s t i g a t i o n has expanded t h a t a r e a by de te rm in in g th e n a t u r a l
co nvection h e a t t r a n s f e r between a s e t o f f o u r is o th e rm a l c y l i n d e r s and
a c ub ic a l e n c l o s u r e .
The r e s u l t s have p o i n t e d out t h e importance o f th e
geometric e f f e c t s f o r m u l t i p l e bodies and t h e accuracy w ith which h e a t
t r a n s f e r can be c o r r e l a t e d f o r many i n n e r and o u t e r body geometric
combinations usi ng only a few simple e q u a t i o n s .
Of t h e geometries s t u d i e d t h e r e appeared t o be no a p p r e c i a b l e
e f f e c t due to r o t a t i o n of t h e i n n e r body about i t s v e r t i c a l a x i s .
How­
e v e r , a s i g n i f i c a n t d i f f e r e n c e was n o t i c e d between the v e r t i c a l and
h o r i z o n t a l c o n f i g u r a t i o n o f th e s e t o f c y lin d e r s .
The v e r t i c a l p o s i t i o n
showed a d e cr ea se i n t h e h e a t t r a n s f e r r e d r e l a t i v e to t h e h o r i z o n t a l
position.
This was a t t r i b u t e d t o a complex i n t e r a c t i o n between the
boundary l a y e r l e n g t h , th e flow p a t t e r n s which r e s u l t e d from the geome­
t r y , and th e cross s e c t i o n a l a rea exposed t o the upward flow.
These
geometric e f f e c t s were n o t i c e d t o d e c r e a s e w ith i n c r e a s i n g Pr and tl
number.
As th e Pr a n d tl number i n c r e a s e d t h e average p e r c e n t e r r o r
de creased from approxim at el y 16% to 1.8% depending on the e q u a ti o n form
used.
Data a t th e lo w e st Prand tl number ( a i r ) responded t o c o r r e la tio n s
66
b e t t e r when t h e geometric e f f e c t was taken i n t o account by using the
boundary l a y e r le n g th as a c h a r a c t e r i s t i c dimension whereas c o r r e l a ­
t i o n s f o r g l y c e r i n pro vi de d b e s t r e s u l t s usi ng the h y p o t h e t i c a l gap
width f o r a l l i n n e r body p o s i t i o n s .
The midrange Pr a n d tl number
f l u i d s (wa te r and s i l i c o n e ) e x h i b i t e d n e a r l y i d e n t i c a l r e s u l t s f o r a l l
e qu at io n fo r m s .
tra n s fe r data.
Many e q u a t i o n s were developed t o c o r r e l a t e th e he a t
These have been pro vid ed in t a b u l a r form and in n e a r l y
a l l cases are a c c u r a t e t o w i t h i n a few p e r c e n t .
Of most importance i s
th e c a p a b i l i t y t o p r e d i c t t h e h e a t t r a n s f e r over a wide Pr a n d tl number
•range and f o r s e v e r a l g e o m e t r ie s .
The fo ll o w i n g e q u a t i o n :
N u 8 = 0 . 2 7 7 R a B° - 2 7 4 P r 0 , 0 1 2
(5 .I)
p ro v i d e s a c c e p t a b l e r e s u l t s when computing the h e a t t r a n s f e r f o r a s e t
o f c y l i n d e r s t o an e n c l o s u r e .
Computational e f f o r t can be somewhat
reduced by using
Nub = 0.286 Rag0 ' 275
with only 1.0% r e d u c t i o n in accuracy.
( 5 .2 )
S i m i l a r r e s u l t s a re a v a i l a b l e
f o r c or re spo ndi ng e q u a ti o n s based on t h e h y p o t h e t i c a l gap width ( L ) .
Using pre vio us c o r r e l a t i o n s , t h e d a t a were c o r r e l a t e d and the
average d e v i a t i o n s compared.
Warrington [14J concluded t h a t e q ua ti ons
developed in h i s i n v e s t i g a t i o n could be extended t o most geometries
w i t h i n c e r t a i n l i m i t s of Rayleigh number and gap w id th .
In a l l
- 67 -
e q u a ti o n s using t h e gap w i d t h , t h e d a t a here s up port e d t h i s c oncl us io n
with average p e r c e n t d e v i a t i o n s well under th o s e i n d i c a t e d in Table 4 . 7 .
C o r r e l a t i o n s usi ng t h e boundary l a y e r l e n g t h proved t o be somewhat
above t h e average d e v i a t i o n and again p o i n t ou t th e geo metr ic problem
involved when m u l t i p l e bodies a r e used. Evidence, h e r e i n , i n d i c a t e s t h a t
any a p p r e c i a b l e d e p a r t u r e from th e geo me tr ie s s t u d i e d here and p r e v i o u s ­
l y , w i l l provid e t h e b e s t r e s u l t s when t h e c o r r e l a t i o n s based on gap
width a r e u t i l i z e d .
RECOMMENDATIONS
A l o g i c a l p r o g r e s s i o n beyond t h i s st ud y would i n c lu d e i n c r e a s i n g
the number o f c y l i n d e r s and v a ry in g t h e i r s p a c i n g , d i a m e t e r , and l e n g t h .
This e x t e n s i o n would move th e a p p l i c a b i l i t y o f th e c o r r e l a t i o n s to
d i r e c t i n d u s t r i a l a p p l i c a t i o n . For p r a c t i c a l use o f t h e d a t a base f o r
he a t t r a n s f e r in e n c l o s u r e s , c o n s i d e r a t i o n o f o t h e r m u l t i p l e geometric
shapes such as p l a t e s needs to be i n v e s t i g a t e d .
The he a t t r a n s f e r problem f o r bodies in e n c l o s u r e s i s a s s o c i a t e d
with a f l u i d flow problem.. In many c as e s t h e he a t t r a n s f e r problem can
be s olv ed e x p e r i m e n t a l l y w it h o u t a complete u n d e rs ta n d in g o f the flow
problem. However, as t h e geo metrie s become more complex, a more d e t a i l ­
ed s tu dy o f th e f l u i d problem must be c o n s i d e r e d .
APPENDICES
;
APPENDIX’ I
HEAT LOSSES FROM THE INNER BODY SUPPORT SYSTEM
'
Since the primary purpose o f t h i s st ud y was to deter mine th e h e a t
t r a n s f e r from f o u r i s o th e r m a l c y l i n d e r s a s o l u t i o n f o r t h e h e a t l o s s e s
a t t r i b u t e d t o the s u p p o r t system was r e q u i r e d .
Although t h i s system
was designed t o be as small as p o s s i b l e the amount, o f d e s i r e d i n s t r u ­
m e nta tio n was th e dominant c r i t e r i o n in s i z i n g the s u p p o r t system.
As
p o in te d ou t in Chapter I I 5 a l l m a t e r i a l s used in th e c o n s t r u c t i o n o f . t h e
s u p p o r t system were chosen t o minimize th e h e a t t r a n s f e r from t h a t system
to th e t e s t f l u i d .
Convection, c o n duc ti on, and r a d i a t i o n l o s s e s were
c ons ide red.- Since r a d i a t i o n and conduction l o s s e s f o r a i r were d i s c u s s e d
in Chapter I I I , only th e conduction and c onvec tio n l o s s e s f o r the o t h e r
f l u i d s a re d e t a i l e d h e r e .
I n i t i a l l y , i t was assumed t h a t only a small amount o f h e a t would be
c onvec te d from the s u p p o r t syst em , e s p e c i a l l y from th e s u p p o r t s p h e r e .
However, ex perimental i n v e s t i g a t i o n i n d i c a t e s t h a t th e s p h e r e was being
he ated by n a t u r a l co nve c tio n from th e t e s t f l u i d in t h e chamber.
A stud y was conducted by embedding two thermocouples i n the sup­
p o rt sphere.
The f i r s t 0.125 cm from th e o u t e r s u r f a c e and a second
1.5 cm i n t o t h e s p h e r e .
A comparison of th e two t e m p e ra t u re s i n d i -
• c a te d t h a t t h e s u r f a c e of th e sphere was an average o f 0 . 8k h o t t e r
than th e i n s i d e and reached a maximum o f 1 . 1 K h o t t e r a t h i g h e r AT1S 9
thus p r o v i d i n g ev ide nc e t h a t the sp her e was heat ed by the f l u i d r a t h e r
than by the s u p p o r t system.
70 -
Flow v i s u a l i z a t i o n s t u d i e s a l s o i n d i c a t e d t h a t the sp he re was
being h e a te d .
Figures 4 .5 and L I show th e flow p a t t e r n s around the
s u p p o r t system.
tion.
Figure 4 .5 r e p r e s e n t s t h e body i n the h o r i z o n t a l p o s i ­
There was no p e r c e p t i b l e f l u i d movement around th e main su p p o rt
stem, th e bottom o f th e s p h e r e , or t h e top o f t h e s p h e r e .
The flow
around t h e rods which connected th e c y l i n d e r and t h e sp h e r e can be a t ­
t r i b u t e d p r i m a r i l y t o the h i g h e r speed flow from th e c y l i n d e r .
The
flow p a t t e r n l e a v i n g the top of t h e lower c y l i n d e r was observed to bend
toward t h e s p h e r e .
However, t h i s phenomenon appears common as i t can
be seen i n r e f e r e n c e [ 3 ] where the s u p p o r t system did n o t i n t e r f e r e with
th e flow p a t t e r n s .
This i s pro bably caused by th e cool a i r being drawn
i n t o t h e co n v ec ti v e plume as d e s c r i b e d by o t h e r s i m i l a r flow v i s u a l i z a ­
t i o n s t u d i e s [ 1 0 , 11J.
The r e s u l t i n g flow p a t t e r n s around th e upper
c y l i n d e r i n d i c a t e t h a t th e s u p p o r t sp he re did no t a p p r e c i a b l y a f f e c t
the o v e r a l l f l u i d movement.
Figure L I r e p r e s e n t s th e flow p a t t e r n in th e v e r t i c a l pla ne cen­
t e r e d between two a d j a c e n t v e r t i c a l c y l i n d e r s .
The. flow around the.
sphere was induced by th e h i g h e r speed flow which r e s u l t e d from h e a t i n g
by th e c y l i n d e r s .
The c e n t e r sp he re appeared to d i r e c t th e flow where­
as a more t u r b u l e n t eddy would prob ab ly e x i s t in i t s absenc e.
Although
t h i s e f f e c t i s not co mpletely p r e d i c t a b l e , th e m a j o r i t y of h e a t t r a n s ­
f e r occurs w i t h i n the high speed flow re g io n along t h e wall o f th e c y l ­
i n d e r and th e p resen ce o f an eddy in th e c e n t e r of t h e geometry would
- 71
I
I
Figure I . I
Flow P a t t e r n Between C yl in de rs
in the V e r t i c a l P o s i t i o n .
•-
72 -
have l i t t l e e f f e c t on t h e o v e r a l l h e a t t r a n s f e r as evi den ce d by p r e ­
vious i n v e s t i g a t i o n [1 4 ] .
These ex per im en ta l o b s e r v a t i o n s and r e s u l t s s u p p o r t th e co ncl u­
si o n t h a t t h e s u p p o r t system i s being he a te d by co nvec tio n from the
cylinders.
T h e r e f o r e , t h e r e i s l i t t l e o r no h e a t t r a n s f e r from the
s u p p o rt system t o the t e s t f l u i d .
A stu dy o f th e p o s s i b l e h e a t l o s s in
the absence o f th e e f f e c t s o u t l i n e d p r e v i o u s l y p r e s e n t s a major d i f f i ­
culty.
However, th e fo ll o w i n g s i m p l i f i e d approach i n d i c a t e s t h a t in t h e ■
absence o f h e a t i n g o f th e s u p p o r t s p h e r e , l e s s than 0.4% o f t h e t o t a l
i n p u t power would be convected by the s p h e r e .
The f o ll o w i n g assumptions
a re r e q u i r e d in t h i s a n a l y s i s :
(1) That t h e s pher e i s is oth e rm a l and i t s te m p e r a t u r e i s t h a t
measured by the o u t e r s u r f a c e thermocouple.
(2) That a l l o f t h e h e a t le a v i n g t h e c y l i n d e r s through th e con­
n e c t i n g rod i s conducted t o the s p h e r e .
(3) That t h e h e a t le a v i n g t h e sph ere through th e main su p p o rt
stem i s conducted t o th e o u t e r body.
From Figure 1.2 t h e r e s u l t i n g h e a t bal anc e i s :
Qconv out = Qcond in - Qcond out .
(i .i )
The thermal c o n d u c t i v i t i e s and a re as a r e known and by usi ng ex per im ent al
val ues f o r t h e t e m p e r a t u r e s , QcONV OUT can
calculated.
These c a l c u ­
l a t i o n s were c a r r i e d o u t and i t was determined t h a t '
Qconv out
Qtotal
X 100 < 0.4%
( 1. 2)
- 73 -
Cyli n d e r
rod
rod
COND(in)
X ( 4 rods)
Sphere
QCONV(out)
^stem ^stem
COND(out)
^ stem
rh
qCOND(Out)
Outer Body
Figure 1.2
rfi
.
S t e a d y - S t a t e Heat Balance f o r
th e Support Sphere.
^0 ^
-
74 -
In th e absence o f h e a t i n g o f th e s u p p o r t s pher e the h e a t t r a n s f e r from
th e sphere t o the t e s t f l u i d could be c o n s id e re d n e g l i g i b l e .
Conduction h e a t lo s s from the t e s t a ppar at us was c a l c u l a t e d
using
a one-dimensional a n a l y s i s o f the conduction o f h e a t from th e s u p p o rt
sphere to the o u t e r body through the main s u p p o r t stem and t h e w ir in g
which i t contained'.
The fol lo wi ng e q u a ti o n was used in t h i s c a l c u l a - '
ti on:
Qcond = M s + kccA cc + kMlfrrtIi (T _ T )
Al
which was ad apted from r e f e r e n c e [1 4 ] .
v S
(1.3)
O'
This c o r r e c t i o n was n o t n e c e s ­
s ar y f o r a i r s i n c e t h e c a l i b r a t i o n f o r r a d i a t i o n l o s s e s in c l u d e d the
conduction l o s s e s through t h e main s u p p o r t stem.
APPENDIX II
DATA IN THE PARTIALLY REDUCED FORM
The fo ll o w i n g d a ta were p a r t i a l l y reduced using a-computer program
on th e Xerox Sigma VII computer system:
TERMINOLOGY
NGEOM i s t h e body p o s i t i o n i d e n t i f i e r
NGEOM = I , Tubes Ho rizon tal and O0 R ot a tio n
NGEOM' = 2 5 Tubes Ho rizontal and 45° R ot a ti o n
NGEOM = 3, Tubes V e r t i c a l and O0 R ota tio n
NGEOM = 4 9 Tubes V e r t i c a l and 45° R ot a ti o n
JFI i s t h e f l u i d i d e n t i f i e r
JFI = I 5 Air
JFI = 2, 99% Aqueous G lyc erin by Weight
JFI = 3, D i s t i l l e d Water
JFI = 4, Dow Corning Dime th ylp oly si lo xa ne (20 c e n t i poise
a t 25°G)
TAVGI i s
th e aver age i n n e r
body
te m p e ra t u re in Kelvin
TAVGO i s
t h e aver age o u t e r
body
t e m p e ra t u re in Kelvin
QCONV i s
t h e he a t t r a n s f e r
in Watts
from
t h e tu be s by n a t u r a l con vec tion
QLOSS i s th e he a t l o s s due to c onduc ti on and r a d i a t i o n in Watts
m
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
I
- 76
FJI
I
I
I
I
I
I
I
I
I
I
I
. I
I
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
TAVGI
300.8911
320.953 I
320.0317
335.5232
3 6 9 = I 1.21
291.8083
351=5647
' 3 12.6440
326.522?
3 6 0 =7 2 1 4
341=5488
2 9 4 =7 5 17
306.2866
303.7593
290=5134
297.3137
308.8235
312=6350
324.9409
330.9578
335.0024
356=3562
'374.7368
339.8726
368.4402
359=7625
317=3740
287=1511
295=6953
305=7207
312.4058
319.2529
301.8118
291.4773
308.0322
317=9199
340=5874
370=5811
365.6000
3 6 0 = 1775
3 5 4 =6 0 0 6
302.5850
312.1499
TAVGO
275.8293
276.2092
2 7 6 =1453
276.2732
276.5120
277 . 7 7 6 1
276.7847
.277.1636
276.5759
278.0186
277.1040
276.6909
2 7 6 =7847
279=1641
278.4729
279=4055
281.3201
280.7588
284.0703
286.3162
288.2358
294.688?
299.3350
289.5476
298=5015
291=3179
281=0544
279.7654
283.1172
288=7095
292.6829
296.2654
286.7583
281=4172
280=0361
280=7419
283.6633
292.4072
293=6790
292.6709
292.4978
279=6553
281=0840
OCONV
10.8257
21.8596
21.2224
30.6386
53=6764
4=8666
41.0307
16=3060
24.8598
45=6148
33=8734
7.0222
13=0315
102=0632
34.4389
64.2133
132.0711
162.8939
263.7048
317.5320
347.6680
6 I 1.4028
900.5037
402.3882
769.8862
684.5491
205.765?
145=2637
327.2454
5 6 9 = 7 9 17
721.3015
910.4226
429.7764
227.9674
201.9373
301=9424
531=7197
829.9453
743=3945
679.6592
610=5542
160.1241
239.4488
QLOSS
4=5495
8=7719
■
8.5714
12.2767
21.7092
2=7856
16.5462
6.9135
10=0692
1 9 . 130 I
13=8197
3=3645
5.5967
1.1178
=5 6 4 4
.8142
1.2931
I .4852
I =9065
2.0786
2=1959
2.8630
3=4736
2=3336
3,2268
3=1033
I .6954
=3 9 3 4
=6 4 0 3
.8966
1.0205
1=1814
.7676
=5 1 4 3
1.4762
1.9580
3=0245
4.0854
3.7669
3=5273
3=2534
1=2175
1=6637
- 77 -
JFI
NGEOM
4
I
4
I
4
I
4
I
4
I
4
I
2
I
2
I
I
2
2
I I
2
I
2
I
2
2
I
2
I
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
2
3
2
• 3
2
3
2
3
2
3
2
3
2
3
4
2
2
4
4
2
4
2
2
4
2
4
4
2
2
4
3
I
I
.3
3
I
TAVGO
283.6130
TAVGI
3 2 4 o6 2 3 5
33 I » 2 6 8 8
337.7014
2 8 8 » I 125
291-7405
296.0330
319.6648
2 9 9 . 8 1 13
347.5947
331.0989
2 8 9 . 1899
310.0530
314.6541
303.2754
292.8750
323.4746
337.5393
296.5020
309.4761
318» 1292
330.6033
305.2446
352.3059
362.9478
290.4514
293.8064
299.4487
305.3650
310.4653
292.9460
296.7964
304.3848
287.3447
296.0232
320.8269
310.7966
344=0618
328.7891
353.7625
37 1 . 6 7 0 7
318.6970
335.6062
3 5 0 . 1421
•
QCONV
QLOSS
343.4346
2 . 1734
285.6982
393.8181
287.3083
279.8755
280»1040
279.2913
279.9177
2 8 1 . 1008
279.5200
278.0269
2 7 7 .6572
280.1799
278.4941
279.5029
278.9268
284.1331
285.8862
281.8723
280.6448
284.1836
286.7375
283.5879
289.9326
293.3992
282.2388
281.1768
282.3479
285.765 I
287.8286
281.9902
263.1807
286.7625
279.0793
280.5225
284.5857
283.4495
287.3291
282.3818
285.9780
293.5759
280.2349
279.9475
281.0332
457.6252
2.4187
2.6674
.4388
.6070
. 884 I
8.3035
4.1319
15.2181
I 1.0470
2.3144
6.1914 .
7.2411
4.8593
2.9309
1.8399
2.3603
.6937
1.3565
I .6995
2.0301
1.0373
2.8257
3» 1 4 4 1
.4304
C6 4 4 8
.8925
1.0346
I . 1863
.5735
.7224
.9285
.4401
.8321
1.9365
1.4569
3-0162
2.4628
3.5772
4.1015
19.4012
29.9733
39.9838
43.8588
65.3452
108.3142
23.0483
8.4386
44.8582
33.2602
4.8014
15.7129
20.5415
I 1.9163
6.1841
244.6227
389.9072
52.8402
139.7749
193.7490
302.4075
97.6798
560.0740
7 0 4 . 100 I
182.8057
309.6936
494.4407
634.1375
797.5591
268.0454
3 7 2 . 1580
558.9868
43.3143
96.5606
293.9106
200.8837
527.4548
396.609 I
664.4192
629.9480
13.8114
22.0749
28.6585
- 78 -•
NGEOM JFI
3
3
3
3
3
3
3
3
3
I
I
I
I
I
I
I
I
I
4
I
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
I
I
I
I
I
I
I
I
3
3
3
3
4
4
4
4
■4
4
4
4
4
4
3
I
2
2
2
2
2
2
CO CO CO
2
2
2
2
' 2
2
2
2
2
2
2
2
2
2
3
3
3
3
TAVGI
363.3445
315=4006
2 8 7 = 1294
289=2085
2 9 6 =5 5 7 I
301=8481
308=206 I
313=9126
332=2793
331=4878
338=9121
2 8 7 =8 2 8 6
293=1897
299=5920
306=3049
347=8640
312=6084
321=4600
357=3816
299=0647
306=3892
322=6948
329=7690
3 5 1=9438
317=5190
356=3040
344=4456
337=0203
3 1 0 =7 1 6 I
310=4563
295=6399
303=0332
339=6428
349=2419
314=2004
320=7068
327=3267
333=6094
355=2363
301=4688
305=0125
288=7729
29 I = 0 9 6 4
TAVGO
2 8 1 =2 9 9 I
281=1428
QCONV
35=8345
280=8137
1=2154
3=2904
3=8881
6=3676
9=3484
11=5619
20=1363
20=2283
23=7913
2=6401
4=4281
6=5169
8=7636
28=0481
12=2537
15=0014
33=6312
67=2570
106=6698
216=6268
277=1313
520=0908
170=223 I
603=5659
4 2 9 =5 9 I I
340=4197
132=7079
131=9876
62=6259
107=7386
433=8677
525=2690
172=4135
223=6218
27 9= 1997
341 = 1960
605=6628
42,0=6980
506=5647
146=9450
201= 199 I
278=7485
281=3118
280=5520
280=0532
280=3787
280=3577
279=3420
279=5369
278=6807
278=8081
279=2468
279=6130
280=2561
278=7358
279=9939
279=943 I
282=6301
282=5563
285=6233
287=3833
292=1392
286= 14 1 I
289=7793
2 9 0 =4 1 2 8
290=1853
282=9326
282=9746
277=3340,
276=8955
279=6203
285=3135
280=1716
281=2695
282=8066
263=4744
286=5537
285=9573
287=5078
281=1851
281=6025
I I =6 6 2 6
QLOSS
49=8613
17.4667
2=9484
3=9473
7=3987
10=0791
13=4702
16=6358
27=7850
27=3581
32=2577
3=3185
5=8361
8=9935
12=4840
38=4469
15=9819
2 I =1026
45=4242.
=8 4 4 4
1=2141
1=8451
2=0588
2=8736
1=5451
3=1702
2=6027
2=2685
I =3648
1=3619
=9 0 0 1
I =2999
2=8463
3=0003
1=6599
1=9189
2 = 1456
2=3929
3=2239
=8 3 7 8
=9 3 8 8
=4 2 9 8
=5 327
.79
\IGEOM J F I
3
3
3
3
3
3
3
3 .
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
■3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
3
4
4
3
4
3
4
3
4
3
4
3
4
■ 4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
.4
4
4
TAVGI
TAVGO
294.2039
31 1 . 0 0 2 4
309.2036
299.4063
312.391 I
310.2861
307.4050
295*2683
290.4421
292.7297
297.6902
30 1. 1409
290.6375
299.7778
309.3889
2 9 4 . 4 7 17
320.0229
.332.3250
338.8718
345.7432
303.4990
313.9126
324.8999
350.8672
359.8857
313.6216
360.7100
29 1 . 4 3 4 1
299.1226
308.4937
319.4910
331.6375
312.8015
296.3948
325.3047
305.3259
344.9695
282.6887
288.7380
289.9534
286.3120
290.6938
291.0413
290=4543
2 8 4 . 1206
283.2729
283.6550
284.4138
285.5645
280.1335
281.5396
282.6677
2 8 0 . I 040
283=5542
285.4182
286.508 I
287.8032
281=0417
282.1628
283=2393
287.0830
2 9 0 =4 0 0 I
281=5437
290.8716
280=6196
281.067 I
2 8 1 .4678.
282.9075
284.5940
280.6997
281.9187
283.4368
282.8655
287.2000
QCONV
265.4355
718.7839
604.7532
332.0325
713=4114
607.6108
509.338 I
263=8301
146.0833
200=8616
335.0520
420.7029
56.1322
108.5977
179.8355
80.1735
270=4939
376=2566
435= 1294
500.9907
143.5284
225=7859
320.7139
568*2803
643=3804
224.4350
643=2002
56.0972
106=4021
179.8742
268.9871
376.3459
224=6364
8 I =3 3 9 3
321=3735
144.4204
499=1257
QLOSS
.6297
I =1941
1=0303
=7 0 1 9
I .1699
1=0336
.9107
.5994
.3999
.5089
.7292
.8429
.5922
I .0027
I .4919
=7 9 8 2
2 =0 17 3
2.5769
2 =867 6
3= 17 37
1.2508
1.7662
2.2886
3=5009
3=8056
I =7595
3.8168
=5 9 5 3
I =0082
I =4766
2=0068
2=5782
1.7675
.7880
2 =2 9 8 I
1=2498
3 = 1745
BIBLIOGRAPHY
1. .
McAdams, W.H . , Heat T r a n s m i s s i o n , Third Ed., McGraw-Hill, New
York, 1954, Chapter 7.
2.
G e b h a r t , B . , Heat T r a n s f e r , Second E d . , McGraw-Hill, New York,
1971, Chapter 8.
3.
Holman, J . P . , Heat T r a n s f e r , Th ir d E d . , McGraw-Hill, New York,
1972, Chapter 7.
4.
F u j i i , T. and F u j i i 5 M., The Dependence o f Local N u s s e lt Number
on Pr and tl Number in the Case o f Free Convection Along a
V e r t i c a l Su r fa ce with Uniform Heat Flux , I n t e r n a t i o n a l J o u r ­
nal o f Heat and Mass T r a n s f e r , Vol. 19, pp. 121-122, 1975.
5.
Roy, S . , High- P r a n d t I -Number Free Convection f o r Uniform Surfa ce
Heat F lu x, Journal of Heat T r a n s f e r , V ol. 9 5, pp. 124-126,
1973,
6.
C h u r c h i l l , S.W. and Ozoe, H., A C o r r e l a t i o n f o r Laminar Free Con­
v e c t i o n from a V e r t i c a l P l a t e , T r a n s a c t i o n s of th e American
S o c i e t y o f Mechanical E n g i n e e r s , V o l. 95, pp. 540-541, 1973.
7.
Le in h a rd , J . H . , On t h e Commonality o f Equations f o r Natural Con­
v e c t i o n from Immersed Bodies, I n t e r n a t i o n a l J o u rn a l o f Heat
arid Mass T r a n s f e r , Vol. 16, pp. 2121-2123, 1973.
8.
E c k e r t , E.R.G. and Soehngen, E.-, S t u d i e s on Heat T r a n s f e r in
Laminar Free Convection with the Zehnder-Mach I n t e r f e r o m e t e r ,
Technical Report No. 5747, U.S.A.F. Air M a te r ia l Command,
Dayton, Ohio, 1948.
9.
Li e b e m a n , J . and G e b h a r t, B . , I n t e r a c t i o n s in Nat ura l Convection
from an Array o f Heated El ements, E x p e r i m e n t a l , I n t e r n a t i o n a l
J ourn a l of Heat and Mass T r a n s f e r , V o l . 12, pp. 1385-1395,
1969.
10.
Brodowic z , K. and K i e r k u s , Wf T . , Experimental I n v e s t i g a t i o n o f
Laminar Free Convection Flow i n Air Above H o ri z o n ta l Wire with
Constant F l u x , I n t e r n a t i o n a l J ourn a l of Heat and Mass Trans' f e r , Vol. 9 , pp. 81-94, 1966.
11.
Forstrom, R .J . and Sparrow, E.M., Experiments on t h e Buoyant
Plume Above a Heated H oriz on ta l Wire, I n t e r n a t i o n a l Journal
' o f Heat and Mass T r a n s f e r , V o l. 10, pp. 321-330, 1967.
- SI -
12.
13. '
Ti ll m a n , E . S . , Natural Cpnvection Heat T r a n s f e r from Ho rizon tal
Tube B und le s, ASME Heat T r a n s f e r Co nference, August 1976,
p a p er 76-HT-35.
Davis, L.P. and P e r o n a 5 J . J . , Development o f Free Convection
Axial Flow Through a Tube Bundle, I n t e r n a t i o n a l Jou rnal o f
Heat and Mass T r a n s f e r , Vbl. 16, pp; 1425-1438, 1973.
14.
War ring ton, R.O., Natural Convection Heat T r a n s f e r Between Bodies
and T h e i r E n c l o s u r e s , Ph.D. D i s s e r t a t i o n , Montana S t a t e Univer­
s i t y , 1975.
15.
Kuehn, T.H. and G o l d s t e i n , R . J . , C o r r e l a t i n g Equations f o r Natu­
r a l Convection Heat T r a n s f e r Between H oriz on ta l C i r c u l a r C yli n­
d e r s , ' I n t e r n a t i o n a l - ' Journal of Heat and Mass T r a n s f e r , V o l. 19,
pp. 1127-1134, 1976.
16.
D u t t o n , J. C. and W el ty , J . R . , An Experimental Study o f Low Pr a n d tl
Number Natural Convection in an Array o f Uniformly Heated
V e r t i cal C y l i n d e r s , ASME Heat T r a n s f e r D iv i s i o n , F e b r u a r y 1975,
p a p er 75-HT-AAA.
17.
Weber, N ., Natural Convection Heat Transfer. Between a Body and i t s
S p h e r ic a l E n c l o s u r e , Ph.D. D i s s e r t a t i o n , Montana S t a t e Univer­
s i t y , 1971.
18.
Bi s h o p , E .H ., Heat T r a n s f e r by Natural Convection Between I s o t h e r ­
mal C o n ce n tr i c S p h e r e s , Ph.D. D i s s e r t a t i o n , U n i v e r s i t y o f
Texas, A u s t i n , 1964.
19.
Scanl an, J . A . , Bi s h o p , E.H. , and Powe, R .E ., Nat ura l Convection
Between Co nce ntri c S p h e r e s , I n t e r n a t i o n a l J o u rn a l o f Heat and
Mass T r a n s f e r , Vol. 13, pp. 1857-1872, 1970.
I
#
cop. 2
C r u p p e r , G ordon
N a t u r a l c o n v e c t io n
h e a t t r a n s f e r b e tw e e n a
fix e d a r r a y o f c y lin d e rs
and i t s c u b ic a l e n c lo s u r e
inT
^ mbr a r V loan