Natural convection heat transfer within enclosures at reduced pressures
by Peter Kevin Brown
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Peter Kevin Brown (1980)
Abstract:
Natural convection heat transfer in air within enclosures has been investigated over the pressure range
of 2670-86,180 Pa (20-646.4 mm Hg), Dimensionless correlations have been generated from the data;
The best correlation found included a correction for the air density: Nul = .342 RaL^1/4 (ρ/ρatn,)^.129
where L is the hypothetical gap width. The Rayleigh number in the experiments ranged over 1x10^3 2x10^6. The geometries used were cylinder-cube (inner body-outer body) and cube-cube, with the
bodies mounted concentrically in both cases. Temperature profiles at four positions (0°, 34°, 80°, and
160° from the upward vertical) were measured for the cylinder-cube case. The thickening of the
boundary layer at low pressures and the region of constant temperature between the bodies at high Ra
were clearly observed. STATEMENT OF PERMISSION TO COPY
In p re s e n tin g th is
th e s is in p a r t i a l f u l f i l l m e n t o f the r e q u ir e ­
ments f o r an advanced degree a t Montana S ta te U n iv e r s ity , I agree th a t
th e L ib r a r y s h a ll make i t f r e e l y a v a ila b le f o r in s p e c tio n .
I fu rth e r
ag ree th a t perm ission f o r e x te n s iv e copying o f t h is th e s is f o r s c h o l­
a r l y purposes may be g ra n te d by my m ajor p ro fe s s o r, o r ,
by th e D ir e c t o r o f L ib r a r ie s .
It
is understood th a t any copying or
p u b lic a tio n o f t h is th e s is f o r f in a n c ia l
w ith o u t my w r it t e n p e rm is s io n .
in his absence,
gain s h a ll not be allow ed
NATURAL CONVECTION HEAT TRANSFER WITHIN
ENCLOSURES AT REDUCED PRESSURES
by
PETER KEVIN BROWN
A th e s is subm itted in p a r t i a l f u l f i l l m e n t
o f th e req u irem en ts f o r th e degree
of
MASTER OF SCIENCE
in
Mechanical E n g in ee rin g
Chai rp e rs o h , LgfacMate Cotfinflttee
Head, M ajor Department
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
Septem ber, 1980
ACKNOWLEDGEMENTS
The a u th o r wishes to thank D r. R, L, Mussulman f o r re a d in g the
o r ig in a l ty p e s c r ip t and e s p e c ia lly D r, R, 0 , W arrington f o r guidance and
suggestions thro u g h o u t th e com pletion o f t h is w ork.
S in c e re thanks are
a ls o due to C harlene Townes f o r ty p in g th e f i n a l copy o f th e th e s is .
TABLE OF CONTENTS
C hapter
Page
V IT A '........................................................' ....................... .................................................. ....
ACKNOWLEDGEMENT
..............................................
11
til
LIST OF TABLES.............................................. .... .................................................. V . .
v
LIST OF F IG U R E S ..............................................
vi
NOMENCLATURE . . .........................................
v ii
ABSTRACT................................................................................................
x
I.
I
II.
III.
IV .
V.
V I.
INTRODUCTION.............. .... ............................................................................................
LITERATURE REVIEW ............................................................
i . .
EXPERIMENTAL APPARATUS AND PROCEDURE ................................
. . . .
13
HEAT TRANSFER AND TEMPERATURE PROFILE RESULTS....................... . . . .
A MODIFICATION OF THE METHOD OF RAITHBV AND HOLLANDS FOR '
CUBICAL GEOMETRIES ..................................................................... . . . . . .
APPENDIX I I .
APPENDIX I I I .
75
DERIVATION OF THE METHOD OF RAITHBY AND HOLLANDS
. .
DATA REDUCTION PROGRAMS.......................................................
PARTIALLY. REDUCED DATA
. . . .'.
21
57
CONCLUSIONS . ................................................................
APPENDIX I .
3
. . .
.
.' . . . . .
BIBLIOGRAPHY....................... • • ..................... ......................................................................102
77
86
96
V
LIST OF TABLES
Tflble
Pflge
4 .1
Rflnge o f Geometries and C o r r e la tin g Param eter . . . . . . . .
22
4 .2
C o r r e la tio n E quations f o r A ll Data
27
4 .3
F i t o f P res e n t Data to P revious C o r r e l a t i o n s ....................... *. .
4 .4
C o r r e la tio n Equations from the V a r ia b le D e n s ity Data Alone
4 .5
D e n s ity -C o rre c te d C o rre la tio n s f o r th e V a ria b le D e n s ity Data
45
4 .6
D e n s ity -C o rre c te d C o rre la tio n s f o r A ll
47
5 .1
C o r r e la tio n s
5 .2
C o r r e la tio n s ( 5 . 8 ) and ( 5 . 9 ) w ith AU
........................................................
35
.
Data . . . . . . . . .
( 5 . 8 ) and ( 5 . 9 ) w ith th e V a ria b le D e n s ity Data .
Data
. . . . . . . . .
36
62
63
Vt
LIST OF FIGURES
F ig u re
3.1
Psge
Schem atic o f Heat T ra n s fe r'A p p a ra tu s w ith S upporting
In s tru m e n ta tio n . . . .
. . . . . . . . . . . . . . . . . . . .
14
I
4 .1
Local Gap W idth a f o r th e C y lin d er-C u b e G e o m e t r y .......................
25
4 .2 .
A ll o f th e
Heat T ra n s fe r D ata:
Nu l
26
4 .3
A ll o f th e
Heat T ra n s fe r D ata:
Nub v s . Rab . . . . . . . . .
.30
4 .4
A ll o f th e
Heat T ra n s fe r D ata:
Q v s , RaL . . . . . . . . . .
32
4 .5
V a r ia b le - D e n s ity D a ta , C y lin d e r-C u b e Geometry . . . . . . . .
39
4 .6
V a r ia b le -D e n s ity D a ta , Cube-Cube Geometry ................................
, .
40
4 .7
Al I o f th e V a r ia b le -D e n s ity D ata:
.......................
43
4 .8
4 .9
vs.
RaL .................................
Nul „ v s . RaL„
Tem perature P r o f i l e Data a t Two D if f e r e n t P ressures:
S o lid L in e s — AT=23.3°C ( 4 2 . 0 ° F ) , P re s s u re = !6 .3 hm Hg;
■ Open S ym bols--A T=20.7°C ( 3 7 . 2 ° F ) , P ressu re=641.9 mm Hg
. . .
Tem perature P r o f ile s a t Two D if f e r e n t P ressures:
• S o lid L in e s --A T = 3 0 .2 °C ( 5 4 . 5 ° F ) , P res s u re = 4 8 .0 mm Hg;
Open SymbolS --A T = I8 . 7°C ( 3 3 . 6 ° F ) , P ressu re=149.6 mm Hg ' .
4 .1 0 Tem perature P r o f i l e Data a t Two D i f f e r e n t P ressures:
S o lid L in e s — AT=23.8°C ( 4 2 . 8 ° F ) , P res s u re= 2 4 8.8 mm Hg;
Open S ym bols--A T=22.9°C ( 4 1 . 3 ° F ) , P res s u re= 5 0 5.7 mm Hg
.
50
. .
51
. . .
52
4 .1 1 Tem perature P r o f i l e D ata: AT=22.8°C ( 4 1 . I ° F ) ,
P re s s u re = 3 9 6 .8 mm Hg . . . . ............................ ....................................
.
53
....................... ....
59
..........................................
65
The Shaded Regions are
..............................................
66
5 .1
Nom enclature f o r th e C y lin d er-C u b e Geometry .
5 .2
Nom enclature f o r Flow Around a Cube . .
5 .3
I d e a l i z a t io n o f Flow Around a Cube:
o f Equal Area
A l ,1 Nom enclature o f th e Method o f R a ith b y and H ollands
. . . . .
79
A l . 2 Nom enclature f o r Two-Dimensional o r A xisym m etric Flow . . . .
84
vt i
NOMENCLATURE
Symbol
D e s c rip tio n
a
Any c h a r a c t e r is t ic le n g th
b
Boundary la y e r le n g th on th e in n e r body
C,C1-4
CP
E m p ir ic a lly determ ined c o n stan ts
C onstant p ressu re h e at c a p a c ity
d
Length o f a s id e o f a cube
D i(D o )
D ia m eter o f a sphere o f s u rfa c e area equal to t h a t',o f the
in n e r (o u t e r ) body
g
A c c e le ra tio n due to g r a v it y
9x
G r a v ita tio n a l a c c e le r a tio n in the d ir e c t io n o f th e x
c o o rd in a te
Gr
The G rashof number, g 3 (T .j-T 0 )a 3p2/ y 2
h
Average h e a t t r a n s f e r c o e f f i c i e n t , P1V ( T 1-- T q )
k
Thermal c o n d u c tiv ity o f a i r
L 5L 11L"
Gap w id th s as d e fin e d on p. 2 3-24
m, Oi0 )
Mass flo w r a te in th e in n e r ( o u t e r ) re g io n o f th e boundary
la y e r
mm Hg
A u n it o f p re s s u re , 760 mm Hg = I atmosphere
M
( V
Nu
The N u s s e lt number, h a /k
P
P ressure o f a i r a t e x p e rim e n ta l c o n d itio n s
Pa
A u n it o f p re s s u re , 1 01 ,32 5 Pa = I atmosphere
Pr
The P ra n d tl number, yCp/k
t 5I
Z (V
ts)
Vttt
Symbol
cI o r % onv
D e s c rip tio n
Heat tr a n s fe r r e d by n a tu ra l convection
^cond
Heat t r a n s fe r r e d by conduction through a s ta g n a n t f l u i d
under s im i l a r c o n d itio n s
q"
Heat tr a n s fe r r e d by c o n ve c tio n per u n it a rea
Q
^c onv^cond
r
Radius o f c u rv a tu re
r*
Dim ensionless ra d iu s d e fin e d on p .19
R1 (R0 )
Radius o f a sphere o f volume equal to t h a t o f th e in n e r
( o u t e r ) body
Ra
The R a y le ig h number, G r-P r
S
Length o f th e x c o o rd in a te along a body
T
Tem perature
Ti
(T 0 )
In n e r (o u t e r ) body te m p e ratu re
Tem perature a t th e w a ll
TS
L
Tem perature f a r away from th e w a ll
I*
D im ensionless te m p e ratu re d e fin e d on p .1 9
Tm
Tem perature a t ym o r te m p e ratu re in th e f l u i d
in n e r and o u te r e q u iv a le n t conduction la y e rs
Ti
(T 0 )
Average te m p e ratu re in th e in n e r (o u t e r ) re g io n o f the
boundary la y e r
Average te m p e ratu re between th e in n e r and o u te r e q u iv a le n t
conduction la y e rs
Tm
ATr
.
A re fe re n c e te m p e ratu re d iffe r e n c e
V e lo c ity component in th e d ir e c t io n o f x
U
U
max
between th e
Maximum v e lo c it y in th e p r o f i l e
TX
Symbol
D e s c rip tio n
X
C o o rd in ate along th e body in th e d ir e c t io n o f flo w
y
C o o rd in ate o u t o f th e body normal to x
L o c atio n o f th e v e lo c it y maximum in th e p r o f i l e
ym
Thermal e x p a n s iv ity o f a i r ( l / V ) ( d V / d T l , where V is th e
volum e; f o r an id e a l gas (3 = 1 /T
3
ri
(Io )
Mass flo w r a te per u n it depth in th e in n e r (o u t e r ) re g io n
o f th e boundary la y e r
6
Boundary la y e r th ic k n e s s o r any one o f th re e gap w idths
L , L ' , o r L"
A (x ) o r Ajl(X )
Local e q u iv a le n t conduction la y e r th ic k n e s s
A o r A^
Average e q u iv a le n t conduction la y e r th ic k n e s s
n
D im ensionless y c o o rd in a te , y /6
•
y
. V is c o s ity o f a i r
p .
D e n s ity o f a i r in th e e xp e rim en tal c o n d itio n s
patm
D e n s ity o f a i r a t atm ospheric pressure and standard
te m p e ratu re ( 2 9 8 . 15 °K)
ABSTRACT
N a tu ra l c o n ve c tio n h e a t t r a n s f e r In a i r w ith in
in v e s tig a te d o ver th e pressu re range o f 2670-86„180
Dim ensionless c o r r e la tio n s have been g e n e ra te d 'fro m
c o r r e la t io n found in c lu d e d a c o r r e c tio n f o r th e a i r
Nu l -
.342. R a '/ 4 tp /P a J
e n clo s u re s has been
Pa (2 0 -6 4 6 ,4 ran H g ),
th e d a ta ;
The best
d e n s ity :
- 129
where L is th e h y p o th e tic a l gap w id th .
The R ayleigh number in the e x p e r­
im ents ranged o ver Ix lO 3 - 2x 106. The geom etries used were c y lin d e r-c u b e
(in n e r b o d y -o u te r body) and cube-cube, w it h th e bodies mounted concen­
t r i c a l l y in both cases.
Tem perature p r o f i l e s a t fo u r p o s itio n s ( 0 ° , 3 4 ° ,
8 0 ° , and 160° from th e upward v e r t i c a l ) were measured f o r th e c y lin d e r cube case.
The th ic k e n in g o f the boundary la y e r a t low pressures and
th e re g io n o f c o n s ta n t te m p e ratu re between th e bodies a t high Ra were
c l e a r l y observed.
CHAPTER I
INTRODUCTION
The phenomenon o f n a tu ra l c o n vectio n heat t r a n s f e r w ith in enclosures
has re c e iv e d a g re a t deal o f a t t e n t io n in re c e n t y e a rs .
There is a grow­
in g demand f o r an u nderstand ing o f t h is phenomenon in such areas as
n u c le a r d e s ig n , e le c t r o n ic packag in g , space h e a tin g , and s o la r c o lle c t o r
d e s ig n .
The m a jo r ity o f t h is e f f o r t has been e x p e rim e n ta l.
l i n e a r i t y and c o u p lin g o f the governing d i f f e r e n t i a l
The non­
e q u atio n s o f c o n t i­
n u it y , momentum, and energy have made a n a ly t ic a l s o lu tio n s d i f f i c u l t to
fin d .
Those few t h a t e x i s t a p p ly o n ly to r e l a t i v e l y sim ple g e o m etrie s ,
such as c o n c e n tric c y lin d e r s o r spheres.
In th e p re s e n t s tu d y , n a tu ra l c o n ve c tio n h e at t r a n s f e r in a i r from
an is o th e rm a l
(h e a te d ) in n e r body to an is o th e rm a l
was in v e s tig a te d .
(c o o le d ) o u te r body
More s p e c i f i c a l l y , r a te s o f h e at t r a n s f e r were meas­
ured f o r v a rio u s te m p e ratu re d iffe r e n c e s between the bodies and d i f f e r e n t
pressures o f a i r in th e t e s t space.
The o u te r body was c u b ic a l; the
in n e r bodies were a v e r t ic a l c y lin d e r w ith h e m is p h e rica l end caps and a
cube.
The p rim ary purpose o f t h is
in v e s tig a tio n was to f in d an e m p iric a l
c o r r e la t io n f o r th e h e at tr a n s fe r r e d by n a tu ra l c o n vectio n in a i r from a
body to i t s e n c lo s u re as a fu n c tio n o f a i r pressure ( i . e . ,
vacuum).
3
6
R ayleig h number in these experim ents ranged o ver 10 - 2 x 10 .
Since
e v a c u a tio n o f the c a v it y surrounding a heated d e v ic e , such as a s o la r
c o l le c t o r tu b e , is an e f f e c t i v e way to reduce heat loss [ 1 - 3 ] , such a
The
2
r e la t io n s h ip would be q u ite u s e f u l.
a g a in s t p re vio u s r e s u l t s .
The c o r r e la tio n s found w e re .te s te d
In a d d it io n , a c o r r e la t io n based on th e method
o f R a ith b y and H ollands [ 4 ] was developed and te s te d a g a in s t th e d a ta .
Tem perature p r o f ile s were a ls o measured f o r some o f th e h e at tr a n s ­
f e r c o n d itio n s .
The p r o f i l e s are u s e fu l f o r e lu c id a tin g g eneral tre n d s ,
such as th e th ic k e n in g o f the boundary la y e r a t low p re s s u re .
t i o n , th e y may p ro v id e a means o f v e r i f i c a t i o n
o r a n a ly t ic a l
s o lu tio n s which may be advanced.
In a d d i­
f o r any fu t u r e num erical
CHAPTER I I
LITERATURE REVIEW
N a tu ra l co n ve c tio n phenomena f a l l
v e c tio n in an i n f i n i t e
■
b ro a d ly in to two c a te g o rie s :
con­
f l u i d b a th , o r e x te rn a l c o n v e c tio n , and convection
w ith in an e n c lo s u re , o r in t e r n a l c o n v e c tio n .
The fo llo w in g re vie w is
in te n d e d to p ro v id e a background f o r th e p re s e n t in v e s tig a tio n and is
not a com plete survey o f re se a rch in n a tu ra l c o n v e c tio n .
Dim ensional a n a ly s is has shown [ 5 , 6 ] t h a t e x te rn a l n a tu ra l convec­
t io n may be c o r r e la te d by
Nua = f ( G r a ,P r)
a
a
where a is some c h a r a c t e r is t ic dim ension and
( N u s s e lt num ber),
1S - T T
Sra ■ 9% A? S ' P2
T-I
Pcd
Pr
( G rashof num ber),
( P ra n d tl num ber).
O ften th e e f f e c t s o f Gra and Pr can be combined by
Nua = f( R a a ) , where
d
a
Ra=, = Gra • Pr
a
. a
In t h is in v e s t ig a t io n , Raa w i l l
(R a y le ig h num ber).
be v a rie d la r g e ly by v a r ia t io n s in p ( i .
e . , p ressu re o f th e a i r ) and a ls o by v a r ia tio n s in a and in AT.
4
Elenbaas [ 7 ] developed some e a r ly c o r r e la tio n s f o r e x te rn a l n a tu ra l
co n ve c tio n h e a t t r a n s f e r from a v e r t ic a l c y lin d e r :
1 /3 , 0 , 1 / 1 2
Nud , we x p (-2 /N u diW) - O . S R a ^ R a ' ;
and from a h o r iz o n ta l c y lin d e r :
Nud!Wexp(-2/N Ud>w) = 0.16RaJ(2/g(RadtWj
where g(Rad w) is a fu n c tio n presen ted g r a p h ic a lly .
The s u b s c rip t d
r e fe r s to th e d ia m e te r and h to th e h e ig h t o f the c y lin d e r , and w means
t h a t th e f l u i d
p ro p e rtie s a re to be e v a lu a te d a t the w a ll te m p e ra tu re .
King [ 8 ] c o r r e la te d e x te rn a l n a tu ra l convection from s e v e ra l geo­
m e tric shapes, in c lu d in g ones s im ila r to those used in t h is in v e s tig a ­
t i o n , by
Nug = 0 .6 0 R a g ^ f o r 10^<Rag<10^
where
I
v e r t ic a l dimension
T
h o riz o n ta l dimension '
Holman [ 6 ] re p o rts a c o r r e la t io n f o r a v e r t ic a l c y lin d e r o f
Nuh = O-BORat1/ 4 f o r ^ > ^ 1 /4
b rh
•
where D is th e d ia m e te r and h th e h e ig h t o f th e c y lin d e r .
L ien h ard [ 9 ] found t h a t la m in a r e x te r n a l convection could be c o rre ­
la t e d w e ll by a balance o f buoyancy and viscous fo rc e s on th e body.
5
Using v e l o c i t y and te m p e ratu re p r o f i l e s
f o r a f l a t p l a t e , he ob tain ed
Nub = 0 .5 2 R a b/ 4
where b is th e le n g th o f th e therm al boundary la y e r on th e body.
There have been s e v e ra l e x p e rim e n ta l and a n a ly t ic a l
in v e s tig a tio n s
o f e x te r n a l c o n vectio n to a i r a t low Ra (lo w pressure o f a i r , small
c h a r a c t e r is t ic dim en sion, o r sm all A T ).
Saunders [ 1 0 ] perform ed e a r ly
experim ents on th e pressure dependency o f convection in a i r .
t h a t Nu^ f o r a v e r t i c a l
He found
f l a t p la te f a l l s w e ll above accepted c o r r e la -
5
tio n s f o r Rab < 10 .
This is seen as a le v e lin g o f f o f Nu to near
u n ity as a pure conduction regim e is e n te re d a t low .p re s s u re .
K y te ,.
Madden, and P ir e t [1 1 ] extended low pressure measurements to a vacuum
in which th e mean fr e e path o f th e a i r m olecules becomes comparable to
th e p h y s ic a l body dim ensions.
NUd'
They found f o r v e r t ic a l w ire s th a t
l n [ l + 4 . 4 7 / ( R a D l^ ) 0 - 26]
in th e m o le c u la r flo w regim e.
Here h is th e h e ig h t o f th e w ire ( c y l i n ­
d e r) and D1 i s th e d ia m e te r plus tw ic e the,m ean fr e e path o f th e gas.
G ryzag o rd is [1 2 ] re p o rts t h a t a v e r t ic a l
sures obeys th e s tan d ard c o r r e la t io n
Nub = 0 . SSBRaJ/4
p la te in a i r a t low p re s ­
6
down to Ra^1=TO, in c o n tra s t to Saunders [1 0 ] and to S u rian o and Yang
[1 3 ].
The l a t t e r study suggests t h a t th e N u s s e lt number should f a l l
above t h i s c o r r e la t io n f o r Ra^ low er than about 500.
re a d e r is r e f e r r e d to th e l i t e r a t u r e
The in te r e s te d
fo r fu r th e r d e ta ils .
A n a ly t ic a l and num erical s o lu tio n s o f e x te rn a l n a tu ra l convection
a re based on s o lv in g the coupled d i f f e r e n t i a l equations o f c o n t in u it y ,
momentum, and energy in th e boundary la y e r appro xim atio n [ 5 ] .
The s o lu ­
tio n s u s u a lly ta k e th e form o f s e rie s s o lu tio n s in Ra f o r th e tem p eratu re
and stream fu n c tio n s .
C hiang, O s s in , and T ie n [1 4 ] solved th e case o f
an is o th e rm a l sphere and presen ted g ra p h ic a l r e s u lts o f v e lo c it y and
te m p e ratu re p r o f i l e s and lo c a l N u s s e lt numbers.
Lin and Chao [1 5 ] and
S e v i l le and C h u rc h ill [ 1 6 ] p resented s im i l a r r e s u lts f o r tw o-dim ensional
and a xisym m etric cases.
a v e r t ic a l
Sparrow and Gregg [1 7 ] solved th e eq u atio n s f o r
c y lin d e r .a n d developed a c o n d itio n under which i t may be
c o r r e la te d by th e f l a t p la te r e s u l t :
v
o l/2
- A r7T ‘(h /D ) < 0 . 1 5 .
G ri z ^
Minkowycz and Sparrow [1 8 ] used th e te c h n iq u e o f lo c a l n o n - s im ila r it y to
s o lve th e case o f a v e r t ic a l c y lin d e r .
The cases solved ranged from near
th e f l a t p la te s o lu tio n to a f a c t o r o f fo u r d e v ia tio n from i t .
e r is d ir e c te d to th e l i t e r a t u r e
C h u r c h ill and C h u rc h ill
a ll
f o r f u r t h e r in fo rm a tio n .
The re a d ­
F in a lly ,
[1 9 ] have presen ted a c o r r e la t in g e q u atio n f o r
g eo m etries o f e x te rn a l n a tu ra l c o n v e c tio n , in both la m in a r and tu rb u -
7
le n t flo w :
1 /6
Ra/300
Nu1 /2 = N u y 2 + {
[ T + ( 0 . 5 / P r ) 9 /1 6 ] 1 6 /9
)
Values a re ta b u la te d f o r Nu , the N u s s e lt number in th e l i m i t as Ra
approaches z e r o , and c h a r a c t e r is t ic len g th s a re given f o r v a rio u s
g e o m e trie s .
In a d d it io n , c o rre c tio n s a re o u tlin e d to account f o r non-
Newtonia'n f l u i d s
and sim ultaneous h e a t and mass t r a n s f e r .
The g re a t m a jo r ity o f work in in t e r n a l n a tu ra l c o n vectio n has been
e x p e rim e n ta l.
An e a r ly study by Elenbaas [2 0 ] w ith p a r a l l e l
v e r t ic a l
■
)
p la te s le d to th e fo llo w in g c o r r e la t io n :
NUq -
R<1q[ 1-exp(-35h/DRa|-j)]^Z^
where D is th e p la te spacing and h is th e h e ig h t.
Holman [ 6 ] , f o r the
same geom etry, g iv es th e fo llo w in g c o r r e la t io n :
Nu0 = 0 .1 QTRaJz4 ( D /h ) 1 /9
w ith th e le n g th s d e fin e d as b e fo re .
fo r
6x1O3 < Ra0 < 2x1O5
In an a n a ly t ic a l study o f p a r a lle l
p la t e s , B a tc h e lo r [2 1 ] recommends using
Db
Nud = 0 .48R aJ/ 4 ( h / D ) 3 /4
fo r
^
> {} .
Newell and Schmidt [2 2 ] made an a n a ly t ic a l study o f long re c ta n g u la r
e n clo s u re s w ith a d ia b a tic top and bottom s u rfa c e s and is o th e rm a l w a lls a t
8
two d i f f e r e n t te m p e ra tu re s .
The r e s u lt in g c o r r e la tio n s were
Nud = 0 . 0 5 4 7 6 ^ -397 ,
and
Nu0 = 0 .1 5 5 G rD‘ 3 15 ( h /D ) " 0 - 265 ,
R a n d a ll, M i t c h e l l , and E l-V la k il
sures t i l t e d
jy = I
2 .5 < - ^ < 2 0 .
[2 3 ] experim ented w ith r e c ta n g u la r e n c lo ­
a t v a rio u s angles and re p o rte d
Nud = 0 .1 1 8 [R a D cos2 ((f> -4 5 0 ) ] 0 ' 29 .
In these e x p e rim e n ts , tp, th e angle from th e h o r iz o n t a l, v a r ie d from 4 5 -
>
3
9 0 ° , th e a sp e c t r a t i o h/D v a rie d from 9 -3 6 , and RaD v a rie d from 2 .8 x 1 0 2 .2 x l 0 5 .
F la c k , K o n o p n ic k i, and Rooke [2 4 ] perform ed h e at t r a n s f e r e x p e r i­
ments w ith an is o c e le s t r ia n g u la r e n clo s u re w ith an a d ia b a tic bottom fa c e
and is o th e rm a l r i s i n g w a lls o f d i f f e r e n t te m p e ra tu re s .
was v a rie d so t h a t th e asp ect r a t i o
0 .2 9 - 0 . 8 7 .
The apex angle
(h e ig h t/b a s e w id th ) ranged from
They c o r r e la te d t h e i r r e s u lt s by
Nu^ = CgCGr-j)02 + 1 .5 8 9 /c o s e
where 6 is th e an g le o f th e r is in g w a lls from the h o r iz o n ta l and I is
t h e i r le n g th .
The c o n s ta n ts are ta b u la te d f o r values o f 9 .
The t r i g o ­
n o m e tric term in th e c o r r e la t io n is to account f o r conduction near the
apex, where th e is o th e rm a l s u rfa ce s a re c lo se to g e th e r.
9
S c a n la n , B ishop, and Powe [2 5 ] measured convection between concen­
tr ic
sp h eres.
They d e fin e d an e f f e c t i v e therm al c o n d u c tiv ity
ke f f _ q t V
i i
k
ATrkATr^r0
.
and recommend th e fo llo w in g c o r r e la t io n :
k
- jp
= 0 . IlT R a J - 276 ,
I . AxlO4 < RaL < 2 . SxlO6
.
where L is th e gap w id th
Kuehn and G o ld s te in [2 6 -2 8 ] have a n a l y t i c a l l y and e x p e rim e n ta lly
tr e a te d th e case o f c o n c e n tric and e c c e n tr ic h o riz o n ta l c y lin d e r s .
They
p re s e n t c o r r e la t io n s based on th e r a t i o o f a c tu a l co n ve c tio n heat tr a n s ­
f e r to t h a t which would be conducted through a sta g n an t f i l m
th e same c o n d itio n s .
Q - W
c o n d
la y e r under
For a i r th e y p re s e n t
* 0 - 159toE ' 272 • 2 j x l O4 I
where L is th e gap w id th .
to L I
LZDi =O .8 in these e xp e rim e n ts .
9 - 6,<104
I t o h , F u jit a ,
N is h iw a k i, and H ir a ta [A 3 ] propose a c o r r e la t io n f o r c o n c e n tric c y lin d e rs
based on a d i f f e r e n t c h a r a c t e r is t ic —le n g th :
Nu = 0.2 0 R a , / 4
where
g I r I 1nV
rP
, Ra > 7 . Ix IO 3
_ V o 1ntV
k
rI 1
10
Ra =
and
ggATC/r.rQ InCr0/ ^ ) ] 3 cpp2
—
There e x i s t few s tu d ie s o f in t e r n a l n a tu ra l c o n vectio n in a i r a t
low. p re s s u re s .
Mack and Hardee [2 9 ] have examined t h is problem f o r con­
c e n t r ic s p h ere s , and Mack and Bishop [ 3 0 ] have done so f o r c o n c e n tric
c y lin d e r s .
These a re a n a ly t ic a l s tu d ie s in which power s e r ie s s o lu tio n s
in Ra a re developed f o r th e te m p e ratu re and stream fu n c tio n s .
G raphical
r e s u lts a re presen ted f o r v e lo c it y and te m p e ratu re p r o f i l e s and lo c a l
N u s s e lt numbers.
t io n a l
The re a d e r is r e fe r r e d to th e l i t e r a t u r e
f o r a d d i­
in fo rm a tio n .
Koshmarov and Ivanov [3 1 ] have p u b lis h e d an e xp e rim en tal in v e s tig a ­
t io n o f n a tu ra l c o n vectio n between c o n c e n tric c y lin d e rs a t v ery low
O
A
p re s s u re s .
Over the range o f 10 <GrL<10 th e y c o r r e la t e th e convection
r e s u lts by
where
Tem peratures here a re a b s o lu te .
Kn1- is th e Knudsen number, d e fin e d by
R e c e n tly , M arker and Leal [3 2 ] have re p o rte d an a n a ly t ic a l r e s u lt
f o r c o n ve c tio n in s h a llo w v e r t i c a l a n n u li
(th re e -d im e n s io n a l f lo w ) :
n
I + 2 .7 6 x 1 0 “ 6
where
A =
and
o
y
h
i
*
R a ith b y and H ollands [ 4 ] have developed an a n a ly s is s im ila r t o , but
more r e fin e d th a n , L ie n h a rd 's
[ 9 ] boundary la y e r a n a ly s is .
The method
in v o lv e s a balance o f bouyancy and viscous fo rc e s v ery c lo s e to the w a l l ,
alo ng w ith an assumed form f o r th e te m p e ratu re p r o f i l e .
New c o r r e la tio n s
g en erated by t h is method in c lu d e
0 .3 1 7 In (D 0ZD1 )
f o r c o n c e n tric c y lin d e rs and
Q
f o r c o n c e n tric spheres.
This method w i l l
be gone in t o in g r e a te r d e t a il
l a t e r (see C hapter V ).
The p re s e n t in v e s tig a tio n is e s s e n t ia lly a c o n tin u a tio n o f work by
W a rrin g to n [3 3 ] w ith th e same a p p a ra tu s .
His work in v o lv e d a wide v a r i ­
e ty o f e n c lo s u re geom etries and f l u i d s w ith a la rg e range o f P ran d tl
numbers.
The b e st c o r r e la t io n o v e r a ll from h is data was
12
For th e c y lin d e r-c u b e geometry (in n e r b o d y -o u te r body) th e b e st c o r r e la
t io n was
Nub = 0 .5 9 3 R a b * 2 4 0 (L /R 1 ) 0 ,4 3 4
and f o r th e cube-cube geom etry,
Nul = 0 . 322RaL * 244( L / R i ) 01466Pr0 ' 0185 .
The b e t t e r c o r r e la tio n s f o r a i r in a l l
g eom etries were
Q = 0.4 6 8R a L ' 172 ( L / R . ) 0 "167 ,
Nul = 0.612R aL l207 ( L / R . ) 0 ' 508 ,
and
Nub = 0.570R a0 ’ 228 (LZR1 ) 0 ' 162 .
In th ese e q u a tio n s , L is th e gap w id th between h y p o th e tic a l c o n c e n tric
spheres o f volumes equal to th e a c tu a l volumes o f th e in n e r and o u te r
b o d ie s .
R1 is th e ra d iu s o f such an in n e r sphere,
b is th e d is ta n c e
tr a v e le d by th e therm al boundary la y e r on th e a c tu a l in n e r body, assum­
in g no flo w s e p a ra tio n .
These c o r r e la tio n s w i l l
be t r i e d w ith the
p re s e n t d a ta and compared to any new c o r r e la tio n s found.
CHAPTER I I I
EXPERIMENTAL APPARATUS AND PROCEDURE
The apparatus used in t h is in v e s tig a tio n
(see F ig u re 3 .1 ) was a
c u b ic a l t e s t space 2 6 .6 7 cm (1 0 .5 in ) along an in n e r s id e , fa b r ic a te d
from 1 .2 7 cm ( 0 . 5 in ) t y p e '6061 aluminum.
A w a te r ja c k e t e n clo s u re o f
th e same m a te ria l surrounded t h is cu b e; i t measured 3 5 .5 6 cm (1 4 .0 in )
alo ng an in n e r s id e .
The two enclosures were fa s te n e d to g e th e r w ith
machine screws and s ea le d w ith s ilic o n e ru b b e r s e a la n t.
Access was
gained in t o th e t e s t space by removing th e w a te r ja c k e t l i d
and a 2 5 .4
cm (1 0 .0 in ) c i r c u l a r c o ver in th e top fa c e o f th e t e s t space.
The
w a te r ja c k e t l i d was s ea le d w ith a g a s k e t; th e c ir c u l a r c o ve r was
fla n g e d and s ea le d w ith an 0 - r in g and high-vacuum g re as e .
The 3 .2 cm
(1 .2 5 i n ) channels s e p a ra tin g th e two enclosures c o n ta in ed c o o lin g
w a te r .
The w a te r flo w was s e p a ra te ly a d ju s ta b le along each o f th e s ix
cube fa c e s . ' The c o o lin g w a te r was c o lle c te d from the t e s t apparatus
through a d ra in m a n ifo ld and pumped through a c h i l l e r a p p a ra tu s , in to an
in s u la te d s to ra g e ta n k , and from th e re through th e supply m a n ifo ld and
back in to th e t e s t a p p a ra tu s .
Two heated in n e r bodies were used in t h i s in v e s t ig a t io n :
a c y lin ­
d e r w ith h e m is p h e rica l end caps, 1 1 .4 3 cm ( 4 .5 in ) in d ia m e te r and 22.61
cm ( 8 .9 i n )
in o v e r a ll le n g th ; and a cube m easuring 1 2 .7 0 cm ( 5 .0 in )
along a s id e .
The in n e r bodies were each mounted on a 1 .2 7 cm (0 .5 in )
s ta in le s s s te e l stem o f .159 cm (.0 6 2 5 in ) w a ll th ic k n e s s .
The leads to
th e therm ocouples and h e a te r tapes on th e in n e r body passed through t h is
14
W ater
W ater
Jacket
In n e r
Body
Thermocouple
Probes
Thermocouple Sw itch
f
S upport
Stem
Thermocouple lea d s
from in n e r and o u te r bodies
from h e a te r tapes
to the I h e a te r tapes
V o ltm e te r
M u lti p o s itio n S w itch
DC
Power
Supply
F ig u re 3 .1
V a r ia b le Power
R e s is to rs
Schem atic o f H e a t T ra n s fe r A pparatus W ith
S u p p o rtin g In s tru m e n ta tio n
\
15
stem.
The stem in tu rn passed through c e n t r a l l y lo c a te d holes o f 1 .2 7
cm ( 0 .5 i n ) d ia m e te r in th e bottom faces o f th e t e s t space and w ater
ja c k e t e n c lo s u re s .
The holes were grooved and f i t t e d w ith 0 -r in g s to
seal a g a in s t th e stem.
The stems were in s u la te d w ith h e a t-s h rin k a b le
tu b in g to m in im ize co n ve c tio n h e at loss to the w orking f l u i d .
Flow
v is u a liz a t io n p ic tu re s taken in a s im ila r apparatus (b u t w ith a tr a n s ­
p a re n t fa c e in t o th e t e s t space) have c l e a r l y shown t h is to be e f f e c t i v e
■ [3 3 ].
The stem passed through a 1 .2 7 cm (0 .5 in ) Conax compression
fittin g
beneath th e w a te r ja c k e t e n c lo s u re .
This f i t t i n g
e lim in a te d
leakag e o f a i r between th e s h rin k tu b in g and the stem and a llo w e d the
in n e r body to be p o s itio n e d v e r t i c a l l y w ith in the system.
Heat was s u p p lie d to th e s u rfa c e o f th e in n e r bodies by a s e rie s o f
h e a te r tapes o f .51 cm ( . 2
i n ) th ic k n e s s ,
.32 cm (.1 2 6 in ) w id th , and
2 8 .8 7 ohms/m ( 8 . 8 o h m s /ft) re s is ta n c e per u n it le n g th .
In p u t v o lta g e s
to th e tapes were i n d i v i d u a l l y v a r ia b le by means o f Ohmite rh e o s ta ts
(0 -3 5 ohms, 150 w a t ts , 2 .0 7 amperes maximum) connected in s e rie s w ith
th e ta p e s .
The in n e r bodies were c o n s tru c te d o f sheet c o p p e r, o f e it h e r
.0 6 4 cm (.0 2 5 i n )
( c y lin d e r ) o r .318 cm (.1 2 5 in )
(cube) th ic k n e s s .
In s u la tin g m a te ria l was packed in s id e th e in n e r bodies to c u t down on
co n ve c tio n w ith in them.
There were thus th re e fa c to rs which c o n trib u te d
to m a in ta in in g an is o th e rm a l heated s u rfa c e on th e in n e r b o d ie s:
I.
I n d i v i d u a l l y c o n t r o lla b le h e a te r tapes a tta c h e d to s e p a ra te areas o f the
s u r fa c e s ; 2.
Low c o n v e c tiv e a c t i v i t y w ith in th e b o d ie s; and 3 .
F a b ric a ­
16
t io n from a high therm al c o n d u c tiv ity m a te ria l which promotes a "conduc­
t io n sm earing" e f f e c t .
Tem peratures were m onitored in these experim ents by means o f copperc o n stan tan therm ocouples and a U n ite d Systems C o rp .- D i g i te c 268 d i g i t a l
m iI l i v o ltm e te r .
There were 25 therm ocouples epoxied .31 8 cm (.1 2 5 in )
from the in n e r s u rfa c e o f th e c u b ic a l t e s t space e n c lo s u re .
o f therm ocouples p e r fa c e o f the cube v a rie d from 3 to 7 .
The number
A ll thermo­
couples on a given fa c e were connected in p a r a l l e l , s in c e i t has been
found th a t th e te m p e ratu re across any fa c e v a rie d no more than .8°C
( I . 5 °F ) [ 3 3 ] .
Thermocouples were mounted flu s h on th e in n e r s u rfa c e o f th e c y l i n ­
d r ic a l
in n e r body and .165 cm (.0 6 5 i n ) beneath th e o u te r s u rfa c e o f th e
c u b ic a l in n e r body.
[I
A ll therm ocouples were f la t t e n e d f o r about 2 .5 cm
i n ) alo ng th e in n e r bodies and th e o u te r body to m in im ize conduction
h e at t r a n s f e r along th e leads and r e s u lt a n t e r r o r in te m p e ratu re measure­
ment.
Thermocouples were p o s itio n e d on th e in n e r body so t h a t a t le a s t
one therm ocouple la y w ith in each h e a te r ta p e w in d in g .
Tem perature p r o f ile s
between the in n e r and o u te r bodies were meas-
ured w ith therm ocouples epoxied in th e ends o f .16 cm (.0 6 2 5 in ) o .d .
s ta in le s s s te e l tu b e s .
Four such te m p e ratu re probes were used in t h is
in v e s t ig a t io n , p o s itio n e d a t 0 ° , 3 4 ° , 8 0 ° , and 160° from th e upward
v e r tic a l.
The 8 0 ° probe was mounted in a v e r t ic a l plan e passing d ia ^
g o n a lly through th e c e n te r o f th e cube; th e o th e r th re e probes were in a
I
17
■
v e r t ic a l plan e through th e c e n te r and p e rp e n d ic u la r to two o p p o s ite v e r ­
tic a l
fa c e s .
The s ta in le s s s te e l probes were p o s itio n e d w ith in probe
h o ld e rs , each o f which c o n s is te d o f a .95 cm (.3 7 5 in ) o .d . s ta in le s s
s te e l tube w ith a .15 9 cm (..0625 i n )
i.d .
Conax compression f i t t i n g
welded on th e o u te r end.and .95 cm (.3 7 5 in ) SAE threads c u t in the
in n e r end.
The probe passed through a .1 8 cm (.0 7 in ) h o le in th e in n e r
end o f th e probe h o ld e r , and was sea led and held in p la c e by th e Conax
fittin g .
The c u b ic a l t e s t space e n clo s u re had holes d r i l l e d
ed to a ccep t th e probe h o le s .
These
holes
were seated and sealed
a g a in s t th e h o ld e rs w ith 0 -r in g s a p p ro x im a te ly .25 cm ( .1
in n e r s u rfa c e o f th e cube.
holes in th e s e s ea ts
The probes passed through
and thus in t o th e t e s t space.
passed through 1 .0 2 cm ( . 4
and th re a d ­
in ) from the
.19 cm (.0 7 5 in )
The probe holders
i n ) holes in th e w a te r ja c k e t e n clo s u re and
were sea led th e re w ith 0 - r in g s .
To perform e x p e rim e n ts , th e in n e r body was p o s itio n e d w ith in the
t e s t space, th e therm ocouple leads were connected to th e m i l I i v o ltm e te r
along w ith a standard re fe re n c e a t O0C ( 3 2 ° F ) , and the h e a te r tape
lead s were connected to th e power s u p p ly .
The c ir c u l a r cover and w a te r
ja c k e t l i d were a tta c h e d and s e a le d , and a l l
p o s itio n e d w ith in t h e i r h o ld e r s .
tem p eratu re probes were
The t e s t space was evacuated to the
d e s ire d vacuum by means o f a S argent-W elch mechanical vacuum pump.
Above
I mm Hg, pressures were m onitored by a m ercury U -tube manometer and below
I mm Hg by a G eneral E l e c t r i c Thermocouple Vacuum Gauge.
Power was
18
a p p lie d to th e h e a te r tap es and a d ju s te d u n t i l the d e s ire d iso th erm al
c o n d itio n was reached on the in n e r body.
S im u lta n e o u s ly , c o o lin g w a te r
flo w r a te s were a d ju s te d so t h a t an is o th e rm a l c o n d itio n was o btained
oh th e o u te r body.
T y p ic a lly , te m p e ratu re v a r ia t io n across e it h e r o f
th e bodies ranged from .5 -1 . V C
C l- Z 0F) once s tead y s t a te c o n d itio n s
were reach ed , which took about 2 hours.
The data then taken were:
therm ocouple re a d in g s on th e in n e r and o u te r b o d ie s, v o lta g e and c u rre n t
in each h e a te r tape c i r c u i t , and a i r p re ss u re w ith in the t e s t space.
These raw fig u r e s were converted to in n e r and o u te r body te m p e ra tu re s .
and t o t a l power consumed ( i . e . ,
t o t a l h e a t t r a n s f e r r a t e , assuming n e g l i ­
g ib le power lo s s in th e leads to th e h e a te r ta p e s ).
The n a tu ra l', c o n ve c tio n h e at t r a n s f e r must, be o b ta in e d from the
t o t a l h e a t t r a n s f e r by s u b tra c tin g o f f th e r a d ia tio n between the bodies
and conduction down th e stem .
T h is c o r r e c tio n was made by e va cu atin g th e
system o ver a p e rio d o f two days to pressures in the range o f 3 -1 3 Pa
(2 5 -1 0 0 m icrons Hg) , where convection presumably does n o t occur [ 3 7 ] .
F ifte e n h e a t t r a n s f e r data p o in ts were taken in t h is p re ss u re range w ith
te m p e ratu re d iffe r e n c e s between the bodies ranging from 1 1 .1 -4 7 .2 * 0
(2 0 -8 5 oF ) .
A p l o t o f h e at t r a n s f e r a g a in s t tem peratu re d iffe r e n c e r e ­
s u lte d in a s t r a ig h t l i n e .
The e q u a tio n o f t h is lin e was then used in
subsequent te s ts , to remove r a d ia t io n and stem conduction c o n tr ib u tio n s .
19
S eventy-seven d a ta p o in ts were taken w ith the c y l i n d r i c a l
body and 26 w ith th e c u b ic a l in n e r body.
form shown in Appendix I I I
in n e r
The data were reduced to the
on a Texas Instru m en ts SR-40 c a lc u la t o r .
The
d a ta were then f u r t h e r reduced and c o r r e la te d by a F o rtra n IV program on
th e Xerox Sigma 7 com puter.
T h ir te e n te m p e ratu re p r o f i l e s ets were c o lle c te d a ls o .
In these
measurements, th e te m p e ratu re probe was in s e rte d in to th e t e s t space un­
til
it
c o n ta cted th e in n e r body.
I t was then w ithdraw n in sm all in c r e ­
ments u n t i l th e probe t i p was flu s h w ith th e w a ll o f th e t e s t space.
The therm ocouple v o lta g e and d is ta n c e along the tra v e rs e were recorded
a t each s te p o f th e w ith d ra w a l.
A second F o rtra n IV program was used to
c o n v e rt th e v o lta g e s to tem peratu res and then c a lc u la t e dim ensionless
te m p e ratu re s and r a d i i (see F ig u re 3 . 1 ) :
T - T
" o u te r body w a ll
T
T
in n e r body w a ll o u te r body w a ll
-r*
* _
r
~
d is ta n c e from o u te r body w a ll
lo c a l gap between in n e r and o u te r w a ll
*
P lo ts were then made o f I
*
a g a in s t r
f o r th e v ario u s probe angles and
e x p e rim e n ta l c o n d itio n s .
E x is tin g s u b ro u tin e s [3 3 ] were used to c a lc u la te th e v is c o s it y ,
h e a t c a p a c ity a t c o n s ta n t p re s s u re , and therm al c o n d u c tiv ity o f a i r .
The d e n s ity and c o e f f i c i e n t o f therm al expansion o f a i r were c a lc u la te d
from th e id e a l gas law .
In form ing th e dim ensionless groups used to
20
c o r r e la t e th e d a ta , no c o rre c tio n s were made in the a i r p ro p e rty values
f o r n o n -s ta n d a rd p re s s u re , excep t o f course in the case o f d e n s ity .
T h is is r ig o r o u s ly c o r r e c t f o r the c o e f f i c i e n t o f therm al expansion and
c o n s ta n t p ressu re h e a t c a p a c ity o f an id e a l gas [ 9 ] .
I t has been found
v e ry n e a r ly c o r r e c t f o r v is c o s it y and therm al c o n d u c tiv ity a t pressures
above I mm Hg, where th e flo w is v is c o u s , n o t m o le c u la r [ 3 4 - 3 6 ] .
No
co n ve c tio n h e a t t r a n s f e r was observed a t pressures below I mm Hg, where
p ressu re c o rre c tio n s to p ro p e rty values would become n e ce s sa ry .
p ro p e rty values were e v a lu a te d a t th e a r it h m e t ic mean f l u i d
A ll
te m p e ra tu re :
CHAPTER IV
HEAT TRANSFER AND TEMPERATURE PROFILE RESULTS
The h e a t t r a n s f e r data taken in 'th is in v e s tig a tio n have been c o r r e ­
la t e d in terms o f th e d im en sio n less param eters which are o b ta in e d from
th e nondim ensional form o f the governing eq u atio n s o f mass, momentum, and
en erg y .
The independent dim en sionless param eters used here are th e Ray­
le ig h number, a c o r r e c tio n f o r geom etry, and a dim ensionless d e n s ity c o r­
r e c t io n .
The d a ta in c lu d e 70 data p o in ts taken o ver th e p ressu re range
o f 2 7 6 0 -8 6 ,1 8 0 Pa (.2 0 -6 4 6 .4 ran Hg) by th e p re s e n t a u th o r and 126 data
p o in ts taken a t lo c a l atm ospheric p ressu re (8 6 ,1 8 0 Pa) p re v io u s ly by
W a rrin g to n [ 3 3 ] .
These two data s ets w i l l
be r e fe r r e d to as " v a ria b le
d e n s ity " and "c o n s tan t d e n s ity " d a ta , r e s p e c tiv e ly .
t h a t th e re is some v a r ia t io n
It
should be noted
( . 9 7 - 1 . 0 6 kg/m ) in f l u i d d e n s ity in the
l a t t e r d a ta a ls o as th e te m p e ratu re changes.
This second group o f d a ta
in c lu d e s a t h i r d geom etry, sphere-cube (in n e r b o d y -o u te r b o d y ), in a d d i­
t io n to th e geom etries d e s c rib e d in C hapter I I I .
There a re a ls o s e v e ra l
s iz e s o f in n e r body in c lu d e d in W a rrin g to n 's data f o r each geom etric
c la s s .
The ranges o f c o r r e la t in g p a ra m e te rs , p re s s u re s , and geom etries
used in th e c o r r e la tio n s a re ta b u la te d in Tab le 4 l l ,
r e la t io n s were t r i e d .
Those which w i l l
Many d i f f e r e n t c o r­
be discussed below a re :
(4 .1 )
Nus = C1R a ^
(4.2)
Nus = C^Ras2 U-ZR1I c3
(4 ,3 )
3
Z
(4.4)
n
Nus = C1R a ^
C1Rag2
22
TABLE 4 ,1
RANGE OF GEOMETRIES AND CORRELATING PARAMETER
Type o f Data
V a r ia b le d e n s ity
C onstant d e n s ity
In n e r
Body
C y lin d e r
Sphere
C y lin d e r
Dimensions
. (cm)
1 1. 43 x 22.61
Range o f Ra
1 .7 x 1 0
1,9x10
1 .1 x 1 0
1 .8 x 10
1 1 .4 3
7 .5 x 1 0
5 .1 x 10
1 7 .7 8
4 .6 x 1 0
1 .8x10
22.86
1 ,7 x 1 0
4 .4x10
1 1 .4 3 x 1 6 .1 3
. 9 .0 x 1 0
3 .8x10
1 7 .7 8 x 22.61
3 .4 x 1 0
1.2x10
1 7 .7 8 x 2 0 .5 7
3 .4 x 1 0
1.3x10
11. 43 x 22.61
7 .6 x 1 0
1 0.1 6
9 .8 x 1 0
12. 70
5 .6 x 1 0
1 6 .2 6
2.6x10
1.1x10
23
Nub =
C1R%C2
Nu6 - C1R a ^
(4 .5 )
(L /R .)^
(4 .6 )
cZ
Q = C1Ra6
(4 .7 )
* C2
Q = C,Ra6
(4 .8 )
■ Q = C1Rac2 ( L / R . ) C3
(4 .9 )
(4 .1 0 )
Nu6 =
(4 .1 1 )
Nu6 =
C' Ra« 2 ( - P a t )C 3 (^ )C4
Nu6 = C1R a V 4 CpJ
cZ
(4 .1 2 )
cX /4
(4 .1 3 )
Nu6 = C1R a J '4 C ^ c Z
(4 .1 4 )
Nu6 =
where L is the gap w id th between h y p o th e tic a l
c o n c e n tr ic spheres o f
volumes equal to those o f th e in n e r and o u te r b o d ie s , R- is the radiu s
o f such an in n e r s phere , b i s the d is ta n c e t r a v e l e d by the boundary l a y e r
along th e i n n e r body assuming no flo w s e p a r a t i o n , p and Patm a re the a i r
d e n s it y i n the experim ent and a t standard atmospheric- p r e s s u r e , and 6 is
any one o f th r e e d i f f e r e n t gap w idths de sc rib e d below.
In a d d i t i o n to
these c o r r e l a t i o n s , the data were t e s te d a g a in s t c o r r e l a t i o n s recommended
by W arrington [ 3 3 ] .
Chapter V is devoted to developing and t e s t i n g c o r ­
r e l a t i o n s based on th e method o f R a ith b y and Hollands [ 4 ] .
Three d i f f e r e n t methods were used to determ ine the c h a r a c t e r i s t i c
gap w id th to be used in fo r m u la tin g th e parameters Rag and Nug.
L is as
24
d e fin e d above.
L'
is the average gap w id th obtained by i n t e g r a t i n g the
l o c a l gap w id th over the s u rfa ce o f the in n e r body.
s p h e r ic a l
in n e r b o d ie s , t h i s lo c a l gap i s measured along a r a y o r i g i n a ­
t i n g in the c e n t e r o f the e n c lo s u re .
lo c a l
For c u b ic a l and
For c y l i n d r i c a l
gap i s d e fin e d as diagrammed in Figure 4 . 1 .
in n e r b o d ie s , the
L" i s the gap w idth
between h y p o th e tic a l spheres o f s u rfa c e areas equal to those o f the
in n e r and o u te r b o d ie s .
Thus L emphasizes the bulk s iz e s of. th e b odies,
L 1 emphasizes the d is ta n c e between th e b o d ie s , and L" emphasizes the
areas on which the h e a t t r a n s f e r takes p la c e .
Figure 4 . 2 i s a p l o t o f th e h e a t t r a n s f e r data used i n t h i s i n v e s t i ­
g a tio n ,
in c lu d in g data o f W arrington [ 3 3 ] .
t h i r d o f W a r r in g to n 's data i s p l o t t e d h e re .
For c l a r i t y , o n ly about one
Figure 4 . 2 demonstrates
c o n s id e r a b le geom etric dependence o f h e a t t r a n s f e r , p a r t i c u l a r l y in
W a r r in g to n 's d a ta .
The equal-volum e gap w i d t h , L, was used in fo rm u la ­
t i n g the dim ensionless groups o f F ig u re 4 . 2 ,
squares c u r v e - f i t s o f a l l
Table 4 .2 l i s t s
the data f o r v a rio u s c o r r e l a t i o n s .
le a s tThe N u s s elt
number c o r r e l a t e s the data b e t t e r than th e r a t i o o f cIconvZcIconcI-
0 f the
th r e e gap w idths used, L g iv e s s l i g h t l y b e t t e r r e s u l t s in the Mu c o r r e ­
l a t i o n s , w h ile L" has a s l i g h t edge in Q c o r r e l a t i o n s .
In n e i t h e r case
i s the advantage s i g n i f i c a n t , and L-based c o r r e l a t i o n s w i l l
be re p o rte d
h e r e a f t e r , except where some p a r t i c u l a r advantage o r p h y s ic a l s i g n i f i ­
cance is a s s o c ia te d w ith o th e r gap w id th s .
25
Figure 4.1
Local Gap Width a f o r the Cylinder-C ube Geometry
log Nu
5 c y lin d e r -c u b e
• cube-cube
o sphere-cube
6 c y lin d e r -c u b e
o cube-cube
open symbols:
data o f W arrington
log Ra
F ig u re
4 .2
Al I
of
th e
Heat T ra n s fe r
D a ta :
Nu l v s . RaL
TABLE 4 .2
CORRELATION EQUATIONS FOR ALL DATA
I5
L
E m p iric a l
Equation Form
Ci
17.8 9
1 7 .9 6
1 7 .9 5
7 1 .4 3
7 1 .4 3
7 1 .4 3
.301
1 8 .8 5
6 9 .9 0
.139
.317
1 7 .9 5
7 0 .9 2
.140
.315
1 7 .8 8
7 1 .9 4
L
L'
L"
.133
.137
.130
.317
.315
.319
(4 .4 )
.186
(4 .5 )
(4 .6 )
(4 .3 )
I L1
-
7 1 .9 4
7 1 .9 4
7 0 .9 2
L"
(4 .2 )
C4
17.8 6
1 7 .9 3
17.91
.317
; .315
.31 9
I L"
■
D e v ia tio n
o f data w i t h i n ±20%
o f equation
70.41
6 8 .8 8
7 0 .9 2
.132
.136
.129
I
C3
%
%
18.91
1 8 .9 6
1 8 .9 3
.32 4
.321
.325
L
L'
'
Average
C2
.123
.129
.122
(4 .1 )
Constants
.303
.301
.29 9
.25 3
I
TABLE 4 , 2 [C o ntinued )
E m p iric a l Constants
Average % D e v ia tio n
Equation Form
% o f data w i t h i n ±20%
o f equation
•
.0545
.0509
7 1 .4 3
2 1 .8 0
24.01
21.71
.0649
.0607
.0567
.0542
.0506
6 9 .3 9
2 0 .3 7
5 1.5 3
4 7 .9 6
5 8 .6 7
17.7 0
.31 5
-.0 3 2 2
-.0 3 7 3
20.12
6 6 .8 4
2 8 .6 3
2 7 .9 7
2 8 .9 7
5 4 .5 9
5 5 .1 0
5 4 .0 8
2 1 .4 7
2 1 .9 2
5 9 .1 8
29
As W arrington [ 3 3 ] has p o in te d o u t , even over t h i s wide range o f
e n c l o s u r e ■g e o m e trie s , the h e at t r a n s f e r r e s u l t s are c o r r e l a t e d as w e ll by
an i n f i n i t e atmosphere type o f c o r r e l a t i o n as by one based on the e n c lo ­
sure geometry.
C o r r e la t io n s based on b , the length o f the in n e r boundary
l a y e r , p ro v id e r e s u l t s as s a t i s f a c t o r y as ones based on gap w id th s .
One
o f the b e t t e r c o r r e l a t i o n s , e quation ( 4 . 6 ) , takes i n t o account both the
boundary l a y e r and gap w id th dimensions i n d i v i d u a l l y .
When the two
le n g th s a re combined i n t o a s in g le p a ra m e te r. Rag (e q u a tio n ( 4 . 5 ) ) , the
c o r r e l a t i o n i s o n ly s l i g h t l y worse.
The e nclos ure geometry i s not
e n t i r e l y i n s i g n i f i c a n t , however, as evidenced by the much h ig h e r r i s e in
th e average d e v i a t i o n , from 1 7,9 5 to 1 8.8 5 p e r c e n t, when the enclosure
is ignored in e q u atio n ( 4 . 4 ) .
The c o r r e l a t i o n s fo rm u la te d w ith Q, equations ( 4 . 7 - 4 . 9 ) , are a f f e c ­
te d d i f f e r e n t l y by the i n c lu s io n o f the geometry pa ram e te r.
( 4 . 8 ) , based on Rag, produces f o r a l l
Equation
gap w idths c o n s id e r a b ly g r e a t e r
e r r o r than e i t h e r e q u atio n ( 4 . 7 ) o r ( 4 . 9 ) .
.
When L/R^ i s in c lu d e d as an
independent param eter in e q u atio n ( 4 . 9 ) , the best f i t ,
w ith I " ,
has an
average d e v i a t i o n o f 1 7 .5 4 p e r c e n t, whereas equation ( 4 . 7 ) , a ls o based
on L" but w ith no geom etric c o r r e c t i o n , has an average d e v i a t i o n o f 17.7 9
p e r c e n t.
F u r t h e r , the exponent on the L/R^ f a c t o r in e q u a tio n ( 4 . 9 ) i s
q u i t e s m a ll, which s i g n i f i e s a small geom etric dependence.
Thus in f o r ­
ming the r a t i o o f convection to h y p o th e tic a l conduction h e a t t r a n s f e r ,
the area a v a i l a b l e f o r h eat t r a n s f e r i s the im p o rta n t p a ra m e te r, and the
Nub=.157Rab
1 .5
1.0 log Nub
•
c y lin d e r -c u b e
■
cube-cube
CO
O
o sphere-cube
0 —
A
c y lin d e r -c u b e
o
cube-cube
open symbols:
-0 .5
data o f W arrington
- -
+
+
5 .0
4 .0
6.0
7 .0
8.0
log Rab
F ig u re
4 .3
A ll
o f th e
Heat T ra n s fe r
D a ta :
Nub v s .
Rafa
31
geom etric c o n f i g u r a t i o n i s o f small
im portance.
I t seems l i k e l y t h a t
th e geom etric v a r i a t i o n i s s i m i l a r in i t s e f f e c t upon the r a t e s o f con­
v e c tio n and conduction in the s ta g n an t f l u i d .
An a d d i t i o n a l
i n s i g h t i n t o the s i g n i f i c a n c e o f the geom etric c o r r e c ­
t i o n may be gained by in s p e c tio n o f th e t a b u l a t i o n s in Table 4 . 2 f o r
e q u atio n forms ( 4 . 1 ) ,
in g d i f f e r e n t i a l
(4 .2 ),
( 4 . 1 3 ) , and ( 4 . 1 4 ) .
can be shown [ 5 ] t h a t the expected c o r r e l a t i o n
h e a t t r a n s f e r is
Nu
= C Ray4
Q
a
where a is an a p p r o p r ia t e dimension.
e nclos ure geometry is e q u atio n ( 4 . 1 3 ) .
The corresponding e quation f o r an
Table 4 .2 shows t h a t . t h i s c o r r e ­
l a t i o n does no t work w e ll w it h the p re s e n t d a ta .
much b e t t e r , and i t
number i s
When the govern­
e quations o f c o n t i n u i t y , momentum, and energy a re c a s t
in dim ensionless form , i t
fo r o v e ra ll
(4 .3 ),
Equation ( 4 . 1 ) works
is found t h a t the p r e f e r r e d exponent on the Rayleigh
. 3 2 , r a t h e r than the c l a s s i c a l l y expected . 2 5 .
This d e v ia t io n
is among th e more s i g n i f i c a n t r e s u l t s o f t h i s i n v e s t i g a t i o n , and a t ­
tempts w i l l
be made to e x p la in i t .
At l e a s t p a r t o f t h i s d e v i a t i o n i s due to the geom etric i n f l u e n c e ,
as may be seen i n s e v e ra l ways.
F i r s t , equations ( 4 . 2 ) and ( 4 . 3 ) , which
in c lu d e geom etric c o r r e c t i o n s , e i t h e r in Ra* o r in d e p e n d e n tly , d e v ia te
le s s i n th e Rayleigh number exponent than does equation ( 4 . 1 ) .
A ls o ,
e q u a tio n ( 4 . 1 4 ) , which c o n ta in s a strong independent geom etric c o r r e c t i o n
Q =.0601 Ra
0 .5
0 --
c y lin d e r -c u b e
cube-cube
o sphere-cube
-0 .5 __
A
c y lin d e r -c u b e
D
cube-cube
open symbols:
data o f W arrington
log RaL
F ig u re
4 .4
A ll
of
th e
Heat T ra n s fe r
D a ta :
Q vs.
Ra^
33
but r e t a i n s the
,25 power on Ra, c o r r e l a t e s the data much b e t t e r than
e q u atio n ( 4 . 1 3 ) .
This improvement amounts to most o f the improvement
achieved by a llo w in g the exponent on Ra to vary f r e e l y .
Of the equations
discussed so f a r , the best c o r r e l a t i o n s w ith each dependent dim ensionless
param eter a re :
'
Nu l = .ISZRaL3 17 ( ^ r ) - 303
(4 .1 5 )
Nub = .145Rab315( ^ r ) - 253
(4 .1 6 )
Q = .0 5 0 6 R a -317( ~ - ) ' - 0373
(4 .1 7 )
w it h average d e v ia t io n s o f 1 7 . 8 6 , 1 7 . 8 8 , and 1 7 .5 4 p e r c e n t , r e s p e c t i v e l y
Since t h i s w ork, as e x p la in e d in Chapter I I ,
is a c o n tin u a tio n o f
re se a rch i n i t i a t e d by W arrington [ 3 3 ] , h is recommended c o r r e l a t i o n s f o r
a i r were t e s te d a g a in s t data taken by th e p re se n t a u th o r .
The b e t t e r
c o r r e l a t i o n s were:
Nul = .4 2 5 R a [2 34 ( ^ r ) ' 498
'
(4 .1 8 )
NuL = .6 1 2 R a [207( ^ - ) , 5 ° 8
(4 .1 9 )
Nul = .173Ra*
(4 .2 0 )
Nub = .5 7 0 Ra;2 28 ^ ) ' 162
(4 .2 1 ).
No c o r r e l a t i o n s w it h Q f i t
the p re s e n t data a c c e p ta b ly .
Equation ( 4 . 1 8 )
34
i s a c o r r e l a t i o n recommended f o r a l l
f l u i d s and a l l
o t h e r c o r r e l a t i o n s a re s p e c i f i c a l l y f o r a i r .
geometries..
The
The r e s u l t s o f these c o r r e ­
l a t i o n s a re presented in Table 4 . 3 .
It
e q u atio n ( 4 . 1 8 ) ,
f l u i d s , works b e t t e r here than the
recommended f o r a l l
corresponding e q u a tio n f o r a i r ,
is i n t e r e s t i n g to note t h a t
e quation ( 4 . 1 9 ) .
w ith h ig h e r Ra exponents g ive b e t t e r f i t s
Again th e c o r r e l a t i o n s
to these d a ta .
It
is then not
s u r p r is i n g t h a t e q u atio n ( 4 . 2 0 ) c o r r e l a t e s the data b e st among these.
There a re two p o s s ib le reasons why W a r r in g to n 's c o r r e l a t i o n s do not work
w e ll
on the p re s e n t d a t a , one s t a t i s t i c a l
and the o th e r p h y s i c a l.
W a r r in g to n 's a i r data cover a r e l a t i v e l y small range o f Ra (see Figure
4 .2 ).
Also th e r e is c o n s id e ra b le s c a t t e r in these data [ 3 3 ] . .
Therefore,
th e c o r r e l a t i o n s presented f o r a i r may not be h i g h l y a c c u r a te .
illu s tra tio n
4 .a ),
of th is ,
in W a r r in g to n 's c o r r e l a t i o n s o f the forms ( 4 . 1 -
the exponent on Ra v a r ie s o v er . 1 7 1 - . 2 9 9 .
th is v a r ia tio n is
As an
.3 0 0 -.3 2 5 .
In the p re s e n t d a ta ,
The o th e r p o s s i b i l i t y is t h a t th e r e i s
something p h y s i c a l l y d i f f e r e n t about v a r i a t i o n s in Ra caused p r i m a r i l y
.
by changing the f l u i d d e n s i t y , as in the pre se nt e x p e rim e n ts , and v a r i a ­
ti o n s due to changing the gap w id th a n d /o r tem perature d i f f e r e n c e as
W arrington d i d .
This w i l l
be discussed i n g r e a t e r d e t a i l
below.
C o r r e l a t io n s were a ls o generated from the data taken in t h i s in v e s ­
t i g a t i o n a lo n e .
These c o r r e l a t i o n s a re presented in T a b le 4 . 4 , f o r
e quation forms ( 4 . 1 ) - ( 4 . 9 ) , ( 4 . 1 3 ) , and ( 4 . 1 4 ) .
v a l e n t t o Table 4 . 2 , exc e pt t h a t i t
Table 4 . 4 is thus e q u i ­
i s f o r the v a r i a b l e d e n s it y data
35
TABLE 4 . 3
FIT OF PRESENT DATA TO PREVIOUS CORRELATIONS
Equation
Average % E r r o r
% o f data w i t h i n
±20% o f equation
.
(4 .1 8 )
4 1 .3 9
35.71
. (4 .1 9 )
5 1 .6 5
3 4 .2 9
(4 .2 0 )
1 7 .7 8
75.71 '
(4 .2 1 )
2 4 .3 3
6 4 .2 9
.
TABLE 4 , 4
CORRELATION EQUATIONS FROM THE VARIABLE DENSITY DATA ALONE
E m p iric a l
Constants
Equation Form
-2 .4 8 6
Average % D e v ia tio n
% o f data w i t h i n ±20%
o f e q u atio n
1 0 .9 9
8 4 .2 9
11.02
8 4 .2 9
1 1.6 2
85.71
1 0 .9 8
.310 -2.2681
.0582
310 . 7 . 2 0
1 0 .9 9
8 4 .2 9
1 1 .6 2
85.71
14.31
8 0 .0 0
1 4 .6 3
7 8 .5 7
1 1 .6 2
85,71
6 8 .5 7
(.4.13)
(4 .1 4 )
-2 .4 5
1 7.3 2
8 0 .0 0
37
o n ly .
It
i s s t r i k i n g t h a t , in forms (4 ,1 ) - ( . 4 . 9 ) , any i n c lu s i o n o f the
geom etric e f f e c t i n th e c o r r e l a t i o n a c t u a l l y worsens the r e s u l t s .
C l e a r l y the v a r i a t i o n betw een.the two geom etries used here (see Chapter
IIIl
i s i n s i g n i f i c a n t in the o v e r a l l h eat t r a n s f e r r a t e .
Again the
boundary l a y e r le n g th c o r r e l a t i o n s a re n e a r l y as a c c u ra te as those based
on gap w id th .
This is somewhat more s u r p r is i n g in t h i s case, because
the thermal boundary l a y e r becomes t h i c k e r a t low R ayleigh numbers and
th e i n f i n i t e atmosphere a pproxim ation should become les s a c c u r a te .
But
s in c e th e boundary l a y e r le n g th i s an easy dimension to d e f i n e , a c o r r e ­
l a t i o n based on t h i s le n g th alone is u s e f u l .
The Q c o r r e l a t i o n s were not as s a t i s f a c t o r y f o r the v a r i a b l e d e n s it y
d a ta .
In t h i s cas e , though, accuracy was rem arkably improved by i n c l u ­
ding an independent geom etric param eter in the c o r r e l a t i o n .
The best o f
the above c o r r e l a t i o n s o f each type a r e :
Nul = .142RaL310
'
Nub = .1 5 7 R a '309
Q = .1 4 5 R a [310( ^ r )™7 ,2 0
i
(4 .2 2 )
(4 .2 3 ).
(4 .2 4 )
w ith average d e v ia t io n s o f 1 0 .9 9 , 1 0 . 9 8 , and 11.62 p e r c e n t , r e s p e c t i v e l y .
Equations ( 4 . 2 2 ) and ( 4 . 2 3 ) are p l o t t e d on Figures 4 . 2 and 4 . 3 , respec­
tiv e ly .
38
Equations ( 4 . 2 2 ) - ( 4 . 2 4 ) were t e s te d a g a in s t a l l
data as an a d d i t i o n a l check on t h e i r v a l i d i t y .
in average d e v ia t io n s o f 2 3 . 0 9 , 2 2 . 0 1 , and
tiv e ly .
o f W a r r in g to n 's a i r
These equations r e s u lte d
2206 . ( s i c ) p e r c e n t , respec­
The v ery l a r g e e r r o r f o r e quation ( 4 . 2 4 ) s i g n i f i e s t h a t the
geom etric c o r r e c t i o n c a l c u l a t e d from th e p re se n t data a lo ne i s u n r e l i a b l e
With o n ly two geometries used, the range o f L/R^ was not g r e a t enough to
produce an a c c u ra te dependence.
I t i s recommended t h a t , i n v a r i a b l e -
d e n s i t y problems w ith w id e ly v a ry in g g e o m e trie s , e quation ( 4 . 2 2 ) o r
( 4 . 2 3 ) be used.
These two equations do not f i t
w e ll a t pressures near atm o s p h eric , however.
the data p a r t i c u l a r l y
(W a r r in g to n 's data were
taken a t 8 6 ,1 0 0 Pa (646 mm H g ), which i s lo c a l atmospheric pressure where
these experim ents were p e r fo r m e d .)
It
i s t h e r e f o r e f u r t h e r recommended
t h a t , in problems near atmospheric p r e s s u r e , c o r r e l a t i o n s o f Warrington
[ 3 3 ] be used.
Since e q u a tio n ( 4 . 2 4 ) was e n t i r e l y u n s a t i s f a c t o r y in c o r r e l a t i n g
the pre vious d a t a , the f o l l o w i n g e quation from Table 4 . 4 was t r i e d :
Q = .0 6 0 1 Ra^310
(4 .2 5 )
The r e s u l t i n g average d e v i a t i o n was 1 9 .1 7 p e r c e n t, a somewhat b e t t e r f i t
than e i t h e r e q u atio n ( 4 . 2 2 ) o r ( 4 . 2 3 ) .
Equation ( 4 . 2 5 )
i s p l o t t e d on
Fig u re 4 . 4 .
The p re s e n t data from the c y lin d e r -c u b e and cube-cube experiments
a re p l o t t e d in Figures 4 .5 and 4 . 6 , r e s p e c t i v e l y ,
The recommended
Nul = . 128Ra
lo g Nu
log Ra
F ig u re
4 .5
V a r ia b le -D e n s ity
D a ta ,
C y lin d e r-C u b e
G e om etry
Nul = .184Ra
lo g Nu
O
log Ra
F ig u re
4 .6
V a r ia b le -D e n s ity
D a ta ,
C u b e -C u b e G e o m e t r y
41
c o r r e l a t i o n s f o r the c y lin d e r -c u b e data a r e :
Nul = .1 2 8 R a [316
(4 .2 6 )
Nub = .136Rab316
(4 .2 7 )
Q = .05 1 6R a*316 .
In a l l
(4 .2 8 ).
cases the average d e v ia t io n is 8 .0 8 p e r c e n t, and 92 p e rce n t o f the
data f a l l w i t h i n ±20 p e rc e n t o f the e q u a tio n .
For the cube-cube d a ta ,
the recommended c o r r e l a t i o n s a r e :
Nul = .1 8 4 R a -293
(4 .2 9 )
Nub = .210Rab
(4 .3 0 )
Q = .0885RaL
2Q3
^
.
(4 .3 1 ).
The average d e v i a t i o n in each case is 1 9 .8 5 p e r c e n t, w i t h 80 p e rce n t o f
the d a ta f a l l i n g w i t h i n ±20 p e rce n t o f th e e q u a tio n .
C l e a r l y the c o r r e ­
l a t i o n s used so f a r , w h i c h a re based oh th e assumed smooth boundary
l a y e r flo w over the heated body or on b e h a v io r s i m i l a r to a s p h e r i c a l .
body o f s i m i l a r s i z e , do not work w e ll
on the c ubical
in n e r body.
The
l i k e l y source o f e r r o r i s t h a t buoyant f o r c e s , which a c t v e r t i c a l l y , do
not a c t i n the d i r e c t i o n o f flo w -a lo n g th e top and bottom fa c e s , which
thus produces a s i t u a t i o n q u i t e d i f f e r e n t from t h a t o f a rounded o b j e c t .
T h is s u b je c t w i l l
be taken up in g r e a t e r d e t a i l
in Chapter V.
42
LOW DENSITY RESULTS
Figure 4 . 7 is a p l o t o f a l l
the data taken in the p re s e n t i n v e s t i g a ­
t i o n , over the p re ss u re range o f 2320-85870- Pa ( 1 7 . 4 - 6 4 3 . 4 mm H g).
i s no d i s c e r n i b l e break in the s lo p e .
There
When the data above and below
26.660 Pa (200 mm Hg) a re c o r r e l a t e d s e p a r a t e l y , the f o l l o w i n g r e s u l t s are
o b ta in e d :
Nur
= . 1 4 3 R a ^ 17 ,
P < 26660 Pa
(4 .3 2 )
Nur
= . I B S R a ^ 03 ,
P > 26660 Pa
(4 .3 3 )
w it h average d e v i a t i o n s o f 1 5 .0 3 and 6 .4 2 p e r c e n t, r e s p e c t i v e l y .
It
is
expected t h a t the slope would decrease a t low er Rayleigh numbers ( 1 0 .1 3 )
but t h i s
is not e v id e n t h e r e . ' The s l i g h t in c re a s e in slope observed i s
p ro b a b ly not s i g n i f i c a n t , because the s c a t t e r in the data becomes l a r g e r
as the h e a t t r a n s f e r r a t e decreases.
At low er Ra, the convection heat
t r a n s f e r i s much s m a lle r than the r a d i a t i o n c o n t r i b u t i o n , so when the
r a d i a t i o n i s s u b tra c te d from the t o t a l ,
a f a i r l y larg e e r r o r .
the r e s u l t (c o n v e c tio n ) contains
Since the Ra exponent i s o n ly s l i g h t l y changed,
the range o f data i n Figure 4 .7 w i l l
be considered a s in g le regime h e re ­
a fte r.
It
has been hypothesized above t h a t th e Rayleigh number and geo­
m e tr ic c o r r e c t i o n s alone do not a d e q u a te ly c o r r e l a t e n a tu r a l convection
over th e p re s e n t range o f low er p re s s u re s .
V
To t e s t t h i s h y p o th e s is , a
Nul ..= JSSRa
log Nu
log Ra
F ig u re
4 .7
Al I
o f th e
V a r ia b le -D e n s ity
D a ta :
Nu^,, v s .
Ral ,,
44
dim ensionless d e n s it y param eter was i n s e r t e d i n t o the b a s ic c o r r e l a t i o n s
(e q u a tio n s ( 4 . 1 0 ) - ( 4 . 1 2 ) ] .
The r e s u l t s a re presented in T able 4 . 5 .
Of
p rim a ry importance i s the r e s u l t t h a t e q u atio n ( 4 . 1 2 ) c o r r e l a t e s the data
as w e ll as any o t h e r , indeed s l i g h t l y b e t t e r .
e q u atio n ( 4 . 1 2 ) d i f f e r s
from the c l a s s i c a l
The o n ly way in which,
c o r r e l a t i o n (e q u a tio n ( 4 . 1 3 ) )
is in the in c lu s io n o f the d e n s it y pa ram e te r.
This m o d i f i c a t io n alone
is enough to a f f e c t th e g r e a t improvement in the c o r r e l a t i o n shown in
T able 4 . 5 .
This is the most c om pelling evidence t h a t the d e v ia t io n o f
th e Rayleigh number exponent from the expected .25 value i s due to the
v a r i a t i o n in f l u i d d e n s i t y .
Even e q u atio n ( 4 . 1 0 ) , in which the Rayleigh
number and d e n s it y param eter are each allow ed to v ary f r e e l y , does not
c o r r e l a t e as w e ll as e q u atio n ( 4 . 1 2 ) ,
cal one f o u r t h power.
in which Ra i s f i x e d a t the c l a s s i ­
Equation ( 4 . 1 1 ) ,
in which the geom etric parameter
i s a ls o in c lu d e d , s l i g h t l y worsens the c o r r e l a t i o n .
As noted above,
however, the range o f geom etries in t h i s data is q u i t e s m a l l , and thus
the c a l c u l a t e d geom etric e f f e c t i s not v e ry r e l i a b l e .
The c la s s i c a l c o r r e l a t i o n a lo n e , Nul = .S l l R a ^
w ith an average d e v i a t i o n o f 2 0 .4 7 p e r c e n t.
, c o r r e l a t e s the data
The m o d i f i c a t io n o f the
exponent in eq u atio n ( 4 . 1 ) ,
Nul = .142RaL3 1 0 ,
and the in c lu s io n o f the d e n s it y param eter in equation ( 4 . 1 2 ) ,
Nu l = .342Ra^/ 4 ( p / p atm) ’ 1 2 9 ,
each reduce the average d e v i a t i o n by about h a l f , to 1 0 .9 9 and 10,92
TABLE 4 ,5
DENSITY-CORRECTED CORRELATIONS FOR THE VARIABLE DENSITY DATA
E m p iric a l Constants
Average % D e v ia tio n
Equation Form
%
o f data w i t h i n ±20%
o f equation
84.-29
.1 9 2 !
(4 .1 0 )
L
1 .4 7 5
(4 .1 1 )
L
1 .7 4 7
1 1 .6 2
. 310 - 2 . 4 8 6
85.71
1 1 .5 9
11.88
85.71
(4 .1 2 )
1 0 .9 2
8 4 .2 9
(4 .1 3 )
2 0 .4 7
68.57
17.3 2
8 0 .0 0
(4 .1 4 )
-
-2 .4 5 4
2.201
46
p e r c e n t, r e s p e c t i v e l y .
E i t h e r e quation ( 4 . 1 ) o r ( 4 , 1 2 )
is easy to use,
w ith the advantage going to ( 4 . 1 ) , sin ce i t ' c o n t a i n s one few er param eter.
It
i s the a u t h o r 's o p in io n t h a t both o f these equations may be used con­
f i d e n t l y in re duce d-pre s sure heat t r a n s f e r c a l c u l a t i o n s , as long as
~
pressures a re w i t h i n the p re s e n t range and the geometries a re not r a d i ­
c a l l y d i f f e r e n t from th e s e .
The success o f the d e n s it y c o r r e c t i o n in troduced has been f u r t h e r
t e s te d w ith W a r r in g to n 's d a ta .
These data were taken a t l o c a l atmospher­
i c p re s s u re , which may be viewed as a p a r t i a l
mm Hg, o r .85 a t m ) .
"vacuum" a t 86000 Pa (.645
Al I o f the p re s e n t d a ta and W a r r in g to n 's data were
c o r r e l a t e d w ith equations ( 4 . 1 0 ) - ( 4 . 1 4 ) , and the r e s u l t s a re presented in
Table 4 . 6 .
Equation ( 4 , 1 3 ) c o r r e l a t e d a l l
v i a t i o n o f 2 8 .6 3 p e r c e n t.
the data w ith an average de­
The geom etric c o r r e c tio n i n . e q u a t i o n ( 4 . 1 4 )
improved t h i s f i g u r e to 2 1 .4 7 p e r c e n t, w h ile the d e n s it y c o r r e c t io n in
eq u atio n ( 4 . 1 2 ) d id s l i g h t l y b e t t e r w ith an average d e v i a t i o n o f 19.76
p e r c e n t.
This i s a d d i t i o n a l evidence t h a t the change in d e n s it y is the
m ajor i n f l u e n c e in the exponent o f the R ayleigh number.
The e n t r i e s
in comparison.
in Tables 4 .5 and 4 . 6 f o r equation ( 4 . 1 0 ) are s t r i k i n g
When t h i s equation is used to c o r r e l a t e the v a r i a b l e -
d e n s it y data o n ly (T a b le 4 . 5 ) , the R ay le igh number exponent is
.148.
This dem onstrates t h a t the major v a r i a n t in the R ayleigh number in the
p re se n t experim ents was indeed the d e n s i t y , because the Ra dependence i s
much weaker when the d e n s it y dependence i s included in d e p e n d e n tly .
TABLE 4 .6
DENSITY-CORRECTED CORRELATIONS FOR ALL DATA
!
Equation Form
E m p iric a l
Constants
Average
6
I
C1
c2
C3
%
D e v ia tio n
C4
%
o f data w i t h i n ±20%
o f e quation
18.91
70.41
.303
1 7 .8 6
7 1 .9 4
.31 3
.0256
18,9 7
7 0 .9 2
1.969
.131
.408
1 7 .5 3
70,41
L
.359
.151
1 9.7 6
70.41
(4 .1 3 )
L
.364
2 8 .6 3
5 4 .5 9
(4 .1 4 )
L
.317
2 1 .4 7
5 8 .1 6
(4 .1 )
L
.123
.23 4
(4 .3 )
L
.132
,317
(4 .1 0 )
L
.145
(4 .1 1 )
L
(4 .1 2 )
.36 0
.571
48
In the corresponding e n t r y in Table 4 . 6 , the Ra dependence i s back to
-
.313 power, and the d e n s it y param eter dependence i s q u i t e small a t .0256
power.
This is m e rely an i l l u s t r a t i o n of. the strong s l a n t i n g o f the data
towards the atmospheric case, i . e . , W a r r in g to n 's 126 d a ta p o in ts were a l l
taken a t atm ospheric p re s s u re , whereas o n ly 70 data p o in ts were taken
over the e n t i r e pressure range.
The c o r r e c t i o n f o r both d e n s it y and
geometry, e q u atio n ( 4 . 1 1 ) , does y i e l d f u r t h e r improved r e s u l t s
(see
T able 4 . 6 ) , w ith an average d e v i a t i o n o f 1 7 .5 3 p e r c e n t, b u t i t
is ques­
t i o n a b l e w hether th e a d d i t i o n a l
in d e e d , over e quation ( 4 . 3 )
com putational e f f o r t .
It
gain in accuracy over e q u a tio n ( 4 . 1 0 ) o r ,
(see Table 4 . 2 ) j u s t i f i e s
the a d d i t i o n a l
i s c l e a r t h a t a d d i t i o n a l data w ith o th e r geo­
m e tr ie s over the e n t i r e range o f pressures would be u s e fu l to f u l l y
e v a lu a te the r e l a t i v e
importances o f geom etric and d e n s it y c o r r e c tio n s
to the ba sic c o r r e l a t i o n s .
In summary, and r e a l i z i n g t h a t th e geom etric e f f e c t remains u n c le a r
a t low p r e s s u re s ,
(<26660 P a ) , the f o l l o w i n g c o r r e l a t i o n s a r e recommend­
ed over th e range o f pressures encountered here:
Nu l = .142RaL310
(4 .3 4 )
Nul -
(4 .3 5 ).
.S42 R a V 4 (PZpatm) - 129
I t should be noted t h a t the prim ary mode o f heat t r a n s f e r a t low p re s ­
sures i s r a d i a t i v e , which i s geometry dependent,
T h e r e f o r e , even i f
49
geometry does not s i g n i f i c a n t l y a f f e c t these c o r r e l a t i o n s , i t w i l l
a la r g e e f f e c t on o v e r a l l
have,
ra te s o f heat t r a n s fe r ,
TEMPERATURE PROFILE RESULTS
Temperature p r o f i l e s between the c y l i n d e r and the e n c lo s in g cube
over the range o f exp e rim en tal
4 .1 1 .
pressures a re d is p la y e d in Figures 4 . 8 -
Im m ed ia te ly e v id e n t is th e th ic k e n in g o f the boundary l a y e r on
the in n e r body a t low er pressures (F ig u r e s 4 . 8 , 4 , 9 ) . .
At h ig h e r p re s ­
sures (F ig u r e s 4 . 8 , 4 . 1 1 ) , th e f i v e regio ns re p o rte d by W arrington [3 3 ]
a re observed:
a steep drop a t the in n e r body, an in n e r t r a n s i t i o n
r e g io n , a le v e l
zone w ith perhaps a s l i g h t , t e m p e r a t u r e i n v e r s i o n , an
o u t e r t r a n s i t i o n zone, and a f a i r l y steep drop again a t th e o u te r w a l l .
These f i v e zones a re c o n v in c in g ly p re s e n t o n ly f o r the 34° and 80°
probes (see Chapter I I I ) .
In e very case the O0 and 160° p r o f i l e s have
no p o in ts o f i n f l e c t i o n , and a steep drop a t . t h e in n e r w a ll s lo w ly l e v e l s
o f f to a le s s d r a s t i c monotonic decrease over the rem ainder o f the gap.
There i s never a re g io n o f c onstant te m peratu re as w ith th e 34° and 80°
probes.
This r e s u l t agrees w e ll w ith a n a l y t i c a l
r e s u l t s o f Mack and
Hardee [ 2 9 ] and experim ents o f S canla n . Bishop, and Powe [ 2 5 ] f o r con­
c e n t r i c spheres.
The v e l o c i t y p r o f i l e a t 160° presented by Mack and
Hardee in d i c a t e s very l i t t l e
c o n ve c tiv e a c t i v i t y ; t h i s i s a ls o supported
by flo w v i s u a l i z a t i o n p i c t u r e s o f W arrington [ 3 3 , 4 0 ] .
The 160° tempera­
t u r e p r o f i l e o f Mack and Hardee a t Ra=IOOO is a ls o a m o n o to n ic a lly
50
0. 2« .
Figure 4 . 8
Temperature P r o f i l e Data a t Two D i f f e r e n t
P r e s s u re s :
S o lid L in e s --A T = 2 3.3 °C
( 4 2 . 0 ° F ) , Pressure = 1 5 .3 mm Hg; Open
Symbols--AT = 2 0 .7 °C ( 3 7 . 2 ° F ) , Pressure =
6 4 1 .9 mm Hg.
51
Figure 4 . 9
Temperature P r o f i l e s a t Two D i f f e r e n t
P ressures:
S o lid L in e s --A T = 3 0 .2 °C
( 5 4 . 4 ° F ) , P res s ure= 4 8 .0 mm Hg; Open
S ym b o ls --A T = IS .7°C ( 3 3 . 6 ° F ) , Pressure
=1 49 .6 mm Hg.
52
Figure 4 .1 0
Temperature P r o f i l e Data a t Two D i f f e r e n t
P r e s s u re s :
S o lid L in e s — AT=23.8°C ( 4 2 . 8 ° F ) ,
P res s ure= 2 4 8.8 mm Hg; Open Symbols--AT=
2 2.9 °C ( 4 1 . 3 ° F ) , P ressure=505.7 mm Hg.
53
Figure 4.11
o
80
O
160
Temperature P r o f i l e Data:
AT=22.8°C ( 4 1 . 1 ° F ) ,
P res s ure= 3 9 6.8 mm Hg.
54
d e c r e a s i n g , f u n c t i o n o f th e same form as the 160° Tow pressure p r o f i l e s
in Figures 4 : 8 - 4 . 9 .
At h ig h e r R ay le igh numbers the i n i t i a l
s t e e p e r , and the h ig h -te m p e ra tu re re g io n becomes s m a lle r .
drop becomes
These are
m a n if e s ta tio n s o f the decrease in boundary l a y e r t h i c k n e s s . a t high
Rayleigh numbers.
The monotonic decrease in the tem peratu re p r o f i l e a t
.1 6 0 ° in d i c a t e s t h a t h eat t r a n s f e r a t t h i s p o s it io n is n e a r l y pure con­
d u c tio n [ 3 3 , 2 9 ] .
The form o f the p r o f i l e s a t 0 ° is s i m i l a r to t h a t a t 1 6 0 ° , but f o r
a d i f f e r e n t reason.
al
W arrington [ 3 3 ] re p o rte d e x p la n a tio n s o f McCoy et^
[ 4 1 ] , Bishop e t a I
[ 4 2 ] , and Scanlan e t al
[ 2 5 ] , t h a t th e form o f the
0° p r o f i l e i s due t o a c o rn e r eddy and flo w s e p a ra tio n n e ar the top o f
th e in n e r body.
This e f f e c t i s a p p a r e n tly present over the e n t i r e p re s ­
sure range o f the p r e s e n t e x p e rim e n ts , because the form o f th e 0° p r o f i l e
in Figures 4 . 8 - 4 . 1 1
i s v i r t u a l l y unchanged.
I f the flo w were smooth and
a tta c h e d to th e top o f th e in n e r body, Mack and Hardee's [ 2 9 ] a n a ly s is
would p r e d i c t a downward, r a t h e r than upward, c o n c a v ity on the p r o f i l e
fo r 0°.
W arrington found t h i s to be the case f o r f l u i d s o f higher
P randtl number ( i . e . ,
h ig h e r v i s c o s i t y ) .
The o r d e r in g o f the p r o f i l e s v e r t i c a l l y changes o v e r the range o f
p re s s u re s .
At low p re s s u r e , and e x c lu d in g the anomalous be hav ior o f the
0° p r o f i l e ,
the o r d e r in g is as would be expected:
and 34° p r o f i l e h ig h e s t .
it
flow s up the body.
160° p r o f i l e lowest
This demonstrates the h e a tin g o f the f l u i d as
At h ig h e r p re s s u re s . Figure 4 . 8 , t h i s o rd e rin g is
55
m a in ta in e d o n ly e x tre m e ly c lo s e to the in n e r w a ll and a t d is ta n c e s w e ll
away from th e in n e r w a l l .
The probable e x p la n a tio n is t h a t th e re is no
w e ll-d e fin e d boundary la y e r y e t formed a t th e 160° p o s it io n , and the h e at
t r a n s f e r is la r g e ly c o n d u c tiv e , as s ta te d above.
In a l l
p r o f i l e is above th e 8 0 ° p r o f i l e , as would be exp ected .
c ases, the 34°
.
The re g io n o f h ig h e r te m p e ratu re near the in n e r body is much la r g e r
a t th e 34° p o s itio n than a t th e 8 0 ° p o s itio n .
This corresponds to the
in c re a s e in boundary la y e r th ic k n e s s in th e d ir e c tio n o f flo w , as would
be e xp e c te d .
I t is a ls o p o s s ib le t h a t th e o u te r body has an e f f e c t on
th e 34° p r o f i l e .
The 1 1 .4 x 2 2 ,6 cm ( 4 .5 x 8 .9 in ) c y lin d e r used here f i l l s
most o f th e v e r t i c a l space w ith in th e e n c lo s u re , so t h a t th e flo w a t 34°
on th e in n e r body may w e ll be impeded by r e tu rn flo w along th e top fa c e
o f th e o u te r body.
Flow v is u a liz a t io n d a ta o f W arrin g to n [3 3 ] show t h a t
a t th e 34° p o s itio n th e c o rn e r eddy mentioned above has begun to form .
There is thus a re g io n o f near s ta g n a tio n c lo se to the in n e r body, which
would account f o r th e anom alously l i n e a r in n e r t r a n s it io n a t 34° in
F ig u re 4 .1 1 .
F a rth e r away from th e body th e flo w is ag ain smooth and the
expected le v e lin g o f f o f the p r o f i l e
between the in n e r and o u te r bodies
occurs.
The 8 0° p r o f i l e is th e most id e a liz e d form .
At t h i s p o in t in the
flo w , th e re is c o n s id e ra b le d is ta n c e between the in n e r and o u te r boundary
la y e r s , th e boundary la y e r has had ample space to develop and s t a b i l i z e ,
and th e re is no n e g a tiv e pressure g ra d ie n t and r e s u lt a n t flo w s e p a ra tio n .
56
The s l i g h t te m p e ra tu re in v e rs io n in th e 8 0 ° p r o f i l e near th e o u te r t r a n ­
s i t i o n re g io n is due t o . t h e sm all r o t a t io n a l motion o f th e f l u i d between
th e in n e r and o u te r boundary la y e r s .
CHAPTER V
A MODIFICATION OF THE METHOD OF RAITHBY
AND HOLLANDS FOR CUBICAL GEOMETRIES
The method o f g e n e ra tin g n a tu ra l c o n vectio n heat t r a n s f e r c o r r e la -
.
tio n s developed by R a ith b y and H ollands [ 4 ] is in tro d u c e d in Appendix I .
To summarize, th e assumptions o f th e model a re :
1.
Laminar flo w , w ith c o n s ta n t f l u i d p r o p e r tie s ;
2.
Boundary la y e rs are t h in compared to th e ra d iu s o f c u rv a tu re ;
3.
L in e a r te m p e ratu re p r o f i l e
in th e in n e r re g io n o f the
boundary la y e r ;
4.
I n s ig n i f i c a n t a c c e le r a tio n fo rc e s in th e in n e r re g io n , so
t h a t viscous and buoyant fo rc e s b a lan c e ; and
5.
The r a t i o o f buoyancy in th e in n e r re g io n to t o t a l buoyancy
in th e boundary la y e r depends on the P ra n d tl number only
and no t on geom etry.
Assumptions I and 2 have been removed by f u r t h e r developments in the
model [ 4 , 4 4 ] .
There w i l l
be two s e c tio n s to t h is c h a p te r:
an a d a p ta tio n o f
R a ith b y and H o lla n d s 's r e s u l t f o r c o n c e n tric spheres to th e p re se n t
g e o m e trie s , and a m o d ific a tio n o f th e b a sic th e o ry to account f o r con­
v e c tiv e flo w around a cube.
It
is w orth c l a r i f y i n g a t th e o u ts e t th a t
t h is method always g e n era te s c o r r e la tio n s o f th e form
Nua [ o r Q) =
Raa ^ '
G
(5 .1 ),
58
where G is a c o r r e c tio n f o r geom etry.
CONCENTRIC SPHERES
The s t a r t in g p o in t f o r g e n e ra tin g a c o r r e la t io n based on c o n c e n tric
spheres is e q u a tio n ( A l . 1 6 ):
4
i »o
i >o"
V
’° r i , o ^ ,o
Ci,oRaS.4n r I fS, O Tj Q-T^CX,o) s/3
i.Pls— -Jn I
^
)
I g
'i,o
0
O
I
1/3 4/3
3/4
ri,odxi,oJ
(5 .2 )
where th e s u b s c rip ts i , o
s ig n if y t h a t th e e q u atio n may be a p p lie d to
e i t h e r th e in n e r o r o u te r s u rfa c e (see F ig u re 5 . 1 ) .
T h is form o f
e q u a tio n ( A l . 16) is f o r axisym m etric flo w , which is somewhat o f an ap­
p ro x im a tio n f o r th e c u b ic a l in n e r o r o u te r body.
I t w ill
however, t h a t t h is e f f e c t on o v e r a ll h e a t t r a n s f e r w i l l
be assumed,
be s l i g h t [ 3 3 ] .
The h e a t t r a n s f e r o u t o f th e in n e r body is
ukD
2 i v
v
*1
where Tjn is th e a s - y e t undeterm ined average f l u i d te m p e ratu re between th e
in n e r and o u te r conduction la y e rs and D1 is th e d ia m e te r o f a h y p o th e t­
ic a l
sphere o f s u rfa c e a rea equal to t h a t o f the in n e r body.
( 5 . 2 ) may then be w r it t e n as
Equation
59
x \ \
F ig u re 5.1
Nomenclature f o r th e C ylin d er-C u b e Geometry
'I.
( D7} D1
' C
r i d“ i f
I
] 3 /4 }
(5 .3 )
For a given shape o f in n e r body, a l l
be absorbed in to
terms in braces a re c o n s ta n t and may
, r e s u lt in g in
;i
1
where Ra0 . is based on T1--T m.
Ci <
(5 .4 )
4
The h e a t f l u x is th e n , a f t e r absorbing
tt
in t o C
i
9 = kDi ( Tr Tm) c i Ra0 i
(5.5)
9 ■
A s im ila r a n a ly s is f o r th e o u te r body leads to
,3 q = kD
S ince I
m
is s t i l l
2
(5 .6 )
cJ
^
- ] , / 4 ( Tm„ - Lo') 5/ 4
o oL
ky
undeterm in ed, th ese h e a t flu x e s may be equated under
th e c o n d itio n on Tm such t h a t Cq=C^=C, which leads to
q = kC[
9 6 ^ 1 /4
kf
J
(T r To )5 /4
. ( D - 7Z 5t D- 7/ 5) 5/ 4
(5 .7 )
'
61
In dim en sionless form t h is becomes
1 /4
J /4
IMul
7Z5 ♦ D -7 75 I 5z4 r t f
C^Ra
or
* 1 /4
(5 .8 )
s-
where 5 may be any one o f th e th re e gap w id th s L, L ' , o r L" discussed in
C hapter IV .
C o r r e la tio n
( 5 . 8 ) was te s te d a g a in s t th e v a r ia b le d e n s ity data and
th e r e s u lts a re presen ted in Table 5 .1 .
p resen ted in T a b le 5 .2 .
The r e s u lts f o r a l l
data are
Also in c lu d e d a re r e s u lts from a s im ila r c o rre ­
la tio n :
Nu5 = Cj Ra*02
.
(5 .9 )
Equation ( 5 . 9 ) c o r r e la te s th e data b e t t e r as the R a y le ig h number exponent
in c re a s e s above th e .25 v a lu e ( c f . C hapter I V ) .
L" i s . t h e
b e st ch ara c ­
t e r i s t i c dim ension to use in e q u atio n ( 5 . 9 ) , although no one gap w id th is
s u p e r io r in e q u a tio n ( 5 . 8 ) .
It
is c le a r t h a t th e c y lin d e r-c u b e geometry
is b e t t e r modeled by t h is e q u iv a le n t spheres c o r r e la t io n than the cubecube geom etry.
This was as e x p e c te d , s in c e the c y lin d e r used, w ith hemi­
s p h e ric a l end caps, is a smoothly rounded o b je c t , whereas th e cube is
sharp-edged and has la rg e f l a t s u rfa c e s .
As may be seen by comparison
w ith T a b le 4 . 2 , no improvement in th e o v e r a ll c o r r e la t io n f o r a l l geomet­
r ie s
is e ffe c te d by e it h e r e q u atio n ( 5 . 8 ) o r ( 5 . 9 ) ,
TABLE 5,1
CORRELATIONS [ 5 , 8 ) AND [ 5 . 9 ) WITH THE VARIABLE DENSITY DATA
E m p iric a l Constants
Average % D e v ia tio n
Equati on Form
a ll
[5 .8 )
geom etries
c y lin d e r-c u b e
geometry
cube-cube
geometry
a ll
geom etries
%
o f d a ta w ith in ±20%
o f eq u atio n
1 .6 2 0
1 .6 1 2
1 .6 1 6
2 1 .0 4
2 0 .8 4
7 1 .4 3
7 1 .4 3
7 1 .4 3
1 .5 7 5
1 .5 7 5
1 .57.5
1 8 .6 3
1 8 .6 3
1 8 .6 3
7 0 .0 0
1 .7 5 4
1 .7 5 4
1 .7 5 4
2 4 .6 8
6 0 .0 0
2 4 .6 8
6 0 .0 0
1 .1 1 3
1 .1 0 9
1 .0 8 7
2 5 .2 8
1 5 .1 8
1 4 .9 4
8 2 .8 6
8 1 .4 3
8 4 .2 9
7 0 .0 0
TABLE 5 .2
CORRELATIONS [ 5 . 8 ) AND ( 5 . 9 ) .WITH ALL DATA
Average % D e v ia tio n
% o f d a ta w ith in ±20%
o f eq u atio n
1 .8 5 0
2 3 .1 3
2 3 .2 8
2 3 .4 0
6 2 .2 4
6 2 .2 4
1 .2 0 7
1 8 .5 2
1 8 .7 4
1 8 .3 7
E m p iric a l C onstants
E q uation form
a ll
(5 .8 )
geom etries
1 .8 4 0
(5 .9 )
a l l g eom etries
.
65.31
6 6 .3 3
6 6 .8 4
64
FLOW AROUND A CUBE
The f a i l u r e o f th e method to c o r r e la t e these geom etries is th a t i t
does n o t account f o r treat t r a n s f e r along h o riz o n ta l s u rfa c e s , in th is
case on o u te r o r in n e r cubes.
In t h is s e c tio n th e th e o ry w i l l
be modi­
f i e d to c o r r e c t f o r h o r iz o n ta l flo w , and a new c o r r e la t io n w i l l be d e v e l­
oped which improves th e f i t
to th e cube-cube d a ta .
be developed f o r th e i n f i n i t e
c o n s ta n t.
This w i l l
This c o r r e la t io n w i l l
atmosphere c as e , which, assumes th a t Too is
le a d to some e r r o r (see Appendix I ) ,
but the c o rre ­
sponding i n f i n i t e atmosphere c o r r e la t io n f o r th e cube is a ls o presented,
f o r comparison purposes.
p ro x im a tio n w i l l
Any e r r o r due to th e i n f i n i t e
then be p re s e n t in both c ases, and th e improvement
e ffe c te d by th e new c o r r e la t io n w i l l
th e two.
atmosphere ap­
In form ing th e i n f i n i t e
as (T 1- T 0 )Z Z .
be th e d iffe r e n c e in e r r o r between
atmosphere c o r r e la t io n s ,
.
I
03 is taken
'
The nom enclature f o r flo w around a cube is presented in F ig u re 5 .2 .
W ith r e fe re n c e to e q u a tio n ( A l . 1 4 ) , th e N u s s elt number may be w r it t e n as
C1R a y 4 [ I J0= r W 3 £ , V 3
Nu
d x ]3 /4
IIo r1 dx
( 5 .1 0 )
By symmetry, and assuming th a t th e flo w along each v e r t ic a l cube fa c e is
independent o f t h a t along any o th e r , th e flo w may be d iv id e d in to fo u r
e q u iv a le n t s e c tio n s as shown in F ig u re 5 ,3 a .
Each s e c tio n is thus ap­
p ro x im a te ly tw o -d im e n s io n a l, so t h a t e q u a tio n (.5 .1 0 ) reduces to
65
F ig u re 5 .2
Nomenclature f o r Flow Around a Cube
66
F ig u re 5 .3
Id e a l i z a t io n o f Flow Around a Cube:
The Shaded Regions are o f Equal Area
67
1 /4 .
Nu^ = CgRa^
The in t e g r a l
cube.
1
c
9- 1 /3 d x ]3 /4
( 5 .1 1 )
in e q u a tio n ( 5 .1 1 ) vanishes alo ng the top and bottom o f th e
Thus no heat t r a n s f e r is p re d ic te d on these faces and th e c o r r e la ­
t io n reduces to th e v e r t ic a l
p la te case [ 4 ] :
( 5 .1 2 )
Nud = .523Rad/ 4
T h is c o r r e l a t i o n , when a p p lie d to a l l
o f th e cube-cube d a ta , r e s u lts in
an average d e v ia tio n o f 3 6 .1 3 p e rc e n t, w ith 1 6 .3 9 p e rc e n t o f th e d a ta
f a l l i n g w ith in ±20 p e rc e n t o f the e q u a tio n .
fo llo w s , each one fo u rth o f th e cube w i l l
'
In the c o r r e c tio n which
be modeled as in F ig u re 5 .3 b .
T h is in v o lv e s an ap p ro xim a tio n in t h a t th e flo w is modeled as s t r i c t l y
tw o -d im e n s io n a l.
The model is t h a t th e f l u i d flow s u n ifo rm ly over the
top and bottom faces in F ig u re 5 .3 b r a t h e r than g a th e rin g towards th e
c e n te r o f these fa c es as in F ig u re 5 .3 a .
Assumption #4 above does not a p p ly along th e top and bottom su rfa ce s
I t was th e r e fo r e decided to form a fo rc e balance on these s u rfa ce s be­
tween viscous and a c c e le r a tio n , r a th e r than buoyant, fo rc e s .
Then the
e q u a tio n o f momentum becomes
F o llo w in g th e suggestion o f Bandrowski and Rybski [3 9 ] th e v e lo c it y p ro ­
file
was assumed to be
68
U = UniaxTid
- n )2
where n = y /6 (boundary la y e r t h ic k n e s s ).
(5 .1 4 )
Equation ( 5 .1 4 )
is th e v e lo c it y
p r o f i l e f o r f u l l y developed la m in a r f r e e c o n vectio n between v e r t ic a l
p la te s a t two d i f f e r e n t tem peratu res [ 5 ] ,
The flo w between c o n c e n tric
cubes here is not f u l l y d e ve lo p e d , so th e adoption o f t h is p r o f i l e does
in v o lv e an a p p ro x im a tio n .
6 is then re p la c e d by A ., th e conduction la y e r
th ic k n e s s , which in tro d u c e s some f u r t h e r a p proxim ation (see F ig u re A l . I ) .
But f o r f l u i d s w ith P ra n d tl numbers n o t too f a r removed from u n i t y , such
as a i r , th e momentum and therm al boundary la y e rs a re o f comparable t h i c k ­
ness, so t h a t th e e r r o r is p ro b a b ly n o t too g r e a t.
W ith t h is assumption
th e v e lo c it y p r o f i l e may be r e w r it t e n as
u=
Two f i n a l
file
assumptions w i l l
0 " Mi
)2
■
(5-15)
now be made concerning th e v e lo c it y p ro ­
on th e top and bottom su rfaces':
That dumax/d x=0 and t h a t d rQ/d x = 0 .
The f i r s t o f th ese im p lie s th a t th e flo w is uniform and f u l l y developed
o ver these s u rfa c e s .
T h is is c le a r l y an a p p ro x im a tio n , s in ce by symmetry
th e v e lo c it y grows from ze ro a t the c e n te rs .
But i f th e re g io n o f t h is
growth is la r g e ly n e ar th e c e n te r , t h is a p proxim ation is no t o v e rly
r e s tr ic tiv e .
The second a p p ro xim atio n s ta te s th a t f l u i d
e n te rs the o u te r
re g io n o f th e boundary la y e r from th e s ta g n a n t re g io n a t a r a te which equals th e growth o f th e in n e r re g io n .
The r a te o f growth o f th e in n e r
69
re g io n is determ ined by a balance o f energy absorbed from th e w a ll w ith
t o t a l energy convected in th e boundary la y e r (see e q u atio n ( 5 .1 7 ) b e lo w ).
Flow v is u a l iz a t i o n data [ 3 3 ,4 0 ] lend some j u s t i f i c a t i o n
to t h is assump­
t io n in t h a t f l u i d does appear to e n te r th e boundary la y e r from the
s ta g n a n t re g io n as th e flo w advances.
The p h y sic a l j u s t i f i c a t i o n
of
these two assumptions is tenuous a t b e s t, but th e m o tiv a tio n f o r each o f
them is p ra g m a tic .
The f i r s t assumption leads to a much more t r a c t a b le
d i f f e r e n t i a l e q u a tio n f o r ym (e q u a tio n ( 5 .2 1 ) b e lo w ).
The second assump­
t io n a ll e v ia t e s th e n e c e s s ity o f having to in d e p e n d e n tly e v a lu a te the
param eter Y (T 0 -T 00)Z (T 5-T m) .
R aith b y and H ollands d id n o t have to e v a lu ­
a te t h is p a ra m e te r, because they were a b le to in d e p en d e n tly e v a lu a te th e
param eter
c o r r e la t io n
in which i t appeared by means o f th e standard f l a t p la te
(see Appendix I ) .
la t i o n f o r h o riz o n ta l
above w i l l
ym( x ) .
be made.
flo w o ver h o r iz o n ta l s u rfa c e s , so th e assumption
These two assumptions w i l l
T his exp re s sio n w i l l
more r e a l i s t i c
But th e a u th o r is aware o f no such c o r r e ­
s it u a t i o n .
lead to an e xp ressio n f o r
then be m o d ifie d to conform to a p h y s ic a lly
The r e s u l t , i t
is hoped, w i l l
p a r t l y compen­
s a te f o r u n r e a l i s t i c assumptions made p re v io u s ly .
F o llo w in g th e development o f Appendix I ,
th e mass flo w per u n it
depth is o b ta in e d from an in t e g r a t io n o f u (y ) from 0 to y m:
r i = pUm / y A
- T
+ f )
(5 :1 6 )
70
The energy balance on th e c o n tro l volume in F ig u re A l . I
k <TS - V _ „
d rI
is , fo r I
dT0 ,V
cP w : (T r TJ
+ %
* r
C V V
■
(5 .1 7 )
W ith th e above assumption t h i s becomes
= cP ^ f
(T r U
•
(5 .1 8 )
T1- is found by in t e g r a t io n over th e l i n e a r tem peratu re p r o f i l e :
Jgm T1Udy
I,
1 . Jnym u dy
which becomes
Ti - T-
(5 .1 9 a )
Ts - Tm
where
„ _ I
r 20-30M+12Mt n
B- M' - L
----------5-J •
M
(5 .1 2 b )
30-40M+15M^
Combining ( 5 .1 8 ) and ( 5 .1 9 ) y ie ld s
dr _
_i_
y m dx
k
- C B
P
( 5 .2 0 )
Combining ( 5 .2 0 ) and ( 5 .1 6 ) r e s u lts in
ym &
[p "max MymE] = C ^B
(5 .2 1 a )
where
2M , M2
T
+ T
(5 .2 1 b )
71
The s o lu tio n to e q u a tio n ( 5 .2 1 ) is
r
2kx
-,1 /2
l c PV x p bemJ
( 5 .2 2 )
\
At t h is p o in t th e main p h y s ic a l hypothesis is made:
t h a t th e in n e r r e ­
gion grows from ze ro th ic k n e s s in d e p e n d e n tly along each s u rfa c e o f the
cube, so t h a t y
=0 f o r both th e top and bottom fa c e s .
Along th e v e r t i ­
go
c al fa c e , th e growth o f ym is assumed to be t h a t o f th e v e r t ic a l f l a t
p la te case, e q u a tio n ( A l . 9 ) .
In o th e r w ords, the boundary la y e rs a re
uncoupled to th e e x te n t t h a t th e in n e r reg io n s are indep en d en t.
The
flo w s rem ain coupled in t h a t th e assumed v e lo c it y p r o f i l e on th e top and
bottom s u rfa c e s , e q u a tio n ( 5 . 1 4 ) , is seen as th e product o f flo w o ver th e
e n t i r e cube.
The d is tu rb a n c e in th e flo w caused.by n e g o tia tin g the sharp
co rn ers o f th e cube is modeled as
a c o lla p s e o f the in n e r re g io n .
M a n ip u la tio n o f e q u a tio n ( 5 .2 2 ) leads to th e r e s u l t t h a t th e in n e r
re g io n would grow much more q u ic k ly on th e h o riz o n ta l s u rfa c e s than on
th e v e r t ic a l
fa c e , where e q u atio n ( A l . 9 ) is expected to h o ld .
I t was
decided then to a tte n u a te th e growth r a te o f e q u atio n ( 5 . 2 2 ) , but to keep
th e same fo rm , so t h a t t h e -in n e r re g io n th ic k n e s s a t th e edge o f the
bottom fa c e is th e same as i f
same le n g th .
i t had grown along a v e r t ic a l p la te o f th e
To d e term in e t h is a tte n u a tio n f a c t o r , a s p e c if ic expression
is needed f o r u
.
To p re se rv e u n it y , and to in tro d u c e an expression
w ith independent p h y s ic a l m e r it , Umgx is a ls o chosen.as th e v alu e i t
72
/
would have i f
it
had grown along t h e . v e r t i c a l p l a t e , as w ith ym, i . e . ,
e q u a tio n CA1.4) is used f o r Ufiigx, so t h a t
PS(T«
(5 ,2 3 )
max
where y m^ is th e v alu e d e sc rib e d above, o r f o r the bottom fa c e
ML
^ml =
where L = d /4 .
( 5 ,2 4 )
cLRa1
/ 4
S i m i l a r l y , f o r the top fa c e
.
M(SL)
( 5 .2 5 )
Ml - K
When e q u atio n s ( 5 .2 4 ) and ( 5 .2 3 ) a re s u b s titu te d in to e q u a tio n (5 .2 2 )
under th e d e s c rib e d c o n s t r a in t , the a tte n u a te d form o f e q u atio n ( 5 .2 2 )
becomes
ym
2kx
n1 / 2 .
( 5 .2 6 )
To perform t h is c a lc u la t io n , M was taken as R aithby and H o lla n d s 's recom­
mended v a lu e o f .4 f o r a i r .
W ith e q u atio n ( 5 . 2 6 ) , h e a t flu x e s can be c a lc u la te d .
fa c e , w ith M =.4 ,
.]l/4
1 .0 1 9 [C p p 'W g L
xl / 2
On the bottom
73
„
q
kM(T, - TJ
= --------- v---------
and
4LkM(,Ts - Tte) _ g g A T p C L , / 4
] '/"
I .Q l 9
7 .8 5 1 LkMATRa
.1 /2
x-
r
O
1 /4
( 5 .2 7 )
S im ila r c a lc u la tio n s f o r the top face r e s u l t in
1 /4
q = 6 ,4 2 1 LkMATRa^
For th e v e r t ic a l
fa c e , e q u atio n ( A l .9 ) is used f o r ym, so t h a t
. y,
and
( 5 .2 8 )
1 .0 1 9x
4LkATM |-g3ATCpP -,1 /4
1 .0 1 9
14 .8 1 SLkMATRa
r
r4L v - l / 4 ^
x 7V x
0
1 /4
(5 :2 9 )
Then th e t o t a l h e a t t r a n s f e r f o r t h is one fo u rth o f th e cube becomes
'
■
q = 1 .0 2 8 k d A T R a y 4 .
In dim en sionless form ,
Nud = .868Rad/ 4 .
( 5 .3 0 )
Equation ( 5 ,3 0 ) c o r r e la te d a l l o f th e cube-cube data w ith an average
d e v ia tio n o f 1 6 .0 4 p e rc e n t, w ith 8 2 .1 6 p e rc e n t o f th e data f a l l i n g w ith in
±20 p e rc e n t o f th e e q u a tio n .
The improvement gained by t h is a n a ly s is is
)
74
re m a rk a b le , from 3 6 .1 3 to 1 6 .0 4 p e rc e n t average d e v ia tio n ;
ment could most l i k e l y be improved by using more r e a l i s t i c
The develo p ­
assumptions
about th e maximum v e lo c it y and mass flo w r a t e s , and perhaps by employing
a b e t t e r type o f c o n s t r a in t on the v a lu e o f ym a t th e bottom edge o f the
cube.
D e s p ite th e many a pproxim ations and assumptions in tro d u c e d in t h is
developm ent, th e f i t
o f th e r e s u ltin g c o r r e la t io n o ver a wide range o f
gap w id th s (W a rrin g to n 's d a ta ) and pressures (th e p re s e n t a u th o r's d a ta )
is en co u raging.
It
is a ls o w orth n o tin g t h a t t h is c o r r e la t io n was d e v e l­
oped f o r th e i n f i n i t e
atmosphere c a s e , and s t i l l
any o th e r f o r th e cube-cube d a ta ( c f .
5 .1 ).
the f i t
is as. good as
eq u atio n s ( 4 . 2 9 ) - ( 4 . 3 1 ) and Table
CHAPTER. VI
CONCLUSIONS
This in v e s tig a tio n has dem onstrated th e im portance o f the f l u i d .
d e n s ity in c o r r e la t in g n a tu ra l convection in a i r a t low p re s s u re s .
New
c o r r e la tio n s have been g enerated o ver a wide range o f a i r p re s s u re s .
The two th e a u th o r recommends most h ig h ly are
Nul = .142RaL310
Nul = .342R aW CpZpatm) ' 129 .
The d e n s ity c o r r e c tio n has been shown to be more im p o rta n t than the
g eo m etric c o r r e c tio n a t low p re s s u re s , as dem onstrated by th e good f i t
o f th e d e n s ity -c o r r e c te d c o r r e la tio n s o f T a b le .4 .5 .
It
is th e a u th o r's
o p in io n t h a t e i t h e r o f th e above two c o r r e la tio n s may be used w ith some
co n fid e n ce to p r e d ic t c o n vectio n h e a t t r a n s f e r in a course vacuum
(p re s s u re > 1300 Pa (1 0 mm H g)) when th e space surrounding th e in n e r,
body is not much la r g e r than th e body i t s e l f .
.
In in fin ite -a tm o s p h e r e
typ e s it u a t io n s , where th e gap w id th L becomes la r g e r than th e in n e r
body, th e a u th o r recommends a boundary la y e r le n g th c o r r e la t io n such as
Nufa = .157Rab309
.
I t has a ls o been shown t h a t th e c h a r a c t e r is t ic dim ension chosen in
c o r r e la t in g h e at t r a n s f e r between th re e -d im e n s io n a l o b je c ts is no t c ru ­
c ia l :
dim ensions based on gap w id th , bulk s iz e s , s u rfa c e a re a s , o r in n e r
76
body boundary la y e r le n g th work e q u a lly w e l l .
APPENDIX I
DERIVATION OF THE METHOD OF RAITHBY AND HOLLANDS
In 1975, G. D. R a ith b y and K. G. T. H ollands [ 4 ] p u b lis h e d a method
o f g e n e ra tin g new c o r r e la tio n s f o r n a tu ra l convection h e at t r a n s f e r .
The method is based on form ing a balance among buoyant; a c c e le r a tio n , and
viscous fo rc e s in th e boundary la y e r v e ry c lo s e to th e w a l l .
The fo llo w ­
in g is a b r i e f o u t lin e o f th e p h y s ic a l development o f th e method.
For
g r e a te r d e t a i l s , th e re a d e r is in v it e d to c o n s u lt th e c it e d p u b lic a tio n .
I t was h y p o th esized th a t th e boundary la y e r in n a tu ra l c o n v e c tiv e
flo w could be d iv id e d in t o in n e r and o u te r re g io n s , Y I
(see F ig u re A l . I ) .
Ym and Y > Ym
Under th e assumption o f n e g lig ib le momentum t r a n s f e r
across th e v e lo c it y extremum a t ym, th e fo rc e balance in th e in n e r r e ­
gion may be w r it t e n
in d e p e n d e n tly from th a t in th e o u te r re g io n .
I t was
f u r t h e r assumed t h a t a c c e le r a tio n fo rc e s in th e in n e r re g io n a re n e g l i ­
g ib le .
O th e r assumptions in c lu d e d :
a ll
f l u i d p r o p e rtie s a re c o n s ta n t;
boundary la y e rs a re v e ry sm all r e l a t i v e to th e ra d iu s o f c u rv a tu re o f
th e submerged body; th e te m p e ratu re p r o t i l e
l i n e a r (F ig u re A l . l ) ;
to th e t o t a l
in th e in n e r re g io n is
th e r a t i o o f th e buoyant fo rc e in th e in n e r re g io n
buoyancy o f th e boundary la y e r is a fu n c tio n o f th e P ra n d tl
number o n ly and i n v a r ia n t w ith x , th e p o s itio n along th e body.
Ttie momentum e q u a tio n in th e x d ir e c t io n is
where gx is th e g r a v it a t io n a l a c c e le r a tio n in the d ir e c t io n o f x .
With
78
th e l i n e a r te m p e ratu re p r o f i l e assumption and th e appro xim atio n
Poo - p = p 3 ( I - Too)
one may w r it e
( A l.2)
P00 - p(y) = -PGfToo - Ts ) ( I - M
”m
where
7M ' TS _ ym
is th e lo c a l th ic k n e s s o f a s ta g n an t conduction la y e r w ith h e at tr a n s ­
f e r e q u iv a le n t to th e a c tu a l convectio n h e a t t r a n s f e r .
( A l.2 )
in to
(A l.I),
S u b s titu tin g
one o b ta in s
(T
4 = -PB
ay
- TJ
y) •
"
(Al.3)
"m
In t e g r a t in g t h is e q u a tio n w ith a p p ro p ria te boundary c o n d itio n s a t y=0 and
y=ym y ie ld s
(Al. 4)
Equation ( A l . 4 ) is then in te g r a te d to g iv e th e mass flo w per u n it depth
(.in to th e paper in F ig u re A l . l ) :
T1 = p23" ‘ V
sl
bA
h i -t
)
(Al.5)
79
c o n tro l volume
F ig u re A l . I
Nom enclature o f the Method o f R aith b y and Hollands
80
The energy balance f o r th e c o n tro l volume in F ig u re A l . I
( TM-TS)
dr,
T
is
dr.
cP
( V
t
I) + r
C p ( V T 0)
( A l . 6)
where th e d e f i n i t i o n s
drQ/dx
Y ~ dr^/dx
and
T.
= average in n e r , o u te r f l u i d
I 9O
have been used.
tem p eratu re
is found by using th e l i n e a r te m p e ratu re p r o f i l e a s ­
sumption and e q u atio n ( A l . 4 ) :
I.
1
=
J^m Ti U dy
-----------------
, which becomes
ffsiudy
Ti = Ts + 'Tm - Ts)
T h is may be r e w r it t e n as
T°°~Ti _ I
M
V tS
5 /8 -2 M /5
1 -5 M /8
( A l . 7)
’
Equations ( A l . 5) and ( A l . 7) a re combined to y ie ld
dr.
&
T
- V I c P tH -
+ T %
] }
( A l.8)
Equations ( A l . 5) and. ( A l .8 ) could be solved f o r y , and thus f o r th e h e a t
81
t r a n s f e r , i f th e param eters M and Y (T 00- T 0 )Z (T m-T 5 ) were known.
p o in t th e main h ypothesis o f th e development was advanced:
A t th is
t h a t these
two param eters depend o n ly on th e P ra n d tl number (P r ) o f th e f l u i d and
not on th e geom etry, as long as th e c o n d itio n ym« R ( x )
t u r e ) is f u l f i l l e d .
(ra d iu s o f c u rva ­
W ith t h is h y p o th e s is , th e Pr dependence o f these
param eters may be found f o r any geometry and then a p p lie d to any o th e r
geom etry.
The au th o rs found th e Pr dependence f o r th e case o f th e v e r t ic a l
f l a t p la te .
S p e c i f i c a l l y , th e Pr dependence o f a new s in g le param eter
in which o n ly th e .a b o v e two param eters appear was d e te rm in e d .
The
a n a ly s is begins by combining eq u atio n s ( A l . 5) and ( A l . 8 ) , r e a l i z i n g th a t
gx=g=constant f o r t h is c as e , to o b ta in a f t e r in te g r a t io n
=
Mx
( A l . 9)
1/4
V
aX
where
C
and Rax is based on T00- T 5 .
5/8-2 M/5
L
V
The im portance o f
f i r s t fo r m u la tin g th e h e a t f l u x in to th e w a ll
a
A
k
( - V Ts >
^m
and thus th e lo c a l h e a t t r a n s f e r c o e f f i c i e n t
To , 1
( A l . 10)
may be dem onstrated by
82
k
h
x
AO V^T
Tm~Ts _ k'M
y
T -T
-yHi
<»
“ y
s
Jm
The h e at t r a n s f e r c o r r e la t io n then becomes
C, Ra
Nux =
1 /4
x
The h eat, t r a n s f e r may thus be c a lc u la te d when the Pr dependence o f
alo n e is known.
No knowledge o f th e Pr dependence o f th e two param eters
M and Y(Tco- T 0 )Z (T m- T 5 ) is re q u ire d .
The authors quoted a s im ila r s o lu ­
t io n r e s u l t o f Rohsenow and Choi [3 8 ] f o r C&:
C
= 0 .4 8 [ P r / ( . 8 6 1 + P r ) ] 1 /4
which was then used to re p la c e e q u a tio n ( A l .1 0 ) in th e g en eral case.
In e x te n d in g th e method to tw o-dim ensional and axisym m etric geome­
t r i e s , th e au th o rs f i r s t d e fin e d an e q u iv a le n t conduction th ic k n e s s
Nu
V x)
In steps which resem ble th e v e r t ic a l
x
Ajl(X )
f l a t p la te developm ent, an equa­
t io n f o r T . is found by e lim in a tin g ym from ( A l .8 ) using ( A l . 5 ) :
-.4/3 .
4k r p
3BCp L
577
o-fn , / 3
( A l . 11)
Z
83
where
I
5/8-2M/5
B = M ■ l-S M /T
S o lu tio n o f t h i s d i f f e r e n t i a l
T -T
CO
0
+ Y Tm- T s
( A l . 12)
e q u atio n w ith th e boundary c o n d itio n T1-=O
a t x=0 (see F ig u re A l . 2) y ie ld s
ijjfx )
173
C1 R a y 4
T1
The average conduction th ic k n e s s is found by an in t e g r a t io n o f th e lo c a l
h e at t r a n s f e r o v er th e le n g th o f th e boundary l a y e r , r e s u lt in g in
I rs J
J05 r 1 dx
s
^
[I
In th ese e q u a tio n s , i= 0 and i = l
j-s
( A l . 14)
( / i 9xjl/3 ^ ^ /4
f o r tw o-dim ensional and axisym m etric
g e o m etries, r e s p e c t iv e ly , and C%=(4/3)C%.
The f i n a l m o d ific a tio n o f th e th e o ry f o r a p p lic a tio n to convection
w ith in e n clo s u re s in v o lv e s th e f a c t t h a t Too is no lo n g e r c o n s ta n t w ith
x.
In t h is case th e energy e q u atio n (A T .6 ) becomes
k ( ^ ) - cpjL Er1(V T 1)] ♦ CpjL Er0(V T 0)].
The assumption t h a t B, e q u a tio n ( A l . 1 2 ) , depends o n ly on Pr is somewhat
le s s s a t is f a c t o r y here because o f p o s s ib le v a r ia t io n as Too-Ts changes.
84
I
F ig u re A l . 2
Nomenclature f o r Two-Dimensional
o r Axisym m etric Flow
.
-RJ
R e a liz in g t h is a p p ro xim a tio n and proceeding as b e fo re y ie ld s
l_,x
A
xAT
1( X )
CnRa
I
4 /3
,„4,' & A L , ,/3 AT4/3 dx],/4
g atv
Io Ir".
(A T .15)
174
i gx AT V173
(r
T A T :)
x
where AT^ is some re fe re n c e te m p e ratu re d iffe r e n c e on which Rax is based.
The average conduction th ic k n e s s becomes
v
-
S
C" Ra' / 4
T
rUs(£t
/
/3
£
r1
(t
) 1 /3 r 4 V 3
gyF^"
•
( A l . 16)
dx]
Equations ( A l . 1 3-A 1 .1 6 ) a re then th e s t a r t in g p o in t f o r th e a n a ly s is in
Chapter V.
APPENDIX I I
DATA REDUCTION PROGRAMS
The fo llo w in g a re F o rtra n IV programs used to reduce th e h e at tr a n s ­
f e r and te m p e ratu re p r o f i l e d a ta .
The su b ro u tin e s CURFT and ERROR are
used to c o r r e la t e th e h e a t t r a n s f e r and a re c a lle d when needed by the
m a in lin e program.
The fu n c tio n T is used to c o n ve rt therm ocouple re a d ­
ings to te m p e ra tu re s .
The in p u ts to th e h e at t r a n s f e r program are the
dim ensions o f th e in n e r and o u te r b o d ie s , t h e i r te m p e ra tu re s , th e convec­
t io n h e at t r a n s f e r r a t e , and th e p re s s u re .
The in p u ts to th e tem peratu re
p r o f i l e program a re th e lo c a l gap w id th , th e tem peratu res o f th e in n e r
and o u te r b o d ie s , and th e m icrom eter and thermocouple re ad in g s a t each
s te p in th e t r a v e r s e .
87
DI MENS I OiM OHOPEC 5 0 0 ) , EASTAE C5 0 0 ), XXNUSC 500 ) ,X C6 ,500
DIMENSION ZASTAfiC 5 0 0 ) , GEOC 500) , X P f i C 500),
I
ZZNUSC 5 0 0 ) , XfiAC 5 0 0 ) , ZfiAC 5 0 0 ) , CC6 ) , DEN I C50 0 )
)
C
C***HEAT TfiANSFEfi DATA REDUCTION
C
C ***K IS A COUNTER
K=0
P I=3*1415927
C ***READ IN A NEGATIVE VALUE Ofi ZERO FOR IDB EOfi LAST RUN
50 READ C105,25) I DB,XCUBE,SD,XXX,TAVGI, TAVGO, P ,PfiESMM
25 FORMAT C1 5 ,7F9.4)
C * * * I DB IS THE GEOMETRY IDENTIFIER
C
!-SPHERICAL INNER BODIES
C
2-CYLINDRICAL INNER BODIES
C
3-CUBICAL INNER BODIES
C***XXX IS A RATIO OF CYLINDER AND OUTER BODf DIMENSIONS
C
XXX=CLENGTH OVERALL-SD)/CXCUEE-SD)
C***XCUBE, SD, AND XLEN ARE OUTER AND INNER BODY DIMENSIONS
C***P IS THE CONVECTION HEAT TRANSFER CPTU/hfi)
C * * +PfiESMM IS THE EXPERIMENTAL AIR PRESSURE CMM HG)
C ***SA IS THE INNER BODY SURFACE AREA
C ***R.I IS THE RADIUS OF AN EQUIVALENT VOLUME INNER SPHERE
C ***DI IS THE RADIUS OF AN EQUIVALENT AREA INNER SPHERE
C ***R0 AND DO ARE CORRESPONDING VALUES FOR THE OUTER BODY
C ***FLL IS THE LENGTH OF THE BOUNDARY LAYER ON THE INNER BODY
C ***ALL DIMENSIONS ARE IN INCHES
C
IF CI DB) 1 00, 1 0 0 , 3 0
30 K = K + I
IF CK.EQ.l) OUTPUT I DB,SD,XXX
GO TO C4 ,5 ,6 ) IDB
C ***SPHEfiES, CUBICAL OUTER BODY
4 SA=PI*SD*SD
RO=C C3.*XCUBE + * 3 ) / C 4 . + P I ) ) * * C I . / 3 » )
HI =SDZP.,
GAP=RO-SD/P.
IF CSD.EQ.4.5) GfiAT I O=.9 I 76J GAP I =4.47
IF (SD.ES*7«0> GfiATI O= «801362)GAP I = 3.22
IF CSD.EG.9.0) QfiATIO = . 562401 ) GAP I = 2.22
PLL=P1*SD/P.
DI=SD
GO TO H
88
C * * * C Y L I NDKfcStCUPI CAL OUTKfc PODY
5 XLEN =XXX* (XCUFE-SD)-I-SD
S A = P I * S D * ( X L E N - S D ) + P I * SD*SD
fc0 = ( ( 3 . *XCUPE**3)/( A . * P I ) ) * * ( I . / 3 . )
fcI=(SD/A.) * ( ( 18.*XLEN/SD)-4.) * * ( I . / 3 . )
GAP=FO-KI
IF (XXX.EG.0.3083) CfcATIO = I .22891C-API =4. 58
IF (XXX.EQ.0.5A29) CfcATIO= I . I PA3IGAPl=3•A9
IF (XXX.EQ.0. 31A3) GfcATIO= . 969 I J C-APl =3 .3 1
IF (XXX.EQ.0. 7333) GfcATIO= I .8200J CAP I =4.083
BLL=P I *S D /2.+XLEN-SD
Dl=SQfcKSD*XLEN)
GO TO 11
C ***CUEES.CUBICAL OUTEfc BODY
6 S A = 6 . * SD * S D
fcO= ( ( 3 . * X C U P E * * 3 ) / ( A . * P I ) ) * * ( I . / 3 . )
fcl= (( 3 . *SD**3) / ( 4 • *P I ) ) * * ( ! . / 3 . )
GAP=RO-FI
IF (SD.EQ.4.0) CfcATIO=I.47093 GAP I =4.16
IF (SD.EQ.5.0) QfcAT I O= I . 52355 GAP I = 3.S3
IF (SD.EQ.6.4) QfcATIO = I .6278) L-APl=P.63
BLL=SD*2.
DI =SQF-T( SA/PI >
11 CONTINUE
DO= 14 . 5 10
GAP2=(DO-DI)/2«
C ***DT IS TEMP DIFFERENCE BEThEEN INNER & OUTER BODIES
DT=TAVGI-TAVGO
C ***H IS THE AVERAGE HEAT TRANSFER COEFFICIENT ( B/HR-FT * * 2 - F)
H=P*I 4 4 • / ( DT*SA)
C ***CALCULATE THE MEAN TEMPERATURE (DEG F)
TM=( TAV L I +TAVGO) / 2 .
C ***TAVO, MEAN TEMPE RATURE ( DF G fc)
' TAVG=TM+459•69
C ***CALCULATF THE FLUID PROPERTIES
C***ABSOLUTF VISCOSITY OF AIR ( LBM/hfc-FT)
C9=134.375
C l= 6 .0133834
CP=I.0432299
C3=I . 3347050
VIS =TAV(-**C2/( F.XP( Cl >*<TAVG+C0)**C3)
C ***SPECIFIC HEAT OF AIfc (BTU/LBM-fc)
CO= 2« 2.3c 775/( 10. **b )
C l=•22797749
SH=Cl+C0*TAVC
89
C***THERMAL CONDUCTIVITY 0>
AIR. (BTU/KR-rT-l- )
XP=. I
C0=-8.5964965
C1=34490.69
CS= 868.23837
03=8056583.8
7 KX=XP
f =C0+Cl*FX +C2*KXtKX +C3*KX*KX*PX-TAV(:
KP=Cl+2.+C2*KX+3.*KX*KX*C3
XP=KX-1/ KP
IF (AF5((XP-FX)ZFX)-.0 00 I ) 8 ,8 ,7
■ 8 COND=XP
C ***DENS1TY OK AIR (LPM/KT**3)
DEN=.07625*(PRKSMM/760•>*< 519 . 69/TAVC >
DEN I ( r t ) =DFN
C +THERMAL EXPANSION COEFFICIENT OK AIR (IZR)
BET=I./TAVG
C♦♦♦THE PRANDTL NUMBER
PR=VIS+SH/COND
XPR(K)=PR
C ***THE GRASHOF NUMBER
GR=3 2 .I 74*PK'T*DT*(GAP**3./1 7 2 8 .)*DF N*DEN*3 6 0 0 .*3 6 0 0 ./(VIS +VIS)
C ***THE NUSSE.LT NUMBER
XXNUS ( K ) = H * GAP/ ( COND* I 2 .')
C***THE RAYLEIGH NUMBER
RA=GR*PR
XRA(K)=RA
GEOl=GAP**.2 5 / ) ( D I * * ( - 1 . 4 ) + D 0 * * ( - 1 . 4 ) ) * * 1 . 2 5 * P I * D I * D I )
C+++RASTAR IS THE MODIFIED RAYLEIGH NUMBER ( RA+GAP/RI)
RASTAR( K) =RA*GAP/RI
C ***eSPH IS THE HEAT TRANSFER BY CONDUCTION BETbEEN CONCENTRIC SPHERES
GSPH=PI*COND*XCUEE*SD*LT/(6 . * (XCUPE-SD))
GCOiND=ORATIO*GSPH
C ***QHOPK I S THE FiATIO OK O CONVECTION TO G CONDUCTION
QHOPE(K)=PZQCON d
GEO(K)=GAPZRI
C**+CALCULATE PARAMETERS BASED ON THE BOUNDARY LAYER LENGTH
GR=32«I 74*9ET*DT*(ELL*ELL*BLL/I 7 2 8 .)*DEN*DEN*3 6 0 0 .* 3 6 0 0 ./(V IS*VIS)
XNUS=H*FLL/(C0ND*12.)
RA=GK*PR
ERA(K)=RA
ZASTAR(K)=RA+GAPZRI
ZZNUS(K)=XNUS
GO TO 50
100 CONTINUE
90
C ***OUTPUT DIMENSIONLESS RESULTS
WRITE C108,2000)
2000 FORMAT ( IH I , THRUtN NO. , 3X, I 7HGHOMETEI C FACTORS )
WRITE C108,2001)
2001 FORMAT ClH , 10X,15H BHOPE
GEO)
DO 200 NN= I , K
G=QHOPE(NN)
G=GFO(NN)
WRITE (108,2002) NN, G, G
2002 FORMAT ( IH , I 5 , E 14 . 4 , EI I .4)
200 CONTINUE
WRITE (108,3003)
3000 FORMAT ( 1H0,7HRUN NO .,I I X , I 7HGAP WIDTH RESULTS,
I
I 5X,29HB0UNDARY LAYER LENGTH RESULTS)
WRITE (108,3001)
3001 FORMAT ( IH , I 4X,3HNUS,SX,2HRA,7X,6KRASTAR,I 2X,
I
3HNUS,8X,2HRA,7X,6HRASTAR)
DO 300 NN= I , K
XNUS=XXNUS (NlN)
RAX=XRA(NN)
RASTX=RASTAR(NN)
ZNUS=ZZNUS(NN)
RAZ=ZRA(NN)
RASTZ=ZASTAR(NN)
WRITE (108,3002) NN,XNUS,R A X , EASTX,ZNUS,RAZ,RASTZ
3002 FORMAT ( I H , I 5 ,E I 6 . 4 , 2EI I . 4 , EI 6 • 4 , 2EI I .4)
300 CONTINUE
C***THE FOLLOWING STEPS ARE DIFFERENT FOR
C
DIFFERENT CORRELATIONS
DO 115 N= I , K
X( l»N) = t .
X(2,N)=ALOG10(XRA(N))
I 15 X (3 ,N )=ALOGI 0 (XXNUS(N))
NV=PINOB=K
OUTPUT 'CORRELATION #1'
CALL CURFT(NV,NOP,X,C)
CALL ERRORCNV,NOB,X,C,SSQ)
C l= 1 0 .** C (I ) !OUTPUT Cl
FND
O Q O O
91
SUBROUTINE CURFT(NV>NOP»X,C)
DIMENSION X ( 6 » 2 5 0 ) » C ( 6 ) / S ( 7 , 7 )
DOUBLE PRECISION S, D
LEAST SQUARES - NX/ INDEPENDENT VARIABLES
WRITTEN PY R • E. POWE - MECHANICAL ENGINEERING
Y = X ( N V + 1 , I ) - NOB= NO. Of' OBSERVATIONS
C ( I ) = C O E E F I C l ENT OR X ( I )
M=NV+I
MP=M+I
DO I I = I i M
DO I J = I i M P
1 S (IiJ)= 0.D 0
DO 8 I = UNOB
DO 2 J = I i M
DO 2 K = I i M
2 S (JiK )=S (JiK )+X (Ji I >*X (K iI )
S ( I i M P ) = 1 .DO
IR ( N V - I ) 99 7 i 99 7 i 998
997 S ( I i I ) = S ( I i 2 ) / S ( I i I )
GO TO 999
998 DO 16 L K = I i N V
I I IF (S( I i I ) ) I 3 i I 2 i I 3
12 WRITE ( I 0 6 i 2 0)
20 FORMAT ( ' EQUATIONS IN SUBROUTINE CURFT ARE DEPENDENT !LOWING ERROR A N A L Y S I S ' )
DO 30 I I I = I i N V
30 C d I I )=0.
GO TO 31
13 DO IA J = I i M
IA S ( M i J ) = S ( U J + l ) / S < I i I )
DO I S I = P i N V
D = S d i I)
DO 15 J = I i M
15 S d - I i J ) = S ( I i J + l ) - D * S ( M i J )
DO 16 J = I i M
16 S ( N V i J ) = S ( M i J )
999 DO 3 I = I i N V
C ( I ) = S d i I)
3 WRITE ( I OS i A) I i C ( I )
A FORMAT ( ' ' i 10Xi 'COEFFICIENT OF X ' i I 2 i ' = ' i E l 5•8)
3 1 RETURN
END
IGNORE FOL
O O O O
92
C
SUBROUTINE ERROR(NV,NOB,X,C,SSQ)
DIMENSION X ( 6 ,4 0 0 ) ,C ( 6 ) ,S ( I 0 j 1 0 ) , ANPC I 3 ) , RNRLC4 0 0 ) ,RNRLS<400)
DOUBLE PRECISION YC, T5, TE., YCI
ERROR ANALYSIS - NOB OBSERVATIONS
Y=XCNV+!, I) - SSe=STANDARD DEVIATION
Cd)
- CONSTANTS
WRITTEN BY R• E. POWE - MECHANICAL ENGINEERING
M=NV+I
TS=0.D0
TE=0.D0
EMX=0.
DO 5 1=1,5
5 ANPC I )= 0•
WRITE C1 08,I )
1 FORMAT C' O' , 5X, ' Y EXPERIMENTAL', B X , ' Y CALCULATED ' , 5X,
I
'NUMERICAL ERROR ' , 5X, 'PER CENT ERROR’ )
DO 2 I = I , NOB
YE=XCM,!)
ADD EQUATION FOR YC AT THIS POINT
YC=0.D0
YC=CC I ) +CC 2 )*XC 2,1)
YC1=10.**YC
'YE1=10.**YE
E=YCl-YEl
EP=100.*E/YE1
EPA=ABSCEP)
IF CEPA-EMX) 6 ,6 ,7
7 EMX=EPA
6 DO 8 0=1,5
ACD=0*5
IF CEPA-ACD) 9 ,9 ,8
9 ANPC 0 )=ANPC O) +I •
8 CONTINUE
TS=TS+E*E
TE=TR+AESCEP)
2 WRITE C108,3) YEl, YC I , E,EP,I
3 FORMAT C4EP0.6,I5)
A=NOP
SSC=CTS/A)**.5
TE=TEZA
WRITE C108,4) SSG,TE
4 FORMAT C'0 ',1 O X , 'STANDARD DEVIATION=' , E15.6,5X, 'AVERAGE. PER CENT D
IEVIATION=',El5.8)
WRITE C108,10) EMX
10 FORMAT C' 0 ' , 30X,'MAXIMUM PER CFNT DEVIATION=',E15.8)
93
DO I I 1= 1,5
X,VP=AiMPt I )
XiVP= I 00 • *XtVP/A
ACD=I*5
11 WRITE (168,12) XNP,ACD
12 I ORMAT ( 5X, El 5» 8, 2X, 'PER CEvT Or DATA WI TH I .V' , 2X, EI 5-5, 2X,
I
'PER CEiMT OE EQUATION')
RETURN
END
94
C+++TEMPEFATUfcE PfcOFILE MAINLINE
K=0
20 K=K+I
C+ ++DP. IS GAP SPACING
C+++VHIGHiVLOW AfcE ThEFMOCOUPLE FEADINGS AT ENDS OF TRAVERSE
READ ( 105,2000) DR,VHIGH,VLOW
2000 FORMAT (3F10.4)
C+++READ IN A NEGATIVE OR ZERO FOR DR EOF LAST PROFILE
IF (DR) 100,100,30
30 GO TO (3 1 ,3 2 ,3 4 ,3 6 ) K
31 WRITE (106,4001)
4001 FORMAT ( IHt,'PROBE NO. I ' )
GO TO 40
32 WRITE (108, 4002)
4002 FORMAT ( I H l , 'PROBE NO.2 ')
GO TO 40
34 WRITE (108,4004)
4004 FORMAT (1H1,'PROBE NO. 4*)
GO TO 40
36 WRITE (108,4006)
4006 FORMAT ( I H l , 'PROBE NO. 6 ' )
40 THIGH=T(VHIGH)
TLOW=T(VLOW)
DT=THIGH-TLOW
WRITE (108,3000)
^ 3000 FORMAT ( 1H0,'DIMENSIONLESS DISTANCE
DIMENSIONLESS TEMPERATURE'
C +++READ IN MICROMETER AND THERMOCOUPLE READINGS
45 READ (105,2001) FMICRO,VOLT
•2001 FORMAT (2K10.4)
C +++READ IN A NEGATIVE OR ZERO FOR VOLT FOP LAST READING IN PROFILE
IF (VOLT) 20,20,50
50 RSTAR=(DR-FMICRO)ZDfi
TT=T(VOLT)
TSTAR=(TT-TLOW)ZDT
WRITE ( 108,3001 ) fiSTAR,TSTAR
3001 FORMAT ( IH , F I 6 . 4 , F28.4)8
GO TO 45
100 CONTINUE
END
FUNCTION T(E)
DIMENSION CC8)
DOUPLE PRECISION TOT
C< I >=491.96562
C(C)=46.381864
CCO =- I . 3918864
C(4 ) =0. I 5260798
C ( 5 )= - 0 .020201612
C (6 )= 0 .0016456956
C(7) =-6.6287090/( 10.**5)
C (B )= I.0 2 4 1 3 4 3 /(1 0 .**6)
TOT=0-D0
DO I 1=1,8
I 10T=T0T+C(I ) * ( £ * * ( I - I ))
T=TOT
RETURN
END
APPENDIX I I I
PARTIALLY REDUCED DATA
The f o l l o w i n g is a l i s t i n g o f a l l
the data used in th e heat t r a n s f e r
program, i n c lu d in g t h a t o f W arrington [ 3 3 ] .
The data were reduced to
t h i s form on a Texas Instrum ents SR-40 c a l c u l a t o r .
The dimensions a re
inches on XCUBE and SD, °F on TAVGI and TAVGO, BTU/hr on P, and mm Hg on
PRESMM.
IDB and XXX a re u n i t l e s s .
97
IDB
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
XCUBE
10.5000
10«5000
10.5000
10.5000
10.5300
I 0.5000
10.5030
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10-5000
10.5000
10.5000
10.5000
10.5000
10.5000
10« S P M
10.5000
10.5000
10.5000
10.5000
10.5063
10.5000
10*5000
10.5000
10.5000
10.5030
10.5300
10.5300
SD
4.5000
4*5000
4.5000
4.5000
4.5000
4.5000
4.5000
4•5000
4.5000
4.5000
4*5000
4.5000
4•5000
4.5000
4.5000
4.5300
4.5000
4.5000
4.5030
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4-5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5030
4.5000
4.5000
/4 • 5 0 0 0
4.5000
4.5000
4•5000
4.5000
4.5000
XXX
• 7333
• 7333
• 7333
. 7 3 33
. 7333
• 7333
• 7333
. 7333
. 7333
• 7333
. 7333
. 7333
.7333
• 7333
• 7333
• 7333
• 7333
. 7333
• 7333
. 7333
. 7333
. 7333
. 7333
• 7333
• 7333
. 7333
. 7333
• 7333
• 7333
• 7333
.7333
• 7333
• 7333
• 7333
. 7333
• 7333
• 7333
. 7333
• 73 33
• 7333
. 7333
• 7333
• 7333
• 7333
. 7333
TAVGI
99-804
I 30.174
125.619
103-574
82*299
69.402
68.596
81•703
95.094
121.952
I 19-798
99*920
83.612
75.386
70*929
83.510
106.169
132-035
128.958
106.5(7
92.225
73.122
74.966
91.109
106.382
129.896
125.856
107.952
87.626
69.886
73.156
92.087
I 10.038
125-650
Ip A .ffg
112.045
92-507
73.686
4 . 2 70
89.558
109.518
127.956
106.912
96.926
122.590
TAVGO
50.954
50.470
49.152
46.161
47.192
48.509
45*568
46-545
46.727
49.500
47-649
48.030
50.052
49.431
45.621
46.212
49.210
51.061
49.444
49-319
49-440
48.098
50-871
49.268
A S - 750
51.634
50.149
46.098
4 6 . 6f 7
47.270
51.063
50.773
49-327
46-917
45.750
51.174
46- I f t
47.991
48»046
52.341
48.312
49.575
49.025
50.014
53-667
P
2.646
5.962
9.081
4 . 2 76
3-663
I -982
3.321
5.291
6.015
13.001
16.184
9.457
6.162
3.662
4 - 764
7.959
14.024
24.398
25.624
16.389
10.536
4.156
7.29 6
12.563
18.613
29.386
32.018
23.089
13.196
7.54 1
6.429
I f .991
27-738
39.503
44.022
30.895
19-270
I I •0 5
11.653
16-895
32.653
45*272
4-037
2.816
3.087
PRESMM
19.4
19.4
48.0
48.0
46.0
48.0
99.9
99.9
99.9
99.9
145. 5
145.5
149.6
145.5
199. 7
199.7
199.7
199.7
248-8
248.8
248.8
248.8
303.0
303.0
304. I
305. 8
396.8
396.8
396.8
396.8
505. 7
505. 7
535. 7
505. 7
599.7
599.7
599-7
59». 7
641.9
64 1 . 9
641.9
641.9
29.9
29.9
15 .-9
98
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
10.5000
10.5000
10*5000
10*5000
10.5000
10.5000
10*5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5030
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5030
10.5020
10.5000
10.5030
10.5000
10.5000
10.5000
4.5000
4*5000
4.5000
4.5000
4.5000
5.0000
5.0000
5.0000
5.0000
5.0000
5»0000
5«0000
5.0000
5»0000
5«0000
5* 0000
5* 0000
5»0000
5« 0000
5* 0000
5.0003
5 * 0030
5•0000
5« 0300
5* 0000
4•5000
4.5000
4« 5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4.5000
4« 5000
7* 0000
7.0000
7.0000
7.0000
7.0000
7.0000
. 7333
. 7333
• 7333
.7333
• 7333
.0000
.0000
. 0000
.0000
.0000
.0000
. 0000
• 0000
.0000
.0000
• 0000
.0000
. 0000
.0000
• 0000
. 0000
.0000
. 0000
.0000
. 0000
. 0000
• 0000
• 0000
• 0000
. 0000
.0000
.0000
.0000
• 0000
• 0000
. 0000
.0000
• 0000
. 0000
.0000
• 0003
• 0000
. 0000
• 0000
. 0000
.0000
I 25.194
102.604
67.174
79.606
124.833
I 10.203
I 35.444
I 13.310
62.213
66.181
102.986
166.660
116.551
72.412
68.471
136.972
121.556
80.523
69.632
72.046
I 15.518
88.435
91.797
75.906
124.875
80.364
151.242
69.892
I I 1.854
59.364
95.048
162.340
51.181
120.312
74.095
136.511
58.400
126.508
100.216
86.440
60.456
140.767
64.406
101*575
I 71.621
124.271
52.136
54.310
52.568
52.409
52.477
49.613
47.099
48.258
45.560
45.265
46.682
48.546
50-424
48.144
47.697
48.717
50.673
46.311
47.163
49.451
49.750
46.972
46*242
46.818
48.218
38.650
38.374
39.010
39.706
44.837
37.776
38.052
39.754
38-834
40.167
38.512
35.793
38.634
39.6(2
44.196
41.574
38.282
37.975
38.972
40.197
40.351
3* 967
6» 044
3.827
3.051
11.826
2.289
11.561
5-390
4.354
3.940
10.715
38.782
13.018
6.288
10.461
34.688
30.231
11.374
7*317
13.219
34.750
16.146
25.077
14.666
51.262
12.526
43*023
9.427
22.905
3*673
17.957
49.777
4.800
27.044
9.678
35.025
8.640
30.8.05
16.408
I I .289
23.568
89.369
21.275
47.153
126.974
70.091
16.1
74.0
74.0
74.0
74.0
I 7. 4
36. I
39.2
39.8
93.4
95-6
100.9
9 7.9
186 . I
197.3
201.0
297.4
29 7.4
297-4
449.5
449.5
449.5
643.4
643.4
643. 4
646.4
646-4
646.4
646* 4
646-4
646.4
646.4
646*4
646.4
646.4
646*4
646.4
646.4
646.4
646 * 4
646 * 4
646.4
64 6*4
64 6.4
646-4
646*4
99
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
2
2
2
2
2
2
2
2
8
2
2
2
2
2
2
2
10.5000
10-5003
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5030
10.5000
10.5000
10.5000
10.5030
10-5000
10.5000
10.5000
10.5903
10.5000
10.5000
10.5000
I 0.5000
10.5003
10.5300
10.5,00
10.5300
I 0.503d
10,5006
10.5090
10.5000
10.5600
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5000
10.5030
10.5000
7.0000
7.0000
7.0000
7.0000
7.0000
7-0000
7.0000
7.0000
9.0000
9.0000
9.0000
4.5000
4*5000
4.5000
4.6000
4.5000
4.5000
4•5000
4-5000
4.5000
4.5000
4.5000
4 *5000
7.0030
7 .GOOi
7.
00
7. 0 3 ;0
7.0000
7-0003
7.0000
7.0000
7.0000
7.0000
7-0000
7-0000
7-0000
7-0300
7-0000
7.0000
7.0000
7.0000
7.0000
7.0000
. 0000
.0000
. 0000
.0000
. 0000
. 0000
. 0000
.0000
.0000
.0000
.0000
• 3063
• 3063
• 3063
• 3063
• 3063
• 3083
• 3063
.3063
• 3 3r 3
• 3083
. 3083
• 306 3
. 5429
• 5429
•5429
• 5429
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• 5429
• 54 29
. 5429
.5429
. 5429
•3143
.3143
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.3143
• 3143
• 3143
159.296
57.163
68.766
162-204
73.064
109.764
135.127
96.989
140.445
70.424
93.102
I 16.409
154.612
84.956
95.695
1&2.922
103.967
160.615
131-104
56.2(3
75.74V
(4.443
I 2 7 .c 3?
99.4(3
I 37-236
152.162
163.959
69.116
120.417
82.663
77.949
98.926
170.815
66 «
560
120-523
69.635
64.689
67.863
85.533
96.742
67.799
123.242
148.212
40.886
36.900
38.589
39.432
41.497
37.361
42*032
41.879
44.776
43.710
42.719
38.435
36.623
37.361
40.198
37.591
36.823
37.436
37.436
37-745
37.207
36.262
38.619
40-963
43.5=8
43.558
45.765
39.126
40.274
38.742.
42.337
43-329
39.662
41.268
43.863
38.665
36.054
43.405
45.765
47.663
46*676
43-329
45.613
109.925
20.126
38.166
118.182
22.657
59.744
79.877
40.702
165.541
33.545
73-504
39.496
76.730
22.248
25.075
59.922
3 3 .I 68
75*691
50.100
10.862
18.844
12.210
45.353
26-333
46.725
55.375
65.392
19.764
41.079
21.191
19.034
25.332
79.919
I 7.374
37.614
30.206
24.609
24.749
32.892
37.771
19.586
68.504
96.694
646.4
646.4
646-4
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64C . 4
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100
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2
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• 5000
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2
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4.5000
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4.5000
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4.5000
4.5000
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4.5000
4.0000
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163.352
I 76.632
184.377
104.960
116.500
79.910
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64.236
57.774
129.047
183.07P
102.177
163.236
104.410
I I 7.686
76.394
89-387
133.335
155.109
161.253
83.537
148.435
I 19.P59
55.948
133.624
I 72.706
65.830
106.937
183.893
158.118
73.830
124.265
I 77.293
96.191
100.197
60.008
143.778
I 64.696
71.737
69.370
117.041
130.720
84.628
111.640
47.739
48-799
47.51 I
46-980
45.917
48.724
48.799
39.355
35.053
36.131
39.891
40.351
43.710
38.435
35-285
36.439
38.895
41.498
38.972
41.115
45.613
38.627
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38.013
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38.512
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37.438
37.898
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I 16.331
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23.610
85.741
24.048
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62.138
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33.618
99.714
48.754
64.014
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65.556
107.632
I 7.274
52.109
35.904
6.763
43.569
68.628
10.641
29.329
78.614
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39.402
74.505
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66.405
98.182
19.376
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38.205
41.134
37.745
36.247
37.092
38.167
38.553
37.668
39.128
38.244
37.015
38.569
39.547
33.857
36.823
37.591
81.319
23.351
I 1.055
96.824
67.327
21.642
29.026
I 13.390
143.439
39.027
82.742
47.940
I 55.999
126.942
15.391
74.285
33.772
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MONTANA STATE UNIVERSITY LIBRARIES
762 1001 3126 5
N378
Brown, Peter K
Natu r a l c o n v e c t i o n heat
transfer within enclo­
sures at r e d u c e d prestsures
B81U5
cop. 2
ISSUED
DATE
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