ROTATING BLADE HEAT EXCHANGER
A THESIS
S U B M IT T E D TO THE FAC ULTY
OF
PURDUE
UNIVERSITY
BY
ABRAHAM
EBENEZER
IN P A R T I A L
MUTHUNAYAGAM
F U L F IL L M E N T OF THE
REQ U IREM EN TS FOR THE D E G R E E
OF
DOCTOR OF PHILOSOPHY
J UNE; i •<* '
G r a d School
o n No. 9
t /ised
PURDUE
UNIVERSITY
Graduate School
T h i s i s to c e r t i f y th at t h e t h e s i s p r e p a r e d
By
Abraham Ebenezer Muthunayagam
Entitled
C om plies
Rotating Blade Heat Exchanger______
with
the
U niversity
s t a n d a r d s o f the G r a d u a t e
regulations
School
with
and
respect
that
to
it
m e e t s the a c c e p te d
originality
and
q u a li ty
F o r t h e d e g r e e of:
Doctor of Philosophy_____________________________
S i g n e d by t h e final e x a m i n i n g c o m m i t t e e :
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A p p r o v e d by t h e h e a d o f s c h o o l o r d e p a r t m e n t :
to .
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To the l i b r a r i a n :
T h i s t h e s i s i s not to be r e g a r d e d a s c o n f i d e n t i a l
c'JC
L rP r o f e s s o r in c h a r g e o f t h e t h e s i s
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PFFASL NOTE::
F ig ure pages a r e not o r i g i n a l
copy.
They t e n d t o " c u r l " .
F i l me d
as r e c e i v e d .
U niversity Microfilms, Inc.
ACKNOWLEDGMENT
I w is h to e x p r e s s m y s i n c e r e th a n k s to Dr.
for his guidance,
W. L e i d e n f r o s t
i n t e r e s t a n d e n c o u r a g e m e n t a t a l l s t a g e s of
t h i s i n v e s t i g a t i o n a n d i n t h e p r e p a r a t i o n of t h i s t h e s i s .
I would like to e x p r e s s my a p p r e c i a t i o n and tha nk s
t o D r . R. J . S c h o e n h a l s , D r . V. A . J o h n s o n a n d D r . M . S .
W ebster
f o r t h e i r v a l u a b l e a d v i c e a n d h e l p d u r i n g t h e c o u r s e of m y s t u d y .
T h e a s s i s t a n c e p r o v i d e d by Dr.
Dr.
W.
R. J .
L e i d e n f r o s t in o b t a i n i n g the P u r d u e R e s e a r c h F o u n d a t i o n
g r a n t f o r t h i s s t u d y is g r e a t l y a p p r e c i a t e d .
Dr,
G r o s h and
Y.S.
Touloukian,
R esearch Center,
I a lso thank
D i r e c t o r of t h e T h e r m o p h y s i c a l P r o p e r t i e s
for the f in a n c ia l a s s i s t a n c e d u r i n g m y f i r s t
y e a r of s t u d y a t P u r d u e U n i v e r s i t y .
I a m g r a t e f u l to M e s s e r s .
C.
Hager,
a n d C.
of t h e a p p a r a t u s .
W .Cole,
S.Smith,
H. S u r f a c e ,
C o p e l a n d f o r t h e i r a s s i s t a n c e in t h e c o n s t r u c t i o n
I a lso thank Dr.
W. F .
W aterman,
and oth er
fellow g r a d u a t e s t u d e n t s for t h e i r s u g g e s t i o n s d u r i n g the c o u r s e
of m y s t u d y ,
and M r s .
D ia ne Co ad y for h e r c o o p e r a t i o n and
w o n d e r f u l j o b in t y p i n g t h i s t h e s i s .
Finally,
I w i s h to a c k n o w l e d g e the p a t i e n c e and u n d e r ­
s t a n d i n g ot m y w i f e a n d f a m i l y .
Ill
TABLE OF CONTENTS
LIST O F T A B LE S
Page
................................................................................................v
L I S T O F I L L U S T R A T I O N S ........................................................................
vi
ABSTRACT
viii
INTRODUCTION
...........................................................................................
1
................................................................
3
D e f i n i t i o n of the P r o b l e m
LITERATURE SURVEY
Introduction
Co-ordinate Systems
M o d e l s o f v e l o c i t y p r o f i l e s ..................................................................
A n a l y s i s of r o t a t i n g b l a d e s ..............................................................
A p p l i c a t i o n of p e r t u r b a t i o n p r o c e d u r e .................................
5
6
8
13
15
TH EO R ETIC A L INVESTIGATION
Introduction
.................................................................
S i m p l i f i c a t i o n of t h e f l o w w i t h i n b o u n d a r y l a y e r . . . .
.....................................................................
A p p r o x im a te Solutions
L a m i n a r flow m o m e n t u m i n t e g r a l s o l u t i o n s
..................
T u r b u l e n t flow m o m e n t u m i n t e g r a l s o l u t i o n
......................
17
19
23
26
33
E X P E RIM E N T A L I N V E S T I G A T I O N
Introduction
.......................................................................
D e s i g n of r o t a t i n g a p p a r a t u s
..........................................................
44
D e s c r i p t i o n of t h e a p p a r a t u s ......................................................
45
The M e a s u r e m e n t s
63
Experimental procedure
.
5 3
H e a t T r a n s f e r m e a s u r e m e n t ...............................................
53
Heat T r a n s f e r m e a s u r e m e n t s with t r i p w i r e s . . .
56
Hea t T r a n s f e r m e a s u r e m e n t s withp it c h e d b l a d e s .
58
Torsion m easurem ents
..........................................................
58
R e s u l t s a n d d a t a r e d u c t i o n s .......................................................
60
COMPARISION O F T H E O R E T I C A L AND E X P E R I M E N T A L
RESULTS
H e a t T r a n s f e r m e a s u r e m e n t s on a flat b l a d e
63
44
IV
Heat T r a n s f e r m e a s u r e m e n t s with tr i p w ires
. . . .
H e a t T r a n s f e r m e a s u r e m e n t s with p itched b l a d e s . .
............................................................
Torsion M easurem ents
Experim ental e r r o r analysis
. . .
E x p l a n a t i o n f o r d e v i a t i o n of e x p e r i m e n t a l r e s u l t s . ,
from theoretical analysis
C o m p a r i s i o n w i t h o t h e r h e a t e x c h a n g e r s .........................
IN FLUENCE OF EL EC TR IC A L FIELD
......................
I n f l u e n c e of e l e c t r i c a l f i e l d on f l u i d p r o p e r t i e s
.
and t r a n s p o r t p h a o m e n a
C h a r a c t e r i s t i c s of c o r o n a d i s c h a r g e i n a n a i r
.
stream
E l e c t r i c field c o r o n a d i s c h a r g e r e l a t i o n s
.
H e a t T r a n s f e r e x p e r i m e n t s with E l e c t r i c a l F i e l d
Forced C onvective Heat T r a n s f e r e x p e r i m e n t s
with e l e c t r i c a l field
Natural convective heat tra n s fe r e x p e rim e n ts
with e l e c t r i c a l field
CONCLUSION
66
72
77
83
gg
90
94
. . .
95
. . .
98
. . .
. . .
99
101
103
. .
103
..........................................................................................
106
RECOMMENDATION
...............................................................................
109
T.IST O F R E F E R E N C E S ............................................................................
Ill
APPENDICES
Appendix
Appendix
Appendix
Appendix
Appendix
Appendix
VITA
A:
B:
C:
D:
E:
F:
D e r i v a t i o n s f o r t h e o r e t i c a l i n v e s t i g a t i o n - 116
D e r i v a t i o n f o r e x p e r i m e n t a l i n v e s t i g a t i o n 133
E l e c t r i c f i e l d - c o r o n a d i s c h a r g e r e l a t i o n 137
T a b u l a t i o n of d a t a a n d r e s u l t s .......................... 144
I n s t r u m e n t a t i o n and m a t e r i a l s . . . . . . .
170
Nomenclature
173
.........................................................................................................................176
V
LIST OF TABLES
Table
1.
T o r s io n C a lib ratio n data
2.
Data and r e s u lt s for heat t r a n s f e r
experiments - 0 degree
.......................................
145
3.
Data and r e s u l t s for heat t r a n s f e r
experim ents - trip wires
....................................
146
4.
Data and r e s u l t s for heat t r a n s f e r
exp erim en ts - pitched blades
...................................
153
5.
C o m p a r i s i o n of m e a s u r e d t o t a l t o r q u e a n d
predicted skin-friction torque blade-Type 1
....................
162
6.
A v e r a g e d r a g c o e f f ic ie n t and A v e r a g e
Reynolds num ber
7.
C o m p a r i s i o n of m e a s u r e d t o t a l t o r q u e a n d
p r e d ic te d skin frictio n torque
8.
Reduced power and heat t r a n s f e r coefficient
9.
10.
..............................................
144
163
...................
164
................
167
F o r c e d c o n v e c t i v e h e a t t r a n s f e r e x p e r i m e n t s ................
with e l e c t r i c a l field
168
N atural convective heat t r a n s f e r e x p e r i m e n t
with e l e c t r i c a l field
169
...............
v i
LIST OF ILLU STRA TIO NS
Figure
1.
V e l o c i t y - C o m p o n e n t p r o f i l e a n d w a l l s h e a r ............... 7
s t r e s s c o m p o n e n t s in s t r e a m l i n e c o o r d i n a t e
2.
V e lc i ty C o m p o n e n t p o l a r plot t r i a n g u l a r m o d e l
3.
Co-ordinate System
4.
L am inar velocity profiles
4.
I n c r e a s e in C
o.
I n c r e a s e in C
a n d Nu d u e to t h r e e __
...
11
.....................................................................
18
............................................................
Z9
an d Nu due to t h r e e f, x
x
2
d i m e n s i o n a l e f f e c t iri l a m i n a r fl ow
......................
34
..............................
35
d i m e n s i o n a l e f f e c t in l a m i n a r f l ow
7.
Turbulent velocity profiles
39
8.
I n c r e a s e in C,
a n d Nu d u e t o t h r e e - ..........................
f,x
x
2
d i m e n s i o n a l e f f e c t i n t u r b u l e n t f l ow
41
4.
I n c r e a s e in C
42
a n d Nu d u e t o t h r e e -
...........................
~2~
d i m e n s i o n a l e f f e c t i n t u r b u l e n t f l ow
1o
Experimental apparatus
4b
1i
. S e c t i o n a l d r a w i n g of a p p a r a t u s
47
12.
Heater Element
49
13.
Slip-ring - brush assem bly
49
14.
Instrumentation
51
15.
Torsion m e ter calibration
52
16.
C i r c u i t for m e a s u r e m e n t andc o n t r o l
17.
Trip wire configurations
.............................
54
57
59
18.
B la d e s and C o u n t e r p a r t s
19.
C o m p a r i s i o n of e x p e r i m e n t a l r e s u l t s
..................
64
20.
C o r r e l a t i o n of e x p e r i m e n t a l r e s u l t s
..................
65
21.
Nu V e r s u s R e f o r B l a d e 1 T y p e 1
..................
67
22.
Nu Ve r s u s R e f o r B l a d e 2
Type 1
..................
68
23.
Nu V e r s u s R e f o r B l a d e 3
Type 1
.....................
69
24.
Nu V e r s u s Re f o r B l a d e 4
Type 1
.....................
70
25.
Nu V e r s u s R e f o r B l a d e 1 T y p e 1
......................
73
26.
Nu V e r s u s Re fo r B l a d e 2
Type 1
.....................
74
27.
Nu V e r s u s R e f o r B l a d e 3
Type 1
.....................
75
28.
Nu V e r s u s Re f o r B l a d e 4
Type 1
.....................
76
29.
C o m p a r i s i o n of t o t a l t o r q u e a n d s k i n f r i c t i o n . . . . 79
t o r q u e for B la d es Type 1
10.
A v e r a g e Cd V e r s u s A v e r a g e R e .......................................
81
11.
Drag Coefficient V e rsu s Reynolds n u m ber
81
32.
C o m p a r i s i o n of p r e d i c t e d s k i n f r i c t i o n t o r q u e . . .
and m e a s u r e d total t o r q u e for B la d e s Type 2
33.
34
dg
—
g
Re
Versus
AT
. . .
....................................................................
82
86
Versus N
87
V e r s u s M..................................................................................
87
38.
—Pi
M
36.
Me a t e x c h a n g e s u r f a c e s
......................................................
91
37.
C o m p a r i s i o n of h e a t e x c h a n g e s u r f a c e s .....................
92
38.
Modified e x p e r i m e n t a l a p p a r a tu s
. . . . ...................
102
39
Mlectrode ho ld er
........................................................................
104
40.
C r o a i s e c t i o n of b la d e and h e a t e r
41. . M o d e l f o r f l o w p e r p e n d i c u l a r
42.
Model for
43 .
J.V .: / q )k V e r s u s
44
idiV V
t eZ
............................
135
toe l e c t r i c a l field . .
138
f l o w p a r a l l e l t o e l e c t r i c a l f i e l d ...................
V- rsu8
Kl
IT
140
..........................................................
143
..........................................................
143
ABSTRACT
M uthunayagam, A braham Ebenezer.
U n i v e r s i t y , J u n e 19b5.
P h .D ., Purdue
Rotating Blade Heat E x chan g er.
M a j o r P r o f e s s o r : W. L e i d e n f r o s t .
I n c r e a s e in h e a t
t r a n s f e r c o e f f i c i e n t s o n t h e g a s s i d e of
h e a t e x c h a n g e r s c a n be a c h i e v e d by i n c r e a s i n g g a s v e l o c i t i e s
a n d c h a n g i n g t h e p r o p e r t i e s of t h e g a s .
I n c r e a s i n g f l ow
v e l o c i t i e s c r e a t e s the p r e s s u r e l o s s e s a c r o s s the heat t r a n s f e r
s u r f a c e s and o t h e r flow p a t h s .
This la tte r loss r e p r e s e n t s a
p o w e r w a s t e w h i c h c a n be e l i m i n a t e d by u s i n g r o t a t i n g h e a t
transfer surfaces.
T h e m e r i t s a n d d e m e r i t s of o n e s u c h h e a t
e x c h a n g e r w e r e i n v e s t i g a t e d in t h i s s t u d y .
The b o u n d a r y l a y e r and m o m e n t u m i n t e g r a l e q u atio n s for
t he flow a r o u n d a r o t a t i n g flat b l a d e w e r e d e v e l o p e d in a r o t a t i n g
stream line coordinate system .
These equations w ere simplified
a n d s o l v e d by a s s u m i n g p r o p e r v e l o c i t y p r o f i l e s t o g i v e t h e l o c a l
and a v e r a g e s kin f r i c t i o n c o e f f i c i e n t s fo r both l a m i n a r and t u r ­
b u l e n t flow
T h e h e a t t r a n s f e r c o e f f i c i e n t s w e r e e s t i m a t e d by
m e a n s of R e y n o l d s a n a l o g y .
F o r t h e b l a d e u s e d in t h e p r e s e n t i n v e s t i g a t i o n t h e i n c r e a s e
m Nusselt num ber,
d u e t o t h r e e - d i m e n s i o n a l f l ow e f f e c t s ,
as
c o m p a r e d to t w o d i m e n s i o n a l f l o w, w a s f o u n d t o be a b o u t 5%
f o r l a m i n a r f l o w a n d l e s s t h a n 1% f o r t u r b u l e n t f l ow.
H e a t e d s u r f a c e s i n t h e f o r m of f l a t b l a d e s w e r e r o t a t e d in
still air.
The h e a t t r a n s f e r c o e f f i c ie n t s w e r e e v a l u a te d (local
wi th r a d i u s but a v e r a g e w i th c h o r d ) c o v e r i n g a R e y n o l d s n u m b e r
4
5
r a n g e f r o m 10 t o 2. 3x10 . T h e r e s u l t s w e r e c o r r e l a t e d a n d
c o m p a r e d with t h e o r e t i c a l p r e d i c t i o n s .
T h e y w e r e f o u n d t o be
ix
a b o u t 25% t o 6 0% l e s s t h a n t h e p r e d i c t i o n s of t u r b u l e n t f l o w a n d ,
d e p e n d i n g on R e y n o l d s n u m b e r ,
f r o m 60% l e s s t o 40% g r e a t e r
t h a n the l a m i n a r flow p r e d i c t i o n s .
T r i p w i r e s of d i f f e r e n t c o n f i g u r a t i o n s w e r e t h e n f i x e d t o t h e
s u r f a c e of t h e b l a d e a n d h e a t t r a n s f e r c o e f f i c i e n t s w e r e d e t e r m i n e d .
R e s u l t s i n d i c a t e d t h a t t h e i n f l u e n c e o f r a d i a l f l o w on h e a t t r a n s f e r
r a t e was v e r y s m a l l ,
confirm ing theoretical predictions.
T h e b l a d e s w e r e n e x t p i t c h e d to d i f f e r e n t a n g l e s to s t u d y
the i n f l u e n c e of the a n g l e of i n c i d e n c e and the c o n s e q u e n t
s e p a r a t e d f l o w on h e a t t r a n s f e r .
M e a s u r e m e n t s showed th a t the
r a t e of i n c r e a s e of h e a t t r a n s f e r c o e f f i c i e n t in s e p a r a t e d f l o w w a s
l e s s t h a n t h a t o f t u r b u l e n t f l o w b u t h i g h e r t h a n t h a t of l a m i n a r f l ow.
A to rs io n m e t e r .designed
by t h e a u t h o r ,
w as used to d e t e r m i n e
t h e a c t u a l t o r q u e r e q u i r e d t o r o t a t e t h e b l a d e s . It w a s f o u n d t h a t
t he p r e s s u r e d r a g c o n t r i b u t e s a m a j o r p o r t i o n of the p r o f i l e d r a g
f o r the b l a d e s u s e d in t h e h e a t t r a n s f e r e x p e r i m e n t s .
Torsion
m e a s u r e m e n t s w e r e d o n e on t h r e e s e t s of t h i n n e r b l a d e s w h i c h
s h o w e d t h a t t h e p r e s s u r e d r a g c o u l d b e r e d u c e d c o n s i d e r a b l y by
r e d u c i n g t h e r a t i o of t h i c k n e s s t o c h o r d w i d t h .
I n t e r a c t i o n of e l e c t r i c a l f i e l d s o n f l u i d p r o p e r t i e s a n d c o r o n a
discharge relations w ere reviewed.
In a i r ,
the c o r o n a d i s c h a r g e
w a s found to h a v e s i g n i f i c a n t i n f l u e n c e on n a t u r a l c o n v e c t i v e h e a t
t r a n s f e r and n e g l i g i b l e i n f l u e n c e on f o r c e d c o n v e c t i v e h e a t t r a n s f e r .
Brief e x p e r i m e n t s c o n f ir m e d this conclusion
It w a s c o n c l u d e d t h a t r o t a t i n g b l a d e h e a t e x c h a n g e r s h a v e
a d v a n t a g e of s i m p l i c i t y i n c o n s t r u c t i o n ,
l o w e r initial c o s t ,
t he
high
h e a t t r a n s f e r c o e f f i c i e n t s and a d a p t a b i l i t y to an y s ur r o u n d i n g T h e
p o w e r r e q u i r e m e n t s w e r e found to b e g r e a t e r t h a n the p u b l i s h e d
v a l u e s f o r f r i c t i o n a l p o w e r r e q u i r e m e n t s of c o m p a c t h e a t e x c h a n ­
g e r s ; but t h e p o w e r l o s s e s in the h e a d e r s ,
i n l e t d u c t a n d o t h e r f l ow
p a s s a g e s of s u c h h e a t e x c h a n g e r s w e r e not c o n s i d e r e d .
1
INTRODUCTION
In t he d e v e l o p m e n t of h e a t t r a n s f e r t e c h n o l o g y t h e r e i s a n
i n c r e a s i n g d e m a n d for c o m p a c t h e a t exchangers with m i n i m u m
heat tra n s fe r surface a re a ,
m inim um
v o lu m e , and m i n i m u m
w e i ght for t r a n s f e r r i n g a giv en a m o u n t of h e a t .
by i n c r e a s i n g h e a t t r a n s f e r c o e f f i c i e n t s .
This is achieved
In r e g e n e r a t o r s and
i n t e r c o o l e r s t h e t h e r m a l r e s i s t a n c e of t h e g a s s i d e i s h i g h a n d
p r a c t ic a l ly con tro ls the heat t r a n s f e r r a te .
desirable
Therefore,
it is
to i n v e s t i g a t e t h e p o s s i b i l i t i e s of i n c r e a s i n g t h e h e a t
t r a n s f e r coefficient to a gas.
T h e h e a t t r a n s f e r c o e f f i c i e n t i s a f u n c t i o n of s e v e r a l
v a r i a b l e s s u c h a s f l u i d p r o p e r t i e s , v e l o c i t y of t h e f l u i d , a n d
s u r f a c e c h a r a c t e r i s t i c s of the he a t t r a n s f e r a r e a .
T h e d i m e n s i o n a l a n a l y s i s for f o r c e d c o n v e c t i v e h e a t t r a n s ­
fer gives:
N u = Cj R e " P r m
w h e r e t h e c o n s t a n t s Cj , n, a n d m d e p e n d on t h e n a t u r e of t h e
f l o w a n d t h e g e o m e t r y of t h e s y s t e m .
F o r a give n fluid with c o n s t a n t p r o p e r t i e s ,
the h e a t t r a n s ­
f e r c o e f f i c i e n t c a n be e x p r e s s e d a s
K
..
where C
-
*
r, UL "
L • “ ■' „ I
and n a r e co n stan ts.
H i g h e r h e a t t r a n s f e r c o e f f i c i e n t s c a n b e a c h i e v e d by
i n c r e a s i n g t h e v e l o c i t y of f l o w a n d t h e t h e r m a l c o n d u c t i v i t y of
the f lu i d, a n d by d e c r e a s i n g t h e k i n e m a t i c v i s c o s i t y .
Increasing
t he v e l o c i t y of flow i n c r e a s e s t h e p r e s s u r e l o s s n ot o n l y a c r o s s
the h e a t t r a n s f e r a r e a but a l s o a c r o s s th e d u c t s isading to the
heat tr a n s f e r surface.
This loss a c r o s s the ducts does not
2
i n c r e a s e the heat t r a n s f e r rate .
T h e p o w e r l o s s in the duct ca n
be e l i m i n a t e d by u s i n g a r o t a t i n g h e a t t r a n s f e r s u r f a c e in s t a t i o n a r y
gases.
A lso high v elo cities can be obtained by rotating the s u rfa c e .
C h a n g e s i n t h e p r o p e r t i e s of t h e f l u i d a r e h a r d t o p r o d u c e .
However,
it h a s b e e n o b s e r v e d t h a t t h e i n t e r a c t i o n o f a n e l e c t r i c
field with s o m e liquids and s o m e g a s e s c a n c h a n g e the p r o p e r t i e s .
Th e t e c h n i c a l f e a s i b i l i t y of one s u c h m o v i n g h e a t t r a n s f e r
♦
s u r f a c e w a s i n v e s t i g a t e d b y L e i d e n f r o s t (21) a n d b y W a t e r m a n ( 40) .
T h ey e s t a b l i s h e d the f e a s i b i l i t y but did not fully i n v e s t i g a t e the
fluid flow a n d h e a t t r a n s f e r p h e n o m e n a .
The an aly sis was based
o n t h e e x a c t s o l u t i o n s of F o g a r t y (9) f o r a s i m p l i f i e d b o u n d a r y
l a y e r e q u a t i o n for a flat b la d e r o t a t i n g a t c o n s t a n t s p e e d .
The
pow er r e q u i r e m e n t s w e re p r e d ic te d fro m the heat t r a n s f e r m e a s u r e ­
m e n t s a s s u m i n g R eynolds a n a l o g y which n e g l e c te d the p r e s s u r e
d r a g (wh ich i s t h e m a j o r p a r t of t h e p r o f i l e d r a g in b lu nt b o d i e s ) .
The a v e r a g e h eat t r a n s f e r m e a s u r e m e n t s on a rotating heated
blade did not a g r e e with the t h e o r e t i c a l p r e d i c t i o n s due to the
i n f l u e n c e of t u r b u l e n c e .
Therefore,
further investigations a re
n e e d e d i n t h e a r e a s of f l u i d f l o w a n d h e a t t r a n s f e r f r o m a r o t a t i n g
flat blade.
T h i s is f u n d a m e n t a l l y a p r o b l e m in t h r e e - d i m e n s i o n a l
boundary la y e r r e s e a r c h and has,
in the d e s i g n a n d u s e of f a n s ,
in a d d i t i o n ,
propellers,
practical application
and turbine blades.
M o s t of t h e w o r k i n t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r s h a s
b e e n c o n f i n e d to the flow in n o n - r o t a t i n g s y s t e m s .
Heat t r a n s f e r
m e a s u r e m e n t s on b l a d e s a r e l a c k i n g e x c e p t f o r t h e r e c e n t w o r k
of W a t e r m a n ( 40).
T h e r e s e a r c h o b j e c t i v e s o f t h i s w o r k w e r e th e follo w ing
1)
To i n v e s t i g a t e t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y the fluid
f l o w a n d h e a t t r a n s f e r i n a t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r on a
rotating blade.
*
N u m b e r s in p a r e n t h e s e s r e f e r t o a u t h o r s l i s t e d in R e f e r e n c e s .
3
2)
T o d e t e r m i n e e x p e r i m e n t a l l y t h e i n f l u e n c e of
s e p a r a t e d flow on t h e h e a t t r a n s f e r c o e f f i c i e n t by p i t c h i n g the
b l a d e t o v a r i o u s a n g l e s of i n c i d e n c e .
3)
To d e t e r m i n e e x p e r i m e n t a l l y the a c t u a l p o w e r r e q u i r e d
for r o t a t i n g the blade.
4)
T o i n v e s t i g a t e t h e p o s s i b l e i n t e r a c t i o n of a n e l e c t r i c a l
f ield on fluid flow a n d h e a t t r a n s f e r .
DEFINITION OF THE PR O B L E M
E x a c t s o l u t i o n s o f t h e c o n s e r v a t i o n e q u a t i o n s of m o m e n t u m
an d e n e r g y a r e dif fi cul t f o r m o s t of the two d i m e n s i o n a l s t u d i e s
b e c a u s e they a r e n o n - l i n e a r p a r tia l differential equations.
They
b e c o m e m o r e c o m p l e x f or t u r b u l e n t flow w h e r e t he s h e a r s t r e s s
i s a f u n c t i o n of t h e s t r u c t u r e of t u r b u l e n c e a s w e l l a s m e a n f l o w
v e l o c i t i e s a n d fluid p r o p e r t i e s .
T h r e e - d i m e n s i o n a l stu d ies in tro d u ce additional difficulties.
T h e t h e o r e t i c a l a n a l y s i s c a r r i e d out in t h i s r e s e a r c h w a s b a s e d
on th e m o m e n t u m i n t e g r a l t e c h n i q u e w h ic h w as u s e d to so lve
the b o u n d a r y l a y e r m o m e n t u m e q u a t io n s for a flat pla te ro ta tin g
at c o n s t a n t s p e e d .
Skin f r i c t i o n v a l u e s w e r e d e t e r m i n e d using
suitable v elo city p ro files and wall s h e a r s t r e s s e s .
Heat tr a n s f e r
coefficients w e r e d e t e r m in e d using Reynold's analogy and then
c o m p a r e d with e x p e r i m e n t a l r e s u l t s
T h e e x p e r i m e n t a l w o r k c o n s i s t e d of t h e e v a l u a t i o n o f t h e
heat tra n s fe r coefficients,
l o c a l w i t h r e s p e c t to r a d i u s a n d
a v e r a g e with r e s p e c t to c h o r d
the a u t h o r w a s u s e d to
blade.
The values
A t o r s i o n m e t e r d e s i g n e d by
e v a l u a t e t h e a c t u a l t o r q u e to r o t a t e the
a r e c o m p a r e d with the t h e o r e t i c a l p re d ic tio n
of t he t o r q u e du e to s k i n f r i c t i o n d r a g .
A verage drag coefficients
w ere also evaluated.
T h e flow a r o u n d the r o t a t i n g b l a d e w a s a n a l y z e d
4
)
q u a l i t a t i v e l y by h e a t t r a n s f e r e x p e r i m e n t s .
T r i p w i r e s of
d i f f e r e n t c o n f i g u r a t i o n s w e r e a t t a c h e d t o t h e s u r f a c e of t he
blade and heat t r a n s f e r m e a s u r e m e n t s w e re m ade.
The influence
of e a c h c o n f i g u r a t i o n o n t h e m e a s u r e d r e s u l t s w a s a n a l y e e d t o g a i n
f u r th e r in fo r m a tio n about the p h en o m en a.
To gain high tu r b u l e n c e wake and h ave b a la n c e four blades
w e r e a c t u a l l y m o u n t e d on t h e r o t a t i n g s h a f t .
t h e w a k e of t h e p r e c e d i n g b l a d e .
E a c h b l a d e r a n in
When they w e re given so m e
a n g l e of a t t a c k t h e f l o w o n t h e s i d e f a c i n g d o w n s t r e a m s e p a r a t e d .
Therefore,
the b l a d e s w e r e p it c h e d to v a r i o u s a n g l e s to stu dy the
i n f l u e n c e o f s e p a r a t e d f l o w a n d w a k e on h e a t t r a n s f e r c o e f f i c i e n t s .
As stated e a r l ie r ,
c h a n g e s in fluid p r o p e r t i e s c o u l d i n f l u e n c e
heat t r a n s f e r coefficients.
It h a s b e e n f o u n d t h a t a n e l e c t r i c a l
f i e l d c a u s e d c h a n g e s in p r o p e r t i e s o f s o m e f l u i d s .
fore,
There­
the l i t e r a t u r e r e l a t i n g to th i s i n t e r a c t i o n of an e l e c t r i c a l
f ield on flui d flow a n d h e a t t r a n s f e r w a s r e v i e w e d .
P r e v i o u s w o r k had e s t a b l i s h e d only th a t c o r o n a d isc h a rg e
c a n i n f l u e n c e t h e h e a t t r a n s f e r i n t h e c a s e of a i r a t a t m o s p h e r i c
conditions.
No l i t e r a t u r e ,
re la te d to fo rced co nvective heat t r a n s f e r
i n a i r i n t h e p r e s e n c e of c o r o n a d i s c h a r g e , w a s f o u n d .
F o r this
reason,
c o r o n a d i s c h a r g e r e l a t i o n s w e r e d e r i v e d f o r s i m p lifie d
cases
Brief ex perim en ts w ere conducted
with a 30000 volts
D. C. p o w e r s u p p l y t o v e r i f y t h e t h e o r e t i c a l p r e d i c t i o n s .
5
L IT E R A T U R E SURVEY
Introduction
It i s d e s i r e d t o a n a l y s e t h e f l o w s u r r o u n d i n g a r o t a t i n g
flat b l a d e a n d d e t e r m i n e the e x p r e s s i o n d e s c r i b i n g th e c o e f f i ­
c i e n t s of s k i n - f r i c t i o n a n d h e a t t r a n s f e r f o r b o t h t h r e e - d i m e n ­
s i o n a l l a m i n a r a n d t u r b u l e n t f l o w.
T h e r e f o r e the e x i s t i n g w o rk
on t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r s i s r e v i e w e d h e r e i n .
In t h e c a s e of t h e t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r ,
t h e v e l o c i t y c o m p o n e n t a l o n g t he e x t e r n a l flow is g e n e r a l l y
k n o w n a s m a i n flow a n d the v e l o c i t y c o m p o n e n t n o r m a l to
t h i s d i r e c t i o n i s c a l l e d c r o s s f l o w.
A s a r e s u l t of t h e c u r v e d
e x t e r n a l flow a t t h e e d g e of t h e b o u n d a r y l a y e r a n i n w a r d p r e s s u r e
g ra d ie n t ex ists to balance the cen trifu g al force.
N e a r the s u r f a c e
of t he b o d y t h e fl ui d is r e t a r d e d d u e to v i s c o u s e f f e c t s a n d h e n c e
the u n a lte r e d p r e s s u r e g ra d ie n t e x ce ed s the c en trifu g al force.
Therefore,
t h e f lui d m o v e s i n w a r d s a n d c a u s e s c r o s s flow.
The
v e l o c i t y v e c t o r c h a n g e s both in m a g n i t u d e and d i r e c t i o n f r o m
t h e o u t s i d e of t h e b o u n d a r y l a y e r t o t h e l a y e r c l o s e r t o t h e
surface.
T h i s c h a n g e of v e l o c i t y n o r m a l l y t a k e s p l a c e in one
d i r e c t i o n a n d m a y b e i n f l u e n c e d by t u r b u l e n t m i x i n g .
The t u r ­
b u l e n t m i x i n g a t t e m p t s to m a i n t a i n the m a i n s t r e a m v e l o c i t y
both in m a g n i t u d e an d d i r e c t i o n d e e p e r into the b o u n d a r y l a y e r .
T h e s t u d i e s in t w o d i m e n s i o n a l flow i n d i c a t e t h a t t h i s d o e s h a p p e n
to t h e m a g n i t u d e of t he v e l o c i t y .
In t h r e e d i m e n s i o n a l fl ow s, t h e
t u r b u l e n c e m i g h t i n f l u e n c e t h e d i r e c t i o n of t h e v e l o c i t y v e c t o r .
H e n c e , a s w a s f o u n d by B r e b n e r (4), t h e c r o s s f l o w i n t h e t u r ­
bulent bou n d ary l a y e r is m u c h le s s than in the l a m i n a r boundary
6
layer.
T h e s t u d i e s of t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r s h a v e
b e e n r e v i e w e d b y S e a r s (30) i n 1954 a n d b y M o o r e (26) i n 1 9 5 6 .
T h re e -d im e n s io n a l turbulent boundary la y e r t have been review ed
b y C o o k e (6) i n 1963.
A l l t h e w o r k i n t h i s a r e a c a n be d i v i d e d i n t o
three sep arate groups.
T h e f i r s t g r o u p c o n s i s t s of t he f o r m s of t h e b o u n d a r y
l a y e r e q u a t i o n s a n d t h e d i s c u s s i o n of t h e b o u n d a r y l a y e r b e h a v i o r
as inferred from these form s,
equations.
w ith ou t a c t u a l s o l u t i o n s of t h e s e
S i g n i f i c a n t w o r k i n t h i s g r o u p h a s b e e n d o n e by
H o w a r t h (15),
H a y e s (13) a n d M o o r e ( 24) .
T h e s e c o n d g r o u p c o n s i s t s of t h e g e n e r a l ,
but a p p r o x i m a t e ,
t r e a t m e n t of t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r flow by t h e
m om entum integral method.
T h e v a l i d i t y of t h i s m e t h o d d e p e n d s
t o a l a r g e r e x t e n t o n t h e c h o i c e of t h e v e l o c i t y d i s t r i b u t i o n i n
the boundary la y e r .
T i m m a n (36) a n d M a g e r (22) i n v e s t i g a t e d
the l a m i n a r and tu rb u len t c a s e s re s p e c tiv e ly .
T h e t h i r d g r o u p c o n s i s t s o f t h e s o l u t i o n s of s o m e s p e c i a l i z e d
c a s e s of t h r e e - d i m e n s i o n a l l a m i n a r b o u n d a r y l a y e r f l o w s .
These
fall i nt o t w o c a t e g o r i e s :
1) S y m m e t r i c a l f l o w s i n w h i c h t h e c h a n g e s w i t h r e s p e c t t o
o n e of t h e i n d e p e n d e n t c o o r d i n a t e s a r e z e r o ,
H o w a r t h ( 15) ,
2)
a s d o n e by
etc.
F l o w s for which s o m e c h a r a c t e r i s t i c p a r a m e t e r s e x i s t
v . h i c h c a n be n e g l e c t e d ,
a s d o n e b y F o g a r t y (b).
COORDINATE SYSTEMS
F o r the a n a l y s i s of t h r e e - d i m e n s i o n a l flow two d i f f e r e n t
t y p e s of c o o r d i n a t e s h a v e b e e n u s e d :
1)
S t r e a m line c o o r d in a te s y s t e m ,
a s s h o w n i n f i g u r e 1,
i n w h i c h o n e o f t h e a x e s (x a x i s ) c o i n c i d e s w i t h t h e m a i n s t r e a m ,
1
fvfcu
a
* ST**
r,SS C O
re
ppvO^^1
vA I I**F
0 ^ P ° N f 'N .
l
lT4*T * ‘
lA?OW
qo
1‘
ES
B.0) IInAO
8
t h e s e c o n d a x i s (z a x i s ) i s n o r m a l t o t h e s t r e a m l i n e , a n d t h e
t h i r d a x i s (y a x i s ) i s n o r m a l t o t h e s o l i d s u r f a c e ,
v e l o c i t y c o m p o n e n t s a l o n g x, y,
2)
u, v , w a r e t h e
z axes respectively.
C a r t e s i a n c o o r d in a te s y s t e m with orig in at the leading
e d g e of t h e b o dy ,
one ax is along the leading edge, the o th e r p e r ­
p e n d i c u l a r to it but in t h e s a m e p l a n e a s t h e b o d y , a n d t h e t h i r d
a x i s n o r m a l to the s u r f a c e .
T h i s s y s t e m w a s u s e d b y S e a r s (31) o n a y a w e d i n f i n i t e p l a t e ,
by F o g a r t y (9) o n a r o t a t i n g f l a t p l a t e , a n d b y G r a h a m (10) o n
r o t a t i n g b l a d e s of a r b i t r a r y c r o s s - s e c t i o n f o r l a m i n a r flow
investigations.
T h e i r a n a l y s i s m a d e u s e of t he " i n d e p e n d e n c e
p r i n c i p l e " by w h i c h t h e
f l o w in t h e c h o r d w i s e d i r e c t i o n c a n b e
c a l c u l a t e d w ith ou t a n y r e f e r e n c e to the s p a n w i s e flow.
The
v a l i d i t y of t h i s p r i n c i p l e a n d t h e l i n e a r i t y o f t h e m o m e n t u m e q u a t i o n
in t he d i r e c t i o n of t h e c r o s s flow g r e a t l y s i m p l i f i e d t h e p r o b l e m .
MODELS OF VELOCITY PR O FIL ES
In t h e s t u d i e s of t h r e e - d i m e n s i o n a l t u r b u l e n t b o u n d a r y
layers,
t h r e e g e n e r a l m o d e l s of t h e v e l o c i t y p r o f i l e s h a v e b e e n
proposed.
A l l of t h e m u s e t h e s t r e a m l i n e c o o r d i n a t e s y s t e m .
F o r t h e f i r s t g e n e r a l m o d e l P r a n d t l (28) s u g g e s t e d t h e
assumption
where G
a n d g a r e u n i v e r s a l f u n c t i o n s of
6
and in g e n e r a l
r e s t r i c t e d by t h e b o u n d a r y c o n d i t i o n s :
at y =
6
G
=
1
at y = o
G
= o
g =
g = l
o
9
If t h i s m o d e l i s v a l i d ,
one would ex p ec t that the function
w
y
fT = J P i8 a u n i v e r s a l f u n c t i o n o f ( — ) a s s h o w n b y M a g e r (22)
«U
6
u s i n g t h e d a t a o f G r u s c h w i t z ( 11) . M a g e r (23) d e r i v e d t h e
b o u n d a ry la y e r m o m e n t u m in t e g r a l equation for s tead y , th r e e dim ensional,
incom pressible,
t u r b u l e n t flow in v o lv in g c e n t r i f u g a l
and c o r io lis fo r c e s and obtained two f ir s t o r d e r p a r ti a l d iffere n tial
equation
withs e v e n u nknow ns.
He u s e d a n o r t h o g o n a l c u r v i ­
linear coordinatesystem , rotating
a b o u t a n a r b i t r a r y a x i s in
s p a c e with a c o n s t a n t a n g u l a r v elo city .
The unknowns consisted
of f i v e i n d e p e n d e n t i n t e g r a l t h i c k n e s s p a r a m e t e r s a n d t h e r e l a t i o n ­
s h i p b e t w e e n w a l l s h e a r s t r e s s in s t r e a m w i s e a n d c h o r d w i s e
directions.
M a g e r (22) s u g g e s t e d s p e c i f i c f u n c t i o n s f o r t h e
velocity profiles.
j
g
«=
,
<f,~
( i - f
>*
He a l s o a s s u m e d th e B l a u s i u s f r i c t i o n la w fo r the wall s h e a r
s t r e s s in the s t r e a m w i s e d i r e c t i o n .
i n t r o d u c e d in th e v e l o c ity p r o f i l e s ,
c h o r d w i s e wall s h e a r s t r e s s e s .
The p a r a m e te r t
,
r e la te d the s t r e a m w i s e and
It a l s o r e d u c e d t h e o r i g i n a l
e q u a t i o n s to two f i r s t o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s w hi ch
m u s t be s o l v e d s i m u l t a n e o u s l y f or s t r e a m w i s e m o m e n t u m t h i c k ­
ness
and the p a r a m e t e r t
.
A g ain M a g e r s im p l if ie d the
e q u a t i o n s by a n o r d e r of m a g n i t u d e a n a l y s i s a n d s o l v e d t h e
resulting simplified equations.
with t h e d a t a of G r u s c h w i t z
His p r e d i c t i o n s a g r e e d w e l l
(11) f o r n = 7.
He p o i n t e d o u t t h a t t h e
f u n c t i o n m a y n o t a l w a y s g i v e g o o d r e p r e s e n t a t i o n of t h e t h r e e d i m e n s i o n a l flow.
M o o r e a n d R i c h a r d s o n (27) a f t e r s t u d y i n g t h e i r d a t a
r e t a i n e d t h e g e n e r a l p r o f i l e a s s u m p t i o n o f P r a n d t l (28) b u t
found t h a t G
a n d g c o u l d be e x p r e s s e d b e t t e r in th e f o r m
10
G
.
O i l
x
g =
Still,
S>
x
(1 " f o e ^
i t r e p r e s e n t s a u n i v e r s a l f u n c t i o n of a s i n g l e v a r i a b l e
y
-jw h e r e H = 6^
r e p l a c e s th e p a r a m e t e r n u s e d by M a g e r
Ox
a s a function p a r a m e t e r .
T h i s f o r m u l a t i o n is m o r e g e n e r a l
t h a n M a g e r ' s (22) i n t h a t a v a i i a b l e
e x p o n e n t 2 i n t h e f u n c t i o n f o r g.
t
a n d T ox.
The
r r e p l a c e s the constant
u n k n o w n s a r e H,
r , 0* ,
T h r e e a u x i i a r y e q u a t i o n s a r e n e e d e d in a d d i t i o n
to the m o m e n t u m i n t e g r a l e q u a t i o n s .
Two d im e n s io n a l m od els
m a y p r o v i d e o n e e q u a t i o n r e l a t i n g H a n d To x .
H ow ever, this
m o d e l did not give an y cl u e a s to the o t h e r e q u a t i o n s b e c a u s e
of t h e d i f f i c u l t i e s i n r e l a t i n g t h e e x p o n e n t r t o t h e d y n a m i c
p r o p e r t i e s of t h e p r o b l e m .
In a s e c o n d m o d e l J o h n s t o n (17) a c c o u n t e d f o r a n a p p a r a n t
c o n s i s t e n c y i n t h e b e h a v i o u r of c r o s s f l o w c o m p o n s n t w a s a
f u n c t i o n o f t h e m a i n f l ow c o m p o n a n t
w
U "
w U (t ’
’
u a n d e x p r e s s e d it
as,
u
u}
w h e r e c a n d A a r e p a r a m e t e r s s u c h t h a t « is r e l a t e d to the wall
s h e a r s t r e s s e s a n d A t o t h e m a i n flow t u r n i n g a n g l e .
^ is an
w
ei q a l i c i t f u n c t i o n o f y a n d — i s a n i a » p l i c i t f u n c t i o n of y.
F r o m t h e a n a l y s i s of t h e e x p e r i m e n t a l d a t a J o h n s t o n (17)
s u g g e s t e d t h e s p e c i a l f u n c t i o n a l f o r m a s s h o w n in f i g u r e (2).
which is a p p l i c a b l e to the flows h a v i n g th e foll owi ng r e s t r i c t i o n s :
1)
Flo w is s e c o n d a r y ,
fully t u r b u l e n t ,
incom pressible
b o u n d ary la y e r type.
2)
T h e m a i n flow is s t e a d y ,
3)
V e l o c i t y p r o f i l e u p s t r e a m of t h e r e g i o n o f t u r n i n g i s
colatte ral.
ir r o t a tio n a l and two d im en sio n al.
n
Region 1
Region 2
L o c u s o f t i p of b o u n d a r
Layer v e l o c i t y v e c t o r
1w
U
ii
Tan
o
A
u
= 1
U
u
U
F igu r e 2
VELOCITY-COM PONENT POLAR PLOT,
TRIANGULAR
MODEL
T h e f i g u r e (2) r e p r e s e n t s t h e p o l a r p l o t ( p r o j e c t e d on
t h e p l a n e of t h e w a i l ) of t h e l oc u s of t h e t i p o f t h e b o u n d a r y - l a y e r
u
ve lo c ity v e c to r C,
— - 1 r e p r e s e n t s the f r e e s t r e a m a n d
u
— = o t h e c o n d i t i o n s a t t h e w a l l . I n t h e r e g i o n (1) n e a r t h e w a l l
w
U
u
U
a n d m the r e g i o n 2 n e a r the f r e e s t r e a m
- U "
A ll-- 1
'
u'
W
— fits the m a i n flow an d the wall b o u n d a r y c o n d i t i o n s of z e r o
velocity.
J o h n s t o n ( 1 7) d e r i v e d t h e m o m e n t u m i n t e g r a l e q u a t i o n s
in a s t r e a m l i n e c o o r d i n a t e s y s t e m .
T h i s r e s u l t e d in t w o p a r t i a l
12
d i f f e r e n t i a l e q u a t i o n s of t h e f i r s t o r d e r w it h 7 u n k n o w n s ,
f i v e of t h e m b e i n g i n t e g r a l t h i c k n e s s p a r a m e t e r s a n d t h e o t h e r
two a r e < and A w hi ch a r e i n t r o d u c e d in his p r o f i l e .
B u t it w a s
p o i n t e d o u t b y M a g e r (17) i n t h e r e v i e w o f t h e w o r k t h a t t h e s e
m o m e n t u m e q u a t i o n s w e r e in e r r o r due to the o v e r s i g h t in the
m atrix components.
B a s e d on his t r i a n g u l a r m o d e l J o h n s t o n d e v e l o p e d r e l a t i o n s
for t h r e e of t h e s e p a r a m e t e r s .
supplied an o th e r relation.
The wall s h e a r s t r e s s r e l a t i o n
H ow ever, the in f o rm a tio n n e c e s s a r y
f o r t h e f i f t h a n d t h e f i n a l a u x i i a r y e q u a t i o n w a s no t d e v e l o p e d .
Hence,
J o h n s t o n ' s m o d e l di d n o t g i v e a n y f i n a l r e s u l t b e c a u s e
of t h e m i s s i n g i n f o r m a t i o n a b o u t t h e i n f l u e n c e of t h e p a r a m e t e r s
on t h e v e l o c i t y p r o f i l e s .
T h e t h i r d m o d e l i s p r o p o s e d by C o l e s (5).
He s u g g e s t e d
that the v e l o c ity at an y point h a s two c o m p o n e n t s ; n a m e l y ,
wall
c o m p o n e n t an d wa ke c o m p o n e n t w h ic h a r e r e l a t e d to f r i c t i o n
v e l o c i t y a n d g i v e n b y l a w o f t h e w a l l a n d l a w of t h e w a k e .
Accord­
i n g t o h i m b o t h c o m p o n e n t s a r e v e c t o r s c o n s t a n t in d i r e c t i o n b u t
v a r y i n g in m a g n i t u d e f o r v a r y i n g d i s t a n c e f r o m t he s u r f a c e .
Coles
f o u n d i n a n e x a m p l e t h a t t h e d i r e c t i o n of t h e w a k e c o m p o n e n t w a s
n e a r l y t h e s a m e a s t h e d i r e c t i o n of t h e e x t e r n a l p r e s s u r e g r a d i e n t
at that point c o n c e r n e d , and m a k e s the te n ta tiv e s u g g e s t i o n that
this should hold u n i v e r s a l l y .
T h i s p r o c e d u r e was c a r r i e d out with
s u c c e s s f o r t h e v e l o c i t y p r o f i l e s m e a s u r e d by K e u t h e (20).
But
t h e s e p r o f i l e s d i d n o t a g r e e w i t h t h e m e a s u r e m e n t s of G r u s c h w i t z
(11).
Coles expected that G r u s c h w i tz 's data had s o m e e r r o r s
a n d t h e r e f o r e c o ul d not be u s e d f o r c o r r e l a t i o n .
J o h n s t o n (17)
checked his m e a s u r e d r e s u lt s with Coles m odel u n su ccessfully .
He f e l t t h a t d u e t o t h e s m a l l s k e w i n g o f h i s p r o f i l e s t h e d a t a m i g h t
not give s i g n i f i c a n t r e s u l t s p a r t i c u l a r l y with r e s p e c t to the wake
component.
B l a c k m a n (3) r e s o l v e d t h e w a k e c o m p o n e n t o f t h e
13
v e l o c i t y int o two c o m p o n e n t s , one a l o n g th e d i r e c t i o n of th e
wall s h e a r s t r e s s an d the o t h e r n o r m a l to it, an d found good
a g r e e m e n t with m e a s u r e m e n t s .
C o l e s m o d e l (5) a n d J o h n s t o n ' s
(17) m o d e l f i t t e d t h e m e a s u r e m e n t s of K u e t h e ( 20) .
Johnston's
m o d e l a g r e e d w i t h G r u s c h w i t z (11) a n d h i s o w n e x p e r i m e n t s ,
w h e r e a s C o l e s m o d e l f a i l e d t o a g r e e w i t h b o t h of t h e m .
Finally,
t h e m e a s u r e m e n t s o f B l a c k m a n (3) h e l p e d t o c o n f i r m C o l e s
m odel.
ANALYSIS O F ROTATING BLADES
Investigation
on r o t a t i n g b l a d e
of t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r s
was
d o n e by F o g a r t y (9) a n d G r a h a m .
a r e of s p e c i a l i n t e r e s t to t h i s r e s e a r c h a n d t h e r e f o r e ,
They
they a r e
r evie wed.
Fogarty
to a c a r t e s i a n
(9) d e r i v e d t h e m o m e n t u m e q u a t i o n
with r e f e r e n c e
c o o r d i n a t e s y s t e m (x, y, z) w h i c h r o t a t e d
c o n s t a n t a n g u l a r v e l o c ity Q a b o u t the z axis.
with a
Th e b la d e w a s in
the x y plane with the le ad in g edge along the y a x i s .
U nder the
x 2
a s s u m p t i o n ( — ) <<1, h e r e d u c e d t h e N a v i e r S t o k e s e q u a t i o n s
for a flat b l a d e a s
3u
0x
u —
a V
3u
=
8 z
4W —
3 v
3 2u
V - — _
. . .
3
,
_
3 2v
u ---- 4 w ------= y f T - 2 u flf v --------->
3 x
3 z
3 t 7-
(I)
. . . (II)
a n d t h e c o n s e r v a t i o n e q u a t i o n of m a s s a s
3u
3w
— + —- = o
9 x
9 z
...
Ill
w h e r e u, v, w a r e t h e v e l o c i t i e s i n t h e x, y, z d i r e c t i o n s
respectively.
at z -
o
at z = 6
The boundary conditions a r e
u = v = w = o
u = y 0 ;
v = -x Q
14
T h e f i r s t of t h e s i m p l i f i e d N a v i e r S t o k e s e q u a t i o n a n d t h e
c o n t i n u i t y e q u a t i o n a r e n o t f u n c t i o n s o f y.
S t o k e s e q u a t i o n is l i n e a r in v.
The second N avier
The f i r s t and th ird equations
a r e i d e n t i c a l t o t h e e q u a t i o n s g o v e r n i n g t h e m o t i o n of a n i n c o m ­
p r e s s i b l e fluid o v e r a flat p l a t e a n d t h e s o l u t i o n is g i v e n a s :
and
Re
c
-
—
v
F o g a r t y s o l v e d n u m e r i c a l l y t h e s p a n w i s e f l ow e q u a t i o n by
assum ing
g(,|)
a n d u s i n g t h e e x p r e s s i o n s f o r u , v , a n d w.
T h e r e s u l t s of t h i s
solution indicated that
v
. x ,
-•* ( ~)
u
y
a n d t h a t the s p a n w i s e flow is s m a l l in c o m p a r i s o n w i t h the
c h o r d w i s e f l o w.
G r a h a m (10) i n v e s t i g a t e d t h e l a m i n a r b o u n d a r y l a y e r
on r o t a t i n g b l a d e s o f a r b i t r a r y c r o s s s e c t i o n s u s i n g t h e
m o m en tu m integral technique.
The m o m e n t u m equatio ns w ere
i d e n t i c a l t o t h a t of F o g a r t y (9) e x c e p t t h a t t h e a x i s of r o t a t i o n
w a s d i s p l a c e d f r o m t h e l e a d i n g e d g e by a d i s t a n c e d i n t o t h e
free s tre a m .
T h is i n d e p e n d e n c e p r i n c i p l e was u s e d and the
c h o r d w i s e flow w as t r e a t e d s e p a r a t e l y .
T h e d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g t h e s p a n w i s e flow
is l i n e a r .
G r a h a m (10) d e v e l o p e d a s u p e r p o s i t i o n t e c h n i q u e a n d
s i m p l i f i e d th e s p a n w i s e flow v e l o c i t y p r o f i l e .
15
T h i s t e c h n i q u e m a d e it p o s s i b l e t o c o n s i d e r i n d e p e m f c n t l y
the effect of c e n t r i f u g a l fo r c e ,
t h e a x i s of r o t a t i o n .
c o r i o l i s f o r c e a n d p o s i t i o n of
G r a h a m (10) d e r i v e d t h r e e d i f f e r e n t i a l
equations and m o m e n t u m in te g ra l equations which d e s c r i b e
s p a n w i s e flow due to c e n t r i f u g a l f o r c e s ,
p o s i t i o n of t h e a x i s of r o t a t i o n .
c o r io lis f o rc e , and
She a l s o a s s u m e d a velocity
p r o f i l e of one p a r a m e t e r f or e a c h of t h e m .
The spanw ise
f l ow s d u e to t he d i f f e r e n t f o r c e s w e r e c h a r a c t e r i z e d by a
mixed m ean thickness,
b e i n g a f u n c t i o n of t h e p a r t i c u l a r
c h o r d w is e and sp a n w is e p ro files.
T h e s o l u t i o n of t h e s p a n -
w i s e flow m o m e n t u m i n t e g r a l e q u a t i o n d e t e r m i n e d e a c h m i x e d
m e a n t h i c k n e s s a s a f u n c t i o n of k n o w n q u a n t i t i e s .
N umerical
s t e p - b y - s t e p p r o c e d u r e wa s u s e d to s o l v e the s p a n w i s e flow
s i m u l t a n e o u s l y wi th c h o r d w i s e flow.
The c h o r d w is e velocity
p r o f i l e c h a n g e s its s h a p e due to p r e s s u r e g r a d i e n t e f f e c t s .
G r a h a m a n a l y z e d s e v e r a l b la d e s of d i f f e r e n t s h a p e s and a g r e e d
w i t h F o g a r t y (9) t h a t t h e e f f e c t o f s p a n w i s e f l o w w a s s m a l l .
APPLICATION OF PERTURBATION PROCEDURE
T a n (34) a n d M a g e r (23) a p p l i e d t h e p e r t u r b a t i o n p r o c e ­
d u r e for t h r e e - d i m e n s i o n a l b o u n d a r y l a y e r a n a l y s i s .
T a n (3 3) a p p l i e d h i g h e r o r d e r a p p r o x i m a t i o n t o t h e p r o b l e m
a n a l y z e d b y F o g a r t y (9) b y m e a n s o f t h e p e r t u r b a t i o n p r o c e d u r e
w h e r e b y all s u c c e s s i v e a p p r o x i m a t i o n s a r e l i n e a r i z e d .
Both
u a n d v v e l o c i t i e s a r e e x p r e s s e d i n t e r m s of a s i n g l e p a r a r - t e r
a s follows:
16
W h e n the c r o s s w i s e v e l o c ity is s m a l l c o m p a r e d with
the s t r e a m w i s e v e l o c i t y ,
the b o u n d a r y l a y e r will be only
slightly th re e dim ensional.
S m a ll p e r t u r b a t i o n p r o c e d u r e will
be a p p l i c a b l e u n d e r s u c h c a s e s .
M a g e r (23) d e v e l o p e d b o u n d a r y
l a y e r a n d p e r t u r b e d e q u a t i o n s for a c a r t e s i a n c o o r d i n a t e s y s t e m
r o ta t in g a t a c o n s ta n t a n g u l a r v e l o c i t y ab o u t an a r b i t r a r y a x i s in
space.
He a l s o d e v e l o p e d s o l u t i o n s f o r the fo ll o w i n g p r o b l e m s :
1)
L a m i n a r flo w o v e r a thi n c y l i n d r i c a l s h e l l of c i r c u l a r
c r o s s s e c t i o n flying a t a s m a l l a n g l e of yaw a l o n g a c i r c u l a r p a t h
and s i m u l t a n e o u s l y s p i n n i n g about its own axis of r e v o lu t io n .
2)
L a m i n a r flow o v e r a flat s u r f a c e with a c o n t i n u o u s
f o r c e f i e l d in t h e c r o s s w i s e d i r e c t i o n .
17
T H E O R E T IC A L IN V ESTIG A TIO N
IN TRO D U CTIO N
T h e p r e s e n t a n a l y s i s u s e d th e m o d e l p r o p o s e d by
P r a n d t l (28) f o r l a m i n a r a n d t u r b u l e n t f l o w s u n d e r t h e f o l l o w i n g
as sum ption:
1)
B la d e is s e m i - i n f i n i t e in s p a n a n d i n f i n i t e s i m a l l y th i n
a n d r o t a t e s a t a c o n s t a n t a n g u l a r v e l o c i t y a b o u t a n a x i s of r o t a t i o n
p e r p e n d i c u l a r t o t h e p l a n e of t h e b l a d e .
T h e c h o r d of t h e b l a d e i s
s m a l l c o m p a r e d to the r a d i a l d i s t a n c e f r o m th e a x i s .
2)
T h e f l u i d in w h i c h t h e b l a d e r o t a t e s i s s t i l l a i r a t
a tm o s p h e r ic conditions.
3)
T h e fluid is i s o t r o p i c , h o m o g e n e o u s an d v i s c o u s .
4)
T h e fluid h a s c o n s t a n t p r o p e r t i e s a n d is i n c o m p r e s s i b l e .
5)
6
B o u n d a r y l a y e r t y p e f l o w e x i s t s a n d t h e fl o w i s s t e a d y .
) V isco u s d is s ip a tio n effects a r e negligible.
A r o t a t i n g s t r e a m l i n e c o o r d i n a t e s y s t e m i s a s s u m e d in
o r d e r to e l i m i n a t e th e t i m e d e p e n d e n c e of t h e c o n s e r v a t i o n e q u a t i o n s .
T h i s c o o r d i n a t e s y s t e m i s s h o w n in f i g u r e (3) w h e r e t h e X a x i s
p a s s e s alo n g the s t r e a m l i n e ,
t h e Y a x i s i s n o r m a l t o it b u t i n t h e
s a m e p l a n e of th e s o lid s u r f a c e , and th e Z a x i s is n o r m a l to th e
p l a n e o f r o t a t i o n a n d r e p r e s e n t s t h e a x i s of r o t a t i o n .
The chord
of the b la d e is C and th e s p a n is s e m i - i n f i n i t e .
N e w t o n 's s e c o n d la w of m o t i o n c a n be a p p l i e d o n ly w ith
r e s p e c t to an i n e r t i a l s y s t e m .
S in c e th e c o o r d i n a t e s y s t e m of
the p r e s e n t a n a l y s i s is n o n - i n e r t i a l , a d d itio n a l a c c e l e r a t i o n s due
to c e n t r i f u g a l an d c o r i o l i s f o r c e s m u s t be a d d e d to the m o m e n t u m
18
F igure 3
C O -O RD IN A TE SYSTEM
19
equations.
T h e m o m e n t u m e q u a t i o n s f o r t h e l a m i n a r f l o w in
the p r e s e n t a n a ly s is a r e !
u 9u
- "ST +
y 9x
+ v
9u ,
9u ,
vu
v -£ - + w — +
9y
9z
y
9 2u
2
_
Qv
Z 9v 9 2 u
y 2 + y7* 9 x 2 + y z 9 x
9z 7
1 9u
u
.fry7 + y 9y
1 9 2u
9v
9v
u 9v
u
,
2
v -5 — + w -r— + — — -------+ Z ftu - 0, y =
9y
9z
y 9x
y
9 2v
.
9w
W 9z
9v
y 9y
v
7
9w _
9y
_1 9 P
p 9z
8y
_u _9_w
y 9x
9 v
7 " 9v 7
1
1
1 9P
- ——
p 9y
9u
T 7 9x
9 2 -w
.U )
2
1
9w
9 2v
9z
1
9y z + y 9y + 7
(2)
9 2 w 9 2w
9x 7 + 9z 2
. . .
(3 )
T h e c o n t i n u i t y e q u a t i o n is
1 8u
y 9x
v
y
9v
9y
9w
9z
•
(4)
The boundary conditions a r e
at z
= o
a t z — <n
u =v =w =o
u — U » Qyi
v- » q w —o
. . .
(5 )
. . .
(6 )
T h e s e b o u n d a ry c o n d itio n s e x p r e s s th e r e q u i r e m e n t th a t the
v e l o c i t y in a v i s c o u s f l u i d i s z e r o a t t h e w a l l a n d e q u a l t o t h e
t a n g e n t i a l v e l o c i t y a t v e r y l a r g e d i s t a n c e n o r m a l to t h e b l a d e .
S IM P L IF IC A T IO N O F TH E F L O W WITHIN BO U N D A RY L A Y E R
T h e d i f f e r e n t i a l e q u a tio n s d efin in g the fluid flow a r e n o n ­
lin ear
sim ultaneous
second o rd e r p a rtia l d ifferen tial equations.
It is i m p o s s i b l e to s o l v e t h e m a n a l y t i c a l l y .
T herefore
equations
a r e s i m p l i f i e d b y a n o r d e r of m a g n i t u d e a n a l y s i s a s w a s d o n e b y
P randtl.
T he boundary la y e r thickness 6
i s a s s u m e d to b e v e r y
20
s m a l l c o m p a r e d to t h e l i n e a r d i m e n s i o n s of t h e b l a d e ,
u, v a n d y
a r e t a k e n t o b e of t h e o r d e r of u n i t y , a n d w a n d z a r e t a k e n to b e
o f t h e o r d e r of 6 .
In t h e e q u a t i o n s ( 1) to ( 4 ) t h e t e r m s of t h e
o r d e r of 6 a r e
9w
w — i
oz
u 9w
—a -- .
y ox
9w
02 w
v — , a n d v —-T
dy
oz
t h e t e r m s of t h e o r d e r of 6 2 a r e
P
9 u
9y
i
y 9u
„
y 3y
—
>
1/ 9 v
91 v
— > ^ t;—T-r
y -3dy
dy
^
y
> — 7-
9 u
■ 1'■7"
8x
(
v
y
T '
0 2v
p
a—T
< a n<J —T
9x
y
p
~y1
t h e t e r m s of th e o r d e r of 6
92 w
V 9yz
v
y
u
7
y
P
3
9y
9x
. ■■ f
v
P
y1
*
9u
0x
are
v
y"7
9w
9y
92 w
9x
A l l t h e o t h e r t e r m s a r e of t h e o r d e r of u n i t y .
N e g le c tin g a l l the
t e r m s of o r d e r 6 a n d s m a l l e r t h e N a v i e r - S t o k e s e q u a t i o n i s
s i m p l i f i e d to t h e P r a n d t l b o u n d a r y l a y e r e q u a t i o n s a s :
U0u
——
y 9x
+ v
9u
9u
+ w —
9y
9z
+
uv
y
19P
9 u .
Z Ov = ---------—— + p ( - —7- )
P y 9x
9z
(7)
0U
0 V U 0V
U
2
1 9P
v — + w -r—+ - — -------- + 2 flu - Q y * - — ——
9y
8z y 9 x
y
P 9y
* » « ^ v >
I f
■ °
<8>
<’ >
21
T h e co n tin u ity e q u a tio n r e m a i n s the s a m e .
T he boundary
conditions a r e
at z = o
u=o,
v = o
w = o
•••( 10)
at z = 4
u = U= fly
v = o
w =o
...(11)
A c o r r e s p o n d i n g s e t of e q u a t i o n s i s o b t a i n e d f o r t h e t u r ­
b u l e n t f l o w b y i n t r o d u c i n g t h e g r a d i e n t of s h e a r s t r e s s i n s t e a d of
t h e g r a d i e n t of t h e v e l o c i t y g r a d i e n t s ,
th a t is, by s u b s t i t u t i n g
( t ^ X ) f o r v ( -vr 2- r )
-P- Sv T
Z
and
1
—
p
9 ,
. ,
, 9 v
nr~ ( t
) f o r ^fr—r )
9z
zy
9z
T h e b o u n d a r y l a y e r m o m e n t u m e q u a t io n s s h o w t h a t the
v a r i a t i o n of p r e s s u r e i n s i d e t h e b o u n d a r y l a y e r w i t h r e s p e c t to
z is z e r o .
T he p r e s s u r e g r a d ie n ts along x and y d ire c tio n s a r e a lso
z e ro a s shown below .
(7 )
A t t h e e d g e of t h e b o u n d a r y l a y e r e q u a t i o n
r e d u c e s to
since
9 ,
— ( f ly )
ox
9P
9x
= o.
E q u a t i o n ( 8 ) r e d u c e s to
y
♦
;<!u
-o»y .
- i
p
2f
9y
a nd
9P
3y
°
since
U
Thus,
p r e s s u r e is c o n s t a n t e v e r y w h e r e s i n c e
9P
=
=
fly.
9P
= °--5f = °- *
. 9P
IT. ' “•
22
T h e d i f f e r e n t i a l eq u atio , s fo r the m o t io n b e c o m e ,
u 9u
9u
9u
uv
— -r— + v —— + w —
-t----- - 2 ftv
y ox
9y
9z
y
92u
= v — -y )
8z'
. . . (12 )
9v
u 9v u 2
n n2
/ 9 2 v.
+w T ” + — "5“ ------+ 2 flu - fty = i / ( _ _ T )
oz
y Ox
y
9z‘
9v
v T~
ay
1 9u
7 8^
v
7
9v
W
9w
87 =
°
...
. . . (13 )
•••
(14 )
E v en th e se s im p lifie d eq u atio n s ca n not be so lv ed
analytically.
T herefore,
t h e m o m e n t u m i n t e g r a l t e c h n i q u e is
a p p l i e d , a s w a s i n t r o d u c e d by V o n K a r m a n .
E q u a t i o n s (12) a n d
(13 ) a r e i n t e g r a t e d w i t h r e s p e c t to z a c r o s s t h e b o u n d a r y l a y e r
f r o m t h e w a l l to t h e e d g e of t h e b o u n d a r y l a y e r .
T h e d e t a i l s of
t h e i n t e g r a t i o n is p r e s e n t e d in A p p e n d i x ( A - 1) .
The m o m en tum
e q u a t i o n in X d i r e c t i o n r e d u c e s to
u
9^
J
u
( f t y u - u 2 ) dz -
J
6
+ ft y
y
o
u
(uv ) d z - i C
°
(uv) dz
6
v d z + 3 f t j 1 v d z = ——
o
...(1 5 )
T h e m o m e n t u m e q u a t i o n in Y d i r e c t i o n r e d u c e s to
6
6
6
I
o
vidz + 7 £
I
,uv)
o
6
if...
. „
.
\ ( ft y - u) fly dz f
y J
o
dI + 7 I
yi
.
o
o
1 C. . .
27o2 L
— i ( fl y - u ) u d z
---y J
P
o
,..
..................................
(lt>)
T h e follow ing b o u n d a ry la y e r in te g ra l th ic k n e s s p a r a m e t e r s
a r e i n t r o d u c e d to s i m p l i f y t h e a b o v e e q u a t i o n s .
M om entum
t h i c k n e s s in x d i r e c t i o n ,
6
6v
x
=
V
J
.
-t;
y f l
(1 -
“
y f l
) dz
...
(17 )
23
M o m e n t u m t h i c k n e s s in Y d i r e c t i o n of t h e f l o w in Y d i r e c t i o n
<5
A
=
I
o
<
7
5
^
d ‘
< 1 8 >
M o m e n t u m t h i c k n e s s in Y d i r e c t i o n of t h e f l o w ir, X d i r e c t i o n
6
S* y * fo ' 7 5 » (1- 7 5 > d-
"’>
M o m e n t u m t h i c k n e s s in X d i r e c t i o n of t h e f l o w in Y d i r e c t i o n
6
v
yx
J
' yf t ' ' yQ
D i s p l a c e m e n t t h i c k n e s s in X d i r e c t i o n
<5
*
(
u
= ^J D (' 1 ■ _^
y f i7 ) d z
x
{2I)
D i s p l a c e m e n t t h i c k n e s s in Y d i r e c t i o n
6
dy = I o « f f i ) d Z
‘M »
T h e follow ing r e l a t i o n e x i s t s b e tw e e n the i n t e g r a l th i c k n e s s .
\
- V
=t
^
)dz-
7S
»dz f e *
<23 >
With t h e s e p a r a m e t e r s the m o m e n t u m i n t e g r a l e q u a t i o n s
r e d u c e f u r t h e r to
7 7 7 (^
*
, +7 ^
(V
- 7 lV
a- >
' - ^
<2 5 >
A P P R O X IM A T E SOLUTIONS
I n f o r m a t io n r e g a r d i n g the v e lo c ity p r o f i l e s a n d w a ll s h e a r
s t r e s s a r e n e c e s s a r y in o r d e r t o s o l v e e q u a t i o n s ( 24 ) a n d ( 2 5 ) .
As s u g g e s t e d by P r a n d t l (28) th e v e l o c i t y p r o f i l e s a r e a s s u m e d
t 0 b
a n dj
e
7 ,
7y f i
=
g
<
y i = ‘G ( f
6>
7
> * < ! > ’
< 2
6
>
(27>
24
The b oundary conditions a r e
at y * 5
G =1
g = o
( 2 8)
at y = o
G =o
g =
(29)
1
T oy
w h e r e « = ---- *- - t a n o n a n d at i s t h e a n g l e b e t w e e n t h e d i r e c t i o n of
T ox
the r e s u l t a n t w a ll s h e a r s t r e s s a n d the d i r e c t i o n of the m a i n
flow o u ts id e th e b o u n d a r y l a y e r .
S i m i l a r to t w o d i m e n s i o n a l b o u n d a r y l a y e r a n a l y s i s ,
the
f o l l o w i n g r a t i o s a r e d e f i n e d f o r f u r t h e r s i m p l i f i c a t i o n of t h e
m o m e n t u m i n t e g r a l e q u a t i o n s (2 4 ) a n d (2 5)
6
( 1-G) dz
I
H
*
-6
5
s
<3 0 >
x
( 1-G) G dz
o
6
( 1-G) G g dz
J
-
-
i
I
s
<3
I
( 1 - G ) G dz
o
<5
£
-
■
G g dz
4—
I
*
•
X
( * « S ) G dz
o
r. L I .
G g
dz
L
■
-
^
(33 )
X
I
( 1-G) G dz
>
25
w h e r e H, J , K a n d L d e p e p d o n t h e a s s u m e d v e l o c i t y p r o f i l e s .
T h e m o m e n t u m i n t e g r a l e q u a t i o n s (24) a n d (25) r e d u c e a f t e r
i n t r o d u c i n g t h e a b o v e v a r i a b l e s to
1 a ,e . JT€ 8■(■■■0* )+ JT0^x
— r — ( X) +
y 9x
9€
2
y
- — + — j« ©
8y
9
y
x
0~
Kc
p
(
yf i >2
2 8
e
------------ + < L -— ( x ) +
y
9y
, , K -J e
( ---------y
4
8C
9x
X— -
...(34)
, T
8€
2 f L €L— +
* 9y
, K -J
(
y
90x
)c
---------- ---------8x
( H - 1) e
T ox
X = - t
— 2
y
p (yfl)
, . . (35)
It i s d i f f i c u l t to s o l v e t h e s e t w o s i m u l t a n e o u s n o n - l i n e a r
pa r t i a l d i f f e r e n t i a l e q u a t i o n s fo r © a n d € . T o o v e r c o m e t h i s
x
d iffic u lty a n o r d e r of m a g n itu d e a n a l y s i s is done, s i m i l a r to the
b o u n d a r y l a y e r a p p r o x i m a t i o n s of N a v i e r - S t o k e s e q u a t i o n s , a s w a s
d o n e b y M a g e r (22).
The m o m e n tu m th ick n ess
© is a s s u m e d to be s m a l l
x
c o m p a r e d t o t h e l i n e a r d i m e n s i o n of t h e b l a d e a n d o f t h e o r d e r
©
.
Y is a s s u m e d to be of th e o r d e r
wall s h e a r s t r e s s e s
of the o r d e r « .
is a s s u m e d to be s m a l l e r
A s a r e s u l t , a l l d e r i v a t i v e s of
a n d t b e c o m e of t h e o r d e r o f
X
th a n unity and
T h i s is j u s t i f i e d b e c a u s e th e c h o r d of th e b la d e
is m u c h s m a l l e r th a n th e r a d i u s .
0
of u n i t y . T h e r a t i o of t h e
©
1 9 ( 0 x)
and « re sp e c tiv e ly .
*
In t h e e q u a t i o n ( 3 4 ) -------------- i s of t h e o r d e r o f © x a n d
9 ( 6 x)
_ Q 8*
z \
/L*
Kt e x
r L
p.,
J *
r—- . J
x -— , — J« © x a n d ----------- a r e o f t h e o r d e r ©x*
9y
By
y
y
© I
T e r m s of t h e o r d e r
x a r e n e g l e c t e d in c o m p a r i s o n w ith t e r m s
.
A
of th e o r d e r
x and « .
i 2 i 9
In e q u a t i o n (35)
* ,
y
i
a
^_
L *jr- ( 0 )* a n d 2sL.€L— a r » of
dy
*
» dy
26
,k - j .
a ®
J .K -j
_ a<
6L, a n d ( ------ ) t ——-*■ , a n d ( ------ ) 0 x —- a r e o f t h e
H -l
y
®X
y
0X
o r d e r * 0 * a n d ( ------ ) 0X i s o f t h e o r d e r 0 X . T e r m s o f t h e
2
y
order
« 0 X a r e n e g l e c te d in c o m p a r i s i o n w ith t e r m s of the
the o r d e r
order
t
2
and t 0 ^ .
I
iL /a i -
y dx
. K- J.
( y *
Thus,
e q u a t i o n s (34) a n d (35) r e d u c e t o
T ox
^
...(36)
da.
, K - J , _ d<
d x + ( y ) ^*dx
H-1
_
_ ( y >
~ ‘
ox
p ( y Sl) 2
...(37)
T h i s a n a l y s i s i n d i c a t e s th a t th e v a r i a t i o n a l o n g the s p a n w i s e
d i r e c t i o n is s m a l l c o m p a r e d to v a r i a t i o n a l o n g c h o r d w i s e d i r e c t i o n .
E q u a t i o n s (36) a n d (3 7) c a n n o w b e s o l v e d f o r
0X a n d
if
t
w all s h e a r s t r e s s is k now n.
LAM INAR FLO W
M OM ENTUM IN T E G R A L SOLUTION
In o r d e r t o f i n d s o l u t i o n s f o r t h e e q u a t i o n s (36) a n d (37)
th e v e l o c i t y p r o f i l e s i n s i d e the b o u n d a r y l a y e r m u s t be a s s u m e d .
T h e v e l o c i t y p r o f i l e i n t h e X d i r e c t i o n i s s i m i l a r to t h a t o f t w o
d i m e n s i o n a l fl o w o v e r a f l a t p l a t e a n d t h e r e f o r e it i s a s s u m e d a s ;
2
3
7 0
= G
'!»=
*o
*
+
+ *5<f>
T h e b o u n d a r y c o n d itio n s to e v a l u a t e the c o n s t a n t s a r e
u
d^u
"
, ! C
yO
at z - 6
=
oi
u ,
— = 1;
yfl
The X d irec tio n velocity profile,
du
= °
—dz
o
with the c o n s t a n t s ,
in a p p e n d i x ( A - 2 ) i s
yVf i
j ?
!2 « 6! » - 72 < f6 > 3
evaluated
27
F o r l a m i n a r f l o w t h e w a l l s h e a r s t r e s s is d i r e c t l y r e l a t e d to t h e
velocity p rofile.
Hence
T
ox
_
p
8u
p ( y f l ) 2~ p ( y f l ) 2 3 z
(40)
T h e v e l o c i t y p r o f i l e in t h e y d i r e c t i o n i s a s s u m e d a s
z .
, z
[ 1 <T>
. . .
b o + b l 1-f ) + b 2 ( f )2 ]
{41)
T h e b o u n d a r y c o n d i t i o n s u s e d to e v a l u a t e t h e c o n s t a n t s a r e
v
3 v / 0u
- (
at z = o
yfl
3 z y 9z
v
3v
= o
= o
yfl
3z
T h e Y d i r e c t i o n v e l o c i t y p r o f i l e w i t h t h e c o n s t a n t s e v a l u a t e d in
at z = 6
a p p e n d i x ( A- 3)
yf l
-
is
^
2
€
<i>
4
- I ' ! ’3]
[ * - < f >]
• • • ( 4 2)
T h e a s s u m e d v elo city p ro file s a n d the w all s h e a r s t r e s s
p a r a m e t e r s p ro v id e the n e c e s s a r y in f o r m a t i o n fo r the s o lu tio n
of t h e m o m e n t u m i n t e g r a l e q u a t i o n s .
S u b s t i t u t i o n of th e a s s u m e d
v e l o c i t y p r o f i l e ( 3 9 ) i n t o t h e e q u a t i o n ( 1 7) g i v e s t h e r e l a t i o n
b e t w e e n 6 a n d 0X a s s h o w n i n a p p e n d i x ( A - 4 )
39
. (4 3)
^
280
S u b s t i t u t i o n of e q u a t i o n ( 43 ) a nd ( 4 0) i n t o e q u a t i o n {36) g i v e s
1 A ( 39 x t = 3 t
y dx
2 *0
2
v
yfl 6
.
)
(44)
T h e s o l u t i o n of t h i s e q u a t i o n y i e l d s , a s s h o w n in a p p e n d i x ( A - 5 ) ,
6 = 4. 6 4 R e , x
9
x
=
.646 R e x
y ( x + * Q)
1/2
y(x +
xq )
• • • (45)
. . .
S u b s t i t u t i n g e q u a t i o n (36) in e q u a t i o n ( 3 7 ) a n d r e a r r a n g i n g
(46)
28
results
dx
dG
+ -L
09*
dx
r 1 +_ L
I
K -J
— 1
K -J
(47)
'
'
* * ‘
S u b s t i t u t i o n o f e q u a t i o n ( 46 ) i n t o e q u a t i o n ( 47 ) g i v e s
dc
dx
+
c
2 ( x+xc
A s s h o w n in a p p e n d i x ( A - 6 ) t h e s o l u t i o n o f t h i s e q u a t i o n i s
'
1 ( F 3
1
*
<3
T h e v a l u e s of H, J ,
• • • <4 9 *
* F j>
K, a n d L a r e d e t e r m i n e d b y s u b s t i t u t i n g t h e
a s s u m e d v e l o c i t y p r o f i l e s in e q u a t i o n ( 3 0 ) to ( 3 4 ) a s s h o w n in
appendix ( A-7) .
The values a r e
H
*
2.6923
J
=
0 . 39 46
K
=
0.8376
L
=
0 . 1376
e =
1. 4 5 3
T herefore
( x + x Q)
. . .
( 50)
W i t h t h e s e e v a l u a t e d q u a n t i t i e s t h e v e l o c i t y p r o f i l e in X
d i r e c t i o n r e d u c e s to
yfl
= 0 . 3 2 33 (
* Re J 1 )
y ( x + x G) « - x
( 51)
. . .
w h e n e q u a t i o n ( 4 5 ) is s u b s t i t u t e d i n t o e q u a t i o n ( 3 9 ) .
S im ilarly,
t h e v e l o c i t y p r o f i l e in Y d i r e c t i o n r e d u c e s to
r
1/2
n- 7 V 7
yfi(x+X
r = 0. 4 6 9 2 ( _ ? L _ r
Q)
y ( x + x Q) e - x
)
1/
1- 0. 43 10<-7 ^ - — R '*)
[_
y ( x + x Q) e x
l/2
z
R
+ 0.03098 ( _ £ £ - * )
V
1
+ 0.006673 ( _
Z
_
n
‘l
e>x )
\
\/2
- 0.0007191 (
y(*+*D)
)4 1
J
. . .
(52)
w h e n e q u a t i o n s ( 5 0 ) , a n d ( 4 5 ) a r e s u b s t i t u t e d i n t o e q u a t i o n (42).
29
o
00
Velocity
o
Function
o
o
o
o
o
N
X ‘a>i
9 ‘0
(°x +x)X
o
30
T h e e q u a t io n a (51) a n d (52) c o m p l e t e l y d e f in e th e v e lo c ity
p r o f i l e s a s a f u n c ti o n of i n d e p e n d e n t v a r i a b l e s .
T h e g r a p h s of
t h e s e p r o f i l e s a r e p r e s e n t e d in f i g u r e (4) a s u n i v e r s a l c u r v e s .
T h e t o t a l s h e a r s t r e s s i s g i v e n by
T
o
=
E quation s (40),
,.0. 5
( T
2 + T . ■*)
'o x
oy
. . .
(53)
( 4 5 ) , a n d ( 5 0 ) a l o n g w i t h (53 ) y i e l d
p ly 5 > 2=
° - 3 2 3 3 R e , * ° ' 5[ i + « ! J ° ' 5
w here
C =
(54)
y
-21
Tox
C f, X
T h e s h e a r s t r e s s y ie ld s the skin f ric tio n c o e ffic ie n t
based
on f r e e s t r e a m v e l o c i t y .
=p l 7 ^ p
* ° - 3 2 3 3 R p. * ° ‘ 5{ ‘ + [ ‘ ‘ 4 5 3 (* +Xo) |2} /-2 ‘
(55)
T h e a v e r a g e v a l u e of t h i s c o e f f i c i e n t a t a n y g i v e n Y i s e v a l u a t e d
b y i n t e g r a t i n g e q u a t i o n (5 5) a l o n g t h e X d i r e c t i o n a s s h o w n in
a p p e n d ix (A -8) w hich r e s u lts a s
Cf
= 0.6466 R e
2
0.
“ °* 5 - | \ + 0 . 1(2. 9 0 6 x Q)
0 1388(2. 906
xq )
4 + 0 . 0 0 4 8 0 8 ( 2. 9 0 6
- 0. 0 0 2 2 9 8 ( 2 . 9 0 6 x Q) 8 + . . .
where R
e, x 0
=
x Q) 6
. . .
( 5 6)
„
W ith th is in f o r m a tio n ,
t h e t o r q u e r e q u i r e d to o v e r c o m e
the f r i c t i o n a l d r a g a l o n g th e c h o r d w i s e d i r e c t i o n is e v a l u a t e d
(skin f r ic t io n
t o r q u e ) . W ith e q u a t i o n s (40) a n d (45) th e t o r q u e
M is e v a l u a t e d in a p p e n d i x ( A - 9 ) .
M
„
r i
= 0 . 6 4 6 6 P 0 1* 5 v °* 5 j
y 3 ( s i n " 1 ^ ) ° -5d y . .
y l
( 57 )
31
By a s s u m i n g t h e v a l i d i t y of R e y n o l d s a n a l o g y ,
it i s p o s s i b l e
to e v a l u a t e th e h e a t t r a n s f e r c o e f f i c i e n t w ith the i n f o r m a t i o n
obtained so far.
R e y n o ld s a n a lo g y s t r i c t l y h olds t r u e fo r the
tw o d i m e n s i o n a l l a m i n a r an d t u r b u l e n t flow o v e r a flat p la te
w h e n t h e P r a n d t l n u m b e r of t h e f l u i d i s e q u a l t o u n i t y .
In th e
d e r i v a t i o n o f t h e R e y n o l d s a n a l o g y it i s a s s u m e d t h a t t h e r a t i o o f
h e a t flux to th e s h e a r s t r e s s is c o n s t a n t at e v e r y p o in t a c r o s s th e
boundary layer.
T h i s b a s i c s t a t e m e n t c a n be r e d u c e d to th e f o r m
f
Nu = C L
Re
2
In g e n e r a l ,
for b o u n d a r y l a y e r flow s the s h e a r s t r e s s and
th e h e a t flux h a v e n o n - z e r o v a l u e s at th e w all and z e r o v a l u e s at
t h e o u t e r e d g e of t h e b o u n d a r y ,
f l o ws .
b o t h in l a m i n a r a n d t u r b u l e n t
T h e w a l l v a l u e s c o n s i s t of t h e l a m i n a r c o m p o n e n t s s i n c e
the t u r b u l e n t f lu c t u a ti o n s ,
vanish.
p a r t i c u l a r l y the v e lo c ity flu c tu a tio n s ,
At t h e o u t e r e d g e of t h e b o u n d a r y l a y e r t h e g r a d i e n t s
of th e v e l o c i t y and t e m p e r a t u r e a r e z e r o an d th e f l u c t u a t i o n s of
the v e l o c i t y a l s o d ie out.
H e n c e , it is r e a s o n a b l e t o a s s u m e t h a t
b o th s h e a r s t r e s s and h e a t flux h a v e t h e i r m a x i m u m v a lu e at
th e w all a n d d e c r e a s e m o n o t o n i c a l l y to z e r o at the o u t e r e d g e of
the b o u n d a ry l a y e r .
F lx p erim en ts p ro v e the above a s s u m p tio n .
y
It i s o f t e n a s s u m e d t h a t t h e s h a p e s of t h e p r o f i l e s
and
cw
a re identical.
If t h i s a p p r o x i m a t i o n i s t r u e t h e n
q"
Mw
^
^
^
M
^77* - ^ ”
q w q”
assum ption
r
T herefore,
= constant.
The
is a c o n s t a n t is f o un d t o b e t r u e f o r 2 d i m e n s i o n a l
flow s
It is a s s u m e d t o b e t r u e f o r t h r e e d i m e n s i o n a l c a s e s a l s o
Hence,
t h e R e y n o l d s a n a l o g y i s a s s u m e d t o be v a l i d f o r t h e p r e s e n t
analysis.
F o r flu id s with P r a n d t l n u m b e r o t h e r th a n one, th e R e y n o l d s
a n a lo g y is m o d if ie d a s
32
Nu =
^f Pr ^
2~
Re
. . . (58)
T h e h e a t t r a n s f e r e q u a t i o n fo r th e l a m i n a r flow o v e r a r o t a t i n g
f l a t b l a d e i s o b t a i n e d b y s u b s t i t u t i n g e q u a t i o n (5 5) i n t o e q u a t i o n
(58)
as
N u , x = 0. 3 23 3 Re^,"■"* P fr1 /3£ + [ l . 4 5 3 ( x +
• • • ( 59 )
x
T h e a v e r a g e v a lu e of N u s s e l t n u m b e r a t a n y g iv e n y is
o b t a i n e d b y a p p l y i n g R e y n o l d s a n o l o g y t o e q u a t i o n (56).
is
The re s u lt
---0 . 5 1/3 r
>
N u = 0. 6466 R e , Xq P r J 1+ 0. 1 (2. 9 0 6 x b r
- 0 . 0 1 3 8 8 ( 2 . 906xf )4 + 0. 0 0 4 8 0 8 (2. 9 0 6 x ) 6
o
o
- 0 . 0 0 2 2 9 8 (2. 906xf )8 + • • 1
l.
n
flL 2x^
w h e r e R e .' x U
. - ------- ■
—D-
. . . (60)
T h e q u a n t i t i e s i n s i d e t h e b r a c k e t s of t h e e q u a t i o n s (55) a n d
(59)
i s t h e r a t i o of t h e l o c a l N u s s e l t n u m b e r ( o r l o c a l s k i n
f r i c t i o n c o e f f i c i e n t ) fo r th e t h r e e d i m e n s i o n a l flo w to th e c o r r e s ­
p o n d in g v a l u e s of tw o d i m e n s i o n a l flow.
It i s a f u n c t i o n of
x , y, a n d b .
In n o n - d i m e n s i o n a l f o r m it c a n be e x p r e s s e d a s a
x
y
f u n c t i o n o f - a n d f- .
*t>
b
F i g u r e (5) s h o w s t h e p e r c e n t a g e i n c r e a s e d u e t o t h e t h r e e
d i m e n s i o n a l e f f e c t o v e r th e tw o d i m e n s i o n a l flow fo r d if f e r e n t
v a l u e s of t h e p a r a m e t e r
^ .
It i s s e e n t h a t t h e i n f l u e n c e o f t h e t h r e e d i m e n s i o n a l e f f e c t
i n c r e a s e s with d i s t a n c e f r o m the le a d in g edge a l o n g the x d i r e c t i o n
an d d e c r e a s e s w ith d i s t a n c e f r o m th e a x i s of r o t a t i o n .
T w o t y p e s of b l a d e s a r e u s e d i n t h e e x p e r i m e n t s .
is for t o r s i o n m e a s u r e m e n t s a l o n e .
param eters
^
b
are
T y p e (2 )
The v a lu e s of n o n -d im e n s io n a l
33
Type 1
B lade
Type 2
B lade 1
10.3 3 « I « 2 5 . 8 3
b
8.5
^ 18
b
4. 25
B lade 2
b
<
9
Blade 3
For heat tra n sfe r m easu re m en ts
Type 1
B lade 1
k
b
= 1L 33
B lade 2
f = 15. 33
b
B lade 3
i = 19. 33
D
B lade 4
J - 2 2 . 33
b
F r o m f i g u r e (5) it c a n be s e e n t h a t t h e i n c r e a s e i n N 1 d u e t o
u, x
U,
t h r e e - d i m e n s i o n a l e ffe c t for the b la d e u s e d fo r h e a t t r a n s f e r
m e a s u r e m e n t s i s of t h e o r d e r of 5%.
E q u a t i o n s (56) a n d (60) g i v e t h e r a t i o of t h e a v e r a g e
N u s s e lt n u m b e r (or a v e r a g e sk in fric tio n c o efficien t) for t h r e e
d i m e n s i o n a l flow to t h e c o r r e s p o n d i n g v a l u e s of tw o d i m e n s i o n a l
flow .
It i s a l s o a f u n c t i o n o f y a n d b.
A s i m i l a r plot to f ig u r e 5
s h o w s in f i g u r e 6 the in f l u e n c e of the t h r e e d i m e n s i o n a l e f f e c t fo r
d i f f e r e n t v a l u e s of th e p a r a m e t e r
^ . These curves dem onstrate
b
th a t the t h r e e d i m e n s i o n a l e ffe c t d e c r e a s e s h y p e r b o l i c a l l y with
d i s t a n c e f r o m t h e a x i s of r o t a t i o n .
T U R B U L E N T FLOW M O M EN TU M IN T E G R A L SOLUTION
T u r b u l e n t fluid m o t i o n is a n i r r e g u l a r c o n d i t i o n of flow
in w hich the v a r i o u s q u a n t i t i e s sh o w a r a n d o m v a r i a t i o n w ith
t i m e a n d s p a c e s o t h a t s t a t i s t i c a l l y d i s t i n c t a v e r a g e v a l u e s c a n be
discerned.
In g e n e r a l t h e r e a r e f o u r d i f f e r e n t i a l e q u a t i o n s w i t h
te n v a r i a b l e s defining the m o tio n .
E q u a t i o n s of m o t i o n a r e n o n - l i n e a r .
L inearising
34
Figure 5
100
I N C R E A S E IN
f, x
A N D Nu, x D UE T O
T H R E E - D I M E N S I O N A L E F F E C T IN L A M I N A R F L O W
Percentage
in crease
80
40
20
x
N o n - D i m e n s i o n a l D i s t a n c e F r o m C e n t e r of B la d e
35
Figure 6
100
I N C R E A S E IN
f
AND Nu DUE TO
T H R E E - D I M E N S I O N A L E F F E C T IN LAMINA R F L O W
80
P e r c e n ta g e
Increase
70
50
40
30
20
20
I
b
24
36
a p p r o x i m a t i o n s c a n n o t be e x t e n s i v e ly u s e d b e c a u s e the non
lin e a r t e r m s a r e the ones th a t c h a r a c t e r i z e the equations.
In
ad d itio n , the p r o b le m is th r e e d im e n s io n a l.
T h e v e l o c i t y o f t h e f l u i d c o n s i s t s of a t i m e - a v e r a g e d m e a n
v e l o c i t y V and c o r r e s p o n d i n g f lu c tu a tin g c o m p o n e n t s V '.
The
m e a n v e lo c itie s do not s a ti s f y the N a v ie r S to k es eq u atio n ,
in stan t values m ay.
but the
W hen the e q u a tio n s of m o tio n a r e a v e r a g e d
we g et R e y n o l d s e q u a t i o n s of m o t i o n in w h ic h th e t i m e - a v e r a g e d
v a l u e s a r e th e s a m e a s t h o s e of s t e a d y , n o n - f lu c t u a ti n g flow b u t
w ith s o m e a d d itio n a l s t r e s s t e r m s kn o w n a s R ey n o ld s S t r e s s e s
Turbulent S tre s s e s .
T h e s e s t r e s s e s a r e n o t r e a l l y s t r e s s e s in
the continuum s e n se ,
h o w e v e r they r e p r e s e n t the t i m e - a v e r a g e d
or
t u r b u l e n t m o m e n t u m c o n t r i b u t i o n * . In t h e t u r b u l e n t f l o w t h e s h e a r
s t r e s s is e x p r e s s e d a s
V o n K a r m a n (39) s t a t e s t h a t a d e f i n i t e r e l a t i o n e x i s t s b e t w e e n
th e f r i c t i o n a t th e w all an d th e a d j a c e n t v e l o c i t y d i s t r i b u t i o n in
la m i n a r and tu r b u le n t flow s.
By a a e r i e s o f c a r e f u l m e a s u r e m e n t s
in t u r b u l e n t f l o w t h e s h e a r s t r e s s l a w h a s b e e n e s t a b l i s h e d b y
B lausius as,
r = 0. 0 1 2 5 6 ( . £ . ) •
. . . (61)
G r u s c h w i t z (11) d e m o n s t r a t e s t h a t t h e l a w i s v a l i d f o r a t h r e e
dim ensional boundary lay er also .
M a g e r (22) a l s o u s e d t h e
a b o v e l a w f o r w a l l s h e a r s t r e s s in t u r b u l e n t f l o w s .
In t h e
p r e s e n t a n a l y s i s th i s law is a s s u m e d .
T h e m o m e n t u m i n t e g r a l e q u a t io n s d e r i v e d fo r l a m i n a r flow
can be u s e d for t u r b u l e n t flow a l s o ,
s in c e no a s s u m p ti o n w as m a d e
a b o u t th e s h e a r s t r e s s in th e d e r i v a t i o n .
F o r the v e l o c i t y p r o f i l e in X d i r e c t i o n
— , w hich is s i m i l a r
y
o
to th e tw o d i m e n s i o n a l flow o v e r a flat p l a t e , th e o n e - s e v e n t h
37
p o w e r Law i s a s s u m e d .
,-o ■ G ( t >
■ < 1 )I/7
• • •
(62)
T h e v e l o c i t y p r o f i l e in t h e Y d i r e c t i o n i s a s s u m e d a s
'/7
^
= *G(f).
*(f)
2
* « < f > [ = „+<:,( I ) + C 2 l j >
1- •
(63)
T h e b o u n d a r y c o n d i t i o n s u s e d to e v a l u a t e th e c o n s t a n t s a r e
V
at z
=
o
— =
yf l
at z
= <5
v
—yfl
QV
o
—
gz
3v
—— = o
3z
= o
3u
-—
3z
=
(
. . .
(64)
T h e v e l o c i t y p r o f i l e in Y d i r e c t i o n w i t h t h e c o n s t a n t s e v a l u a t e d
ii a p p e n d i x ( A - 10) ,
r e d u c e s to
V
75
2
= ‘ ( 5 >
t 1 ' <^ H
• • •
(65)
The a s s u m e d velo city p ro files a n d w all s h e a r s tr e s s
p a r a m e t e r s p r o v id e the n e c e s s a r y i n f o r m a t i o n f o r the so lu tio n
of m o m e n t u m i n t e g r a l e q u a t i o n s .
S u b s t i t u t i o n of t h e a s s u m e d
p r o f i l e s (62) in to th e e q u a t i o n (17) g iv e s th e r e l a t i o n b e tw e e n
a n d ©x
(5
a s s h o w n in a p p e n d i x ( A - 1 1)
ex
=
~
6
. . .
(66 )
I n t r o d u c t m g t h e B l a u s i u s f r i c t i o n Law f o r t h e w a l l s h e a r
s t r e s s in X d i r e c t i o n g i v e s
1 ^ = 0 . 0 1 2 6 5
< ^ 1
. . .
( 67)
• • '
(68>
a nd s u b s t i t u t i o n i nt o e q u a t i o n (36) y i e l d s
7
£
'
' V
° - 012”
( n^
)0
T h i s e q u a t i o n is s o l v e d in a p p e n d i x ( A - 1 4 a n d t h e r e s u l t is
©x
= 0 . 0 3 6 0 1 R e> x
y ( x + x Q)
. . .
(69 )
y(x+x )
o'
. . .
(70 )
and
b = 0. 3704 R
-
e, x
0 . 2.
38
S u b s t i t u t i n g e q u a t i o n (3 6) i n t o e q u a t i o n (3 7) a n d r e a r r a n g i n g
gives
d c
dx
d
dx (
^
v r,
1 i
t + K -J
H -l
K -J
. . . (47)
I n t r o d u c i n g e q u a t i o n (69) i n t o e q u a t i o n (47) g i v e s
d«
dx
4
5
e
< i +
T »=
K -J
(x+ xo)
T he s o lu tio n of th is e q u a tio n is ,
. H -l . 5
f = ( TT~
K - Jt ') 4
(x+ xo)
(2. 25 +
T h e v a l u e s o f II, J ,
.(71)
H- ‘
K -J
( s e e a p p e n d i x A - 13]
. • •(72)
1 )
K -J
K, a n d L a r e d e t e r m i n e d by s u b s t i t u t i n g
t h e a s s u m e d v e l o c i t y p r o f i l e s in e q u a t i o n s (30) t o ( 34) , a s s h o w n
in a p p e n d i x ( A - 1 4 ) .
The values a re
H r
1. 2 8 5 7
J = 0.5423
T herefore
t
K=
2.6727
L=
1. 128 6
= 0. 0 6 1 6 4 (x+ xo )
. . . (73)
T h e X d i r e c t i o n v e l o c i t y p r o f i l e r e d u c e s to
. 2 -xi
u
z Re, x x | j
1. 152
-:o)
.) J 7
y(x+ xc
yfl
■ (74)
w h e n e q u a t i o n (70) i s s u b s t i t u t e d i n t o e q u a t i o n (62),
d i r e c t i o n v e l o c ity p r o f il e r e d u c e s to
J.
.
v
■tl (x~+ xo)
0. 0 7 1 0 1
7. 2 88 8
z
Re,
The Y
2
Jt
y (x+ x o )
5. 39 96 (
^
y (x+ xo)
x Re, x
y ( x + xo) 1)
...(75)
T h e e q u a t i o n s (74) a n d (75 ) c o m p l e t e l y d e f i n e t h e v e l o c i t y
p r o file s as a function of in d e p e n d e n t v a r i a b l e s .
T h e g r a p h s of
t h e s e p r o f i l e s a r e p r e s e n t e d i n f i g u r e (7) a s u n i v e r s a l c u r v e s .
N e x t, th e to t a l w a ll s h e a r s t r e s s is e s t i m a t e d by
_
r z
_ 2 . 0. 5
T o = [ Tox + To y J
Velocity
o
o
o
o
o
Function
39
oo
o
40
' W i t h e q u a t i o n (67) t h e a b o v e r e l a t i o n b e c o m e s
o
_
,=
(y o ) 2
p
n n i o KE / v
0 .0 1 2 5 5 (_ !:
‘
yq©„
, 0 . 2 5 ,,
) ‘
(1 +
„0.5
‘
...(76)
The s h e a r s t r e s s y ield s the sk in fric tio n co efficien t
c f,x ,
2
b a s e d on f r e e s t r e a m v e l o c i t y .
c
-
t
= —
. =
p (y n)2
2
X. ) ] V
0 . 0 2 8 8 1 R e , x - ° * 2« f l + [ 0 . 0 6 l 6 4 ( x+
\.
o' j j
...(77)
q
T h e a v e r a g e v a l u e of
f, x a t a n y g i v e n y i s e v a l u a t e d by
2
i n t e g r a t i n g e q u a t i o n (77) a l o n g t h e X d i r e c t i o n a s s h o w n in
a p p e n d i x ( A - 15).
It is g i v e n a s
£
f
0 . 0 3 6 0 1 Re , x ‘ °* 2 £l + 0. 142 9 (0. 1233
- 0 . 0 2 0 8 3 (0. 1 2 3 3 x o ) 4 + 0. 00 7 3 5 3 (0. 1233X©)6
- 0. 00 35 5 1 (0. 1233x )8 + . . . 1
L
r>
w here
R«,
2X0
= -Z-----------
T h e t o r q u e M,
...(78)
J
r e q u ir e d to o v e r c o m e the fric tio n a l d r a g
a l o n g the c h o r d w i s e d i r e c t i o n a l o n e is e s t i m a t e d in a p p e n d ix
( A - 16),
s i n c e t h e d r a g a l o n g t h e r a d i a l d i r e c t i o n d o e s not
c o n tr i b u te to the t o rq u e ,
M -
0.0,60.
Z ^ V A z . i n - 1 V
V1
' 8 dy
Y
...179)
A s s u m i n g t h e v a l i d i t y ot R e y n o l d s a n a l o g y t h e h e a t t r a n s f e r
c o e f f i c i e n t s c a n be e v a l u a t e d .
T he lo c a l N u s s e l t n u m b e r is th e n
given a s ,
1/3
N u x = 0. 0 2 8 8 1 R e , x ' 8 P r
j
| l + [0. 0 6 16 4( x+ * h ) ] 2j>
1Z. . . (80)
T h e a v e r a g e N u s s e l t n u m b e r f o r t u r b u l e n t fl o w i s o b t a i n e d
b y a p p l y i n g R e y n o l d s a n a lo g y to e q u a t i o n ( 78) .
T h e r e s u l t is
41
Figure K
.
20
I N C R E A S E IN
f, x A N D Nu, x D U E T O
T H R E E -D IM E N SIO N A L E F F E C T
IN T U R B U L E N T F L O W
0. 0
xo
. 5
1. 0
N o n - D i m e n s i o n a l D i s t a n c e F r o m C e n t e r of B l a d e
42
Figure 9
I N C R E A S E IN C f
A N D Nu DUE TO
2
P e r c e n ta g e
Increase
T H R E E - D I M E N S I O N A L E F F E C T IN T U R B U L E N T FL O W
0. 04
0 . 02
4
8
16
12
1
20
24
43
—
8 1 / 3T
Nu = 0.03601 Re, x
P r Jl + 0. 1429(0 . 1233xo)
)
- 0. 0 2 0 8 3 ( 0 . 1 2 3 3 x o ) 4 + 0. 0 0 7 3 5 3 ( 0 . 1 2 3 3 x o ) 6
- 0. 0 0 3 5 51 (0. 1 2 3 3 x o ) 8 +
J
. . . (81)
T h e q u a n t i t y i n s i d e t h e b r a c k e t s o f t h e e q u a t i o n (80) a n d
(77) i s t h e r a t i o o f t h e l o c a l N u s s e l t n u m b e r ( o r l o c a l s k i n f r i c t i o n
c o e f f i c i e n t ) f o r the t h r e e d i m e n s i o n a l f l o w t o t h e c o r r e s p o n d i n g
q u a n t i t i e s fo r tw o d i m e n s i o n a l flow.
it i s a f u n c t i o n o f x, y , a n d b.
X
S i m i l a r to l a m i n a r f lo w
In n o n - d i m e n s i o n a l f o r m i t c a n b e
V
e x p r e s s e d a s a f u n c t i o n o f — a n d -f- . F i g u r e (8) s h o w s t h e
xo
b
p e r c e r t a g e i n c r e a s e d u e to the t h r e e d i m e n s i o n a l e f f e c t o v e r the
y
tw o d i m e n s i o n a l flow f o r d i f f e r e n t v a l u e s of th e p a r a m e t e r — .
b
It i s s e e n t h a t t h e t r e n d i s s i m i l a r t o t h e l a m i n a r f l o w . B u t
th e v a l u e s a r e c o n s i d e r a b l y s m a l l e r w h ich c a n be e x p e c t e d due
t o t h e e f f e c t of t u r b u l e n t m i x i n g .
In a s i m i l a r
of a v e r a g e Nu a n d
—
in N u a n d
due
w a y , e q u a t i o n s (81) a n d (78) g i v e t h e r a t i o
C
f . F i g u r e (9) s h o w s t h e p e r c e n t a g e i n c r e a s e
~Y
to t h r e e - d i m e n s i o n a l effect o v e r tw o d im e n s io n a l
flow f o r d i f f e r e n t v a l u e s of th e p a r a m e t e r
^ . It i s s e e n t h a t t h e
b
t h r e e d im e n s i o n a l e ffe c t d e c r e a s e s h y p e r b o l i c a l l y w ith d i s t a n c e
f r o m t h e a x i s of r o t a t i o n ,
s i m i l a r to l a m i n a r flow.
44
E X P E R IM E N T A L IN V ESTIG A TIO N
IN TRO D U CTIO N
T h e a s s u m p t i o n s m a d e in the t h e o r e t i c a l i n v e s t i g a t i o n
a r e c h e c k e d e x p e r i m e n t a l l y by c o n s t r u c t i n g a n a p p a r a t u s to
m e a s u r e th e h e a t t r a n s f e r r a t e and d r a g on r o t a t i n g b l a d e s .
The b lade fo r the t h e o r e t i c a l i n v e s t ig a t io n is a s s u m e d
to be s e m i - i n f i n i t e i n s p a n a n d i n f i n i t e s i m a l l y t h i n w i t h t h e
chord sm all,
rotation.
c o m p a r e d t o t h e r a d i a l d i s t a n c e f r o m t h e a x i s of
A l l t h e s e a s s u m p t i o n s c a n n o t be m e t b y t h e a p p a r a t u s
b e c a u s e of t h e d i f f i c u l t i e s in c o n s t r u c t i o n .
T h e r e f o r e , a hollow ,
thin b la d e e n c lo s in g h e a t e r and t e m p e r a t u r e m e a s u r i n g d e v ic e s
is u s e d to m e a s u r e th e h e a t t r a n s f e r c o e f f i c i e n t s ( a v e r a g e w ith
r e s p e c t to c h o r d but l o c a l w ith r e s p e c t to r a d i u s ) .
E r r o r s due
to r a d i a l c o n d u c t i o n a r e e l i m i n a t e d by g u a r d h e a t e r s .
Local skin frictio n coefficien ts m e a s u r e m e n t s a r e v e ry
d if f i c u lt to m a k e on a r o t a t i n g b l a d e ,
but it is p o s s i b l e to
d e s i g n a t o r s i o n a l d e v i c e to m e a s u r e the to ta l e f f e c tiv e t o r q u e
n e e d e d for ro ta tin g the b la d e s .
S u ch m e a s u r e m e n t s will h e lp
to e v a lu a te the a v e r a g e d r a g c o e ffic ie n t
w hich is a m e a s u r e
of t h e p o w e r r e q u i r e m e n t s .
DESIGN O F R O T A T IN G A P P A R A T U S
The r e q u ir e m e n t s for the o p tim u m d e s ig n a r e :
1)
T h e r o t a t i n g b la d e s h o u ld be fla t an d a s th in a s p o s s i b l e .
2)
T h e b l a d e s h o u ld o p e r a t e in a n i n c o m p r e s s i b l e fluid w h ic h
is at r e s t in the r e g i o n s f a r a w a y f r o m th e b la d e .
3)
H e a t l e a k a g e a l o n g th e r a d i a l d i r e c t i o n s h o u ld be
elim inated.
45
4)
A w i d e r a n g e of R e y n o l d s n u m b e r s m u s t b e c o v e r e d
to o b s e r v e th e i n f l u e n c e of l a m i n a r ,
turbulent and tra n s itio n regions
of t h e f l o w .
D ESC R IPTIO N O F TH E A PP A R A T U S
F i g u r e (10) s h o w s t h e p h o t o g r a p h o f t h e a p p a r a t u s t a k e n
from above.
T h e a p p a r a t u s c o n s i s t s of f o u r b l a d e s fix e d to
a h u b w h i c h i s s u r r o u n d e d by a b r a s s t u b e t o r e d u c e t h e d i s ­
t u r b a n c e of th e flow.
The blades a r e e le c tric a lly in su la ted fro m
the hub in o r d e r to a v o id
d i s t r u b a n c e s i n e. m . f. m e a s u r e m e n t s
a n d in o r d e r t o s t u d y t h e e f f e c t of e l e c t r i c a l f i e l d s o n h e a t
t r a n s f e r on e a c h b l a d e i n d e p e n d e n t l y .
The b lad es a r e m ounted
in s u c h a w a y t h a t t h e i r p i t c h c a n b e c h a n g e d t o a n y d e s i r e d a n g l e
i n o r d e r t o d e t e r m i n e t h e i n f l u e n c e o f t h e a n g l e of a t t a c k a n d
s e p a r a t e d f l o w on h e a t t r a n s f e r r a t e .
F i g u r e (11) s h o w s t h e s e c t i o n a l d r a w i n g o f t h e a p p a r a t u s .
T h e h u b is m o u n t e d to th e v e r t i c a l h o ll o w s h a f t A .
This shaft
is p o s i t i o n e d v e r t i c a l l y w ith th e h e l p of a s e l f - a l i g n i n g b e a r i n g
a t t h e m i d d l e a n d a p i l o t b e a r i n g a t t h e b o t t o m of t h e s h a f t .
T he pilot b e a r i n g is fix ed to the h o u s in g of a t o r s i o n m e t e r to
w h i c h t h e h o l l o w s h a f t i s c o n n e c t e d b y m e a n s of a t o r s i o n
spring.
T h e h o u s i n g i t s e l f is fixed to a n o t h e r s h a f t B w h ic h is
p o s i t i o n e d v e r t i c a l l y b y m e a n s of t w o b a l l b e a r i n g s .
A gear
p u l l e y i s a t t a c h e d t o t h e b o t t o m of t h e s h a f t a n d i s d r i v e n by a
D. C . m o t o r m o u n t e d w i t h i t s a x i s p a r a l l e l t o t h e a x i s of r o t a t i o n
of t h e b l a d e s .
T h e d r i v i n g m o t o r d e v e l o p s 5 K. W. a t 1750 r . p. m . a n d 2 5 0
v o l t s D. C.
T h e s p e e d of t h e m o t o r i s v a r i e d b y a l t e r i n g t h e
c u r r e n t th r o u g h the field w in d in g s a n d by a l t e r i n g th e v o lta g e
a c r o s s th e a r m a t u r e w in d in g s w ith th e h e l p of r h e o s t a t s .
In
a d d itio n , tw o d i f f e r e n t p u ll e y s a r e u s e d to a llo w tw o s te p c h a n g e s .
Figure
10
F.X P F R I M E N T A I . A P P A R A T US
47
; 'm ti/i/ir u /r r im .
s
m
w
n
1
m
m
:
W ////M
(ij STREAMLIMNB MASS TUBt
0
TORSION SPRING
(D HUB
0
GRADUATED RIM
(D
INSULATIMB 5EA LIN B
0
PILOT BEA RIN G
0
SCLF - ALMMIMB BEARHM
0
BALL B E A R A M
0
TOROUE
®
DRIVE PU LLEY
INOtCATOR
FIG
II
S E C T IO N A L
DRAWING
OF
APPARATUS
48
The b la d e s u s e d for h e a t t r a n s f e r m e a s u r e m e n t s a r e
m a d e o u t of a 1" O. D. a n d 0. 0 3 5 " t h i c k s t a i n l e s s s t e e l t u b e ,
f l a t t e n e d with th e a id of s p e c i a l l y m a d e m e t a l d i e s .
T h e h e a t e r e l e m e n t i s s h o w n i n f i g u r e ( 12) .
It is m a d e
by w i n d i n g n i c h r o m e w i r e s 0 . 0 0 8 " i n d i a m e t e r o n a
tran site core.
ra d ia l location.
-7 " t h i c k
lo
E ac h blade h as one h e a te r e le m e n t a t a d iffe re n t
T h e h e a t e r e l e m e n t s c o n s i s t of t h r e e h e a t e r s :
th e m a i n h e a t e r w h ic h is 1 in c h w id e, a n d two g u a r d h e a t e r s
0. 5 i n c h w i d e o n e i t h e r s i d e .
T h e l e a d s f r o m t h e h e a t e r s a r e t a k e n t h r o u g h a s e t of
s l i p - r i n g s to the m e a s u r e i n g an d c o n t r o l d e v i c e s l o c a t e d o u t s i d e
the ro ta tin g s y s t e m .
M ain and V e r n i e r r h e o s t a t s a r e c o n n e c te d
in s e r i e s w it h e a c h h e a t e r e n a b l i n g t h e i n d e p e n d e n t c o n t r o l of
the p o w e r input to e a c h h e a t e r .
T e m p e r a t u r e s on the s u r f a c e of th e b la d e a r e m e a s u r e d
by m e a n s of i r c n - c o n s t a n t a n
as reference.
t h e r m o c o u p l e s w ith m e lti n g ice
F o u r t h e r m o c o u p l e s a r e fixed into the s u r f a c e
of e a c h blade at d if f e r e n t lo c a tio n s ,
one a t the top and one at
t h e b o tt o m a t th e c e n t e r of th e m a i n h e a t e r a n d the o t h e r tw o a t
t h e c e n t e r of e a c h g u a r d h e a t e r .
T h is e n a b l e s the r e g u la tio n
of t h e p o w e r i n p u t t o t h e g u a r d h e a t e r in o r d e r t o g e t i s o t h e r m a l
c o n d itio n s at the b la d e s u r f a c e and e lim in a te the ra d ia l h e at
conduction.
T h e t h e r m o c o u p l e s a r e fixed to th e s u r f a c e of the
b l a d e s by d r i l l i n g h o l e s 0 . 0 4 2 i n c h in d i a m e t e r a n d by s i l v e r i n g
t h e w i r e s t o g e t h e r w i t h th e s u r f a c e .
T w o s e t s of s l i p - r i n g s an d b r u s h e s a r e u s e d to c o n n e c t
the r o t a t i n g s y s t e m an d the s t a t i o n a r y m e a s u r i n g and c o n t ro l
system s.
shaft A.
T h e s l i p - r i n g s a r e m o u n t e d o n t h e t o p of t h e
F i g u r e (13) i s a p h o t o g r a p h o f t h e s l i p - r i n g s a n d b r u s h e s .
T h e c o n t a c t s of th e s lip r i n g s u s e d to t r a n s f e r th e p o w e r
t o t h e h e a t e r s a r e m a d e w i t h b r a s s a n d t h e b r u s h e s a r e m a d e of
F i g u r e 1Z
F ig u re
13
HEATER ELEM ENT
SL IP RIN G-BRU SH A SSEM BLY
50
I
sp rin g loaded c o p p e r im p re g n a te d c a rb o n .
T he p o w e r is
t r a n s f e r r e d w ith o u t a n y s p a r k a n d w ith a r e s i s t a n c e of a b o u t
1. 5 o h m s a c r o s s t h e b r u s h s l i p
rin g c o n ta c t w hich c h a n g e s
slig h tly w ith sp e e d and w e a r .
The s lip r in g s u sed to t r a n s f e r th e r m o c o u p le c o n n e c tio n s
a r e 0. 25 i n c h e s i n d i a m e t e r a n d a r e m o u n t e d o n t h e t o p of t h e
other slip rin g s.
T hese slip -rin g contacts
a r e gold p l a t e d .
Tw o s i l v e r p la te d w ir e b u r s h e s c o n t a c t on e i t h e r s i d e of e a c h
slip ring.
The b r u s h a s s e m b l y is m o u n te d on a s t a t i o n a r y s t e e l
f r a m e a s s h o w n i n f i g u r e ( 11 ).
The w hole a p p a r a t u s is
e n c l o s e d i n s i d e a r o o m 7' by 7 ! , o p e n a t t h e t o p
and bottom .
A l l t h e c o n n e c t i o n s a r e t a k e n to a p a n e l o u t s i d e t h e r o o m
( f i g u r e 14) w h e r e a l l c o n t r o l s a n d m e a s u r e m e n t s a r e a c c o m p l i s h e d .
The t o r s i o n m e t e r i n c o r p o r a t e d in the a p p a r a t u s m e a s u r e s
th e a c t u a l t o r q u e r e q u i r e d by th e b la d e ,
w h ic h is u s e d to
c a l c u l a t e th e d r a g c o e f fic ie n t and to c o m p a r e with th e s k in
friction coefficient.
As a lre a d y stated, a to rsio n sp rin g
c o n n e c t s t h e h o u s i n g a n d s h a f t A , a s s h o w n in f i g u r e (11).
U n d e r s t e a d y s t a t e t h e a m o u n t of d e f l e c t i o n b e t w e e n t h e s h a f t A
a n d t h e h o u s i n g is i n d i c a t e d by t h e r e l a t i o n b e t w e e n a p o i n t e r
f i x e d t o t h e s h a f t a n d a g r a d u a t e d r i m m o u n t e d on t h e h o u s i n g .
T h i s a m o u n t of d e f l e c t i o n i s a m e a s u r e o f t h e t o r q u e r e q u i r e d
to r o t a t e th e b l a d e s and th e hub.
The t o r s i o n m e t e r is c a l i b r a t e d u n d e r s t a t i o n a r y c o n d i t i o n s
using a known fo rc e a c tin g at a known ra d iu s .
T a b l e (1) s h o w s
t h e data a n d *-e8ults of t h e c a l i b r a t i o n .
F i g u r e (15) s h o w s t h e r e l a t i o n b e t w e e n t o r q u e a n d t h e
r e l a t i v e d i s p l a c e m e n t w h e r e e a c h u n it of d i s p l a c e m e n t c o r r e s ­
p o n d s t o 25 d e g r e e s .
It i s s e e n t h a t t h e r e i s a l i n e a r r e l a t i o n
b e tw e e n t o r q u e and d i s p l a c e m e n t e x c e p t a t the h ig h e r v a l u e s
Figure
14
INSTRUMENTATION
t
1- - - - 1- - - - 1- - - - 1- - - - r
480
4 00
MO
240
160
HO L-
•
t>
H
Deflection (divisions)
TORSION M E T E R CA LIBRA TIO N
Figure
15
10
53
of d i s p l a c e m e n t w h e r e th e a m o u n t of t o r q u e n e e d e d fo r a g iv e n
displacem ent in c re a s e s .
T h i s i s due to th e a d d i t i o n of th e
s p r i n g t e n s i o n and f r i c t i o n b e t w e e n th e c o i l s to th e t o r s i o n
value.
THE MEASUREMENTS
T h e s e v e r a l q u a n t i t i e s t o be o b s e r v e d a r e m e a s u r e d
a s follow s:
1)
T h e s p e e d of r o t a t i o n of th e b l a d e s is m e a s u r e d w ith
a s t r o b o t a c which h a s a fo llo w e r,
lo c a te d in s id e the r o o m and
f o c u s s e d on t h e g r a d u a t e d r i m .
2)
of a
T h e p o w e r i n p u t to t h e h e a t e r s
i s m e a s u r e d by m e a n s
D. C. a m m e t e r a n d Et C. v o l t m e t e r .
3)
T h e t e m p e r a t u r e of t h e s u r f a c e o f t h e b l a d e s i s m e a s u r e d
with a n i r o n - c o n s t a n t a n
th e r m o c o u p l e with m e lti n g ic e b ath a s
reference tem p eratu re.
T h e o u t p u t of t h e t h e r m o c o u p l e i s
m e a s u r e d w i t h a k - 3 D. C. p o t e n t i o m e t e r a l o n g w i t h a n e l e c t r o n i c
null - d e t e c t o r .
4)
T h e t o r q u e i s m e a s u r e d by t h e t o r s i o n m e t e r a s
d e sc rib e d previously.
A l l e l e c t r i c a l c o n n e c t i o n s a r e m a d e w i t h 22 g a u g e c o p p e r
w ire.
T h e e l e c t r i c a l c i r c u i t for the m e a s u r e m e n t and the
c o n t r o l i s s h o w n in f i g u r e (16),
EX PER IM EN TA L PROCEDURE
Heat T r a n s f e r M e a s u r e m e n ts
In t h e t h e o r e t i c a l i n v e s t i g a t i o n t h e v i s c o u s h e a t i n g
effects a r e neglected.
viscous d issipation.
But in a r e a l s itu a tio n t h e r e is a l w a y s
In o r d e r t o c o m p a r e t h e e x p e r i m e n t a l
v a lu e s w ith t h e o r e t i c a l v a lu e s the h e a t t r a n s f e r c o e f f ic ie n ts
a r e b a s e d on t h e t e m p e r a t u r e d i f f e r e n c e b e t w e e n w a l l t e m p e r a ­
t u r e and the a d ia b a tic w all t e m p e r a t u r e .
54
iLl £4
z O
o
< < <
«o in
<
z
$
Ui
UJ
ft
o
z
ft
ft
_J
CO
a
z<
H
(/)
C/)
UJ
ft
}
flD
<
ft
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o
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H
Z
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DO 0«2
UJ
m
UJ
g
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o
55
T h e b la d e is r o ta t e d a t th e d e s i r e d s p e e d and the
s p e e d of r o t a t i o n is m e a s u r e d w ith th e s t r o b o t a c .
a s t e a d y s t a t e is r e a c h e d ,
A fter
the a d ia b a tic w all t e m p e r a t u r e at
e a c h l o c a t i o n of t h e t h e r m o c o u p l e i s m e a s u r e d .
The h e a te r
p o w e r is t h e n t u r n e d on a n d th e in p u t p o w e r to e a c h g u a r d
h e a t e r is a d j u s t e d w ith the r h e o s t a t s until th e t e m p e r a t u r e
d i f f e r e n c e s b e t w e e n the c e n t e r of the m a i n h e a t e r a n d the
c e n t r e s of t h e g u a r d h e a t e r s w h i c h a r e o n t h e s a m e s i d e
the b la d e a r e z e r o .
of
T hus the r a d ia l c o n d u c tio n is e l im i n a te d .
U n d e r s t e a d y s t a t e th e t e m p e r a t u r e of a l l th e t h e r m o c o u p l e s
are m easured.
T h e a v e r a g e of t h e d i f f e r e n c e s b e t w e e n t h e h e a t e d
b la d e s u r f a c e t e m p e r a t u r e a t th e c e n t e r of th e m a i n h e a t e r an d
the c o r r e s p o n d i n g a d i a b a t i c w all t e m p e r a t u r e for b o th s i d e s
of t h e b l a d e i s u s e d t o c a l c u l a t e t h e h e a t t r a n s f e r c o e f f i c i e n t .
T h e m a x i m u m t e m p e r a t u r e d i f f e r e n c e i s l i m i t e d t o 7 0 ° F in
o r d e r to p r e v e n t h e a t t r a n s f e r by r a d i a t i o n and n a t u r a l c o n ­
vection.
T he c u r r e n t th r o u g h th e h e a t e r is m e a s u r e d w ith an
am m eter.
P o w e r c a n be e v a l u a t e d by m a s u r i n g t h e c u r r e n t
t h r o u g h t h e h e a t e r a n d t h e v o l t a g e a c r o s s t h e h e a t e r o r by
m e a s u r i n g t h e c u r r e n t a n d k n o w i n g t h e r e s i s t a n c e of t h e h e a t e r .
All m e a s u r e m e n t s a r e done only th ro u g h th e slip rin g and
b r u s h units w hich have a finite c o n ta c t r e s i s t a n c e .
r e s i s t a n c e v a r i e s with s p e e d ,
of c a r b o n on t h e s l i p
rings.
T his
w e a r of t h e b r u s h e s a n d d e p o s i t
T herefore,
t h e m e t h o d to
e v a l u a t e t h e p o w e r by c u r r e n t m e a s u r e m e n t w i t h k n o w n
r e s i s t a n c e is u s e d in o r d e r to a v o i d e r r o r s d u e to c h a n g e
in c o n t a c t r e s i s t a n c e s .
On t h e o t h e r h a n d t h i s m e t h o d h a s
the d is a d v a n t a g e of c h an g in g h e a t e r r e s i s t a n c e with c h a n g in g
tem p eratu re.
S in ce the t e m p e r a t u r e r a n g e s in volved a r e s m a l l
56
a n d f a i r l y c o n s t a n t , t h e e r r o r i n d u c e d i s l e s s t h a n 3%.
T h e v o lta g e a c r o s s th e h e a t e r is m e a s u r e d only fo r c h e c k in g
purposes.
T h e p r o c e d u r e is r e p e a te d for a ll s p e e d s .
E ach run
t a k e s a b o u t 45 m i n u t e s a n d a b o u t h a l f a n h o u r i s a l l o w e d
b e t w e e n s u c c e s s i v e r u n s in o r d e r to r e d u c e t h e f r i c t i o n a l
heating at b ru s h and slip ring c o n ta c ts.
The am bient te m p e r a ­
t u r e is a l s o m e a s u r e d a n d found to be c o n s t a n t w ith i n 1°F .
T o c h e c k t h e r e p r o d u c i b i l i t y of t h e m e a s u r e m e n t s a f e w
of the r u n s a r e r e p e a t e d with d i f f e r e n t h e a t e r p o w e r in p u ts .
T h e r e p r o d u c i b i l i t y i s f o u n d t o be a b o u t + 4%.
H ow ever,
the b la d e s a r e d i s a s s e m b l e d and th e n r e a s s e m b l e d ,
r e p r o d u c i b i l i t y i s l e s s t h a n + 8%.
when
th e
T h i s i s m a i n l y d u e to
d i f f i c u l t y in a c h i e v i n g t h e o r i g i n a l s e t t i n g a n d t h e r e f o r e
i d e n t i c a l a n g l e of i n c i d e n c e .
H E A T T R A N S F E R M E A S U R E M E N T S W ITH T R I P WIRES
T h e t h e o r e t i c a l a n a l y s i s p r e d i c t s t h e e x i s t a n c e of
r a d i a l f l o w a n d t a n g e n t i a l fl o w i n s i d e t h e b o u n d a r y l a y e r
ro ta tin g flat b la d e.
of a
It i s s e e n t h a t t h e i n f l u e n c e o f r a d i a l f l o w
i s s m a l l c o m p a r e d t o t a n g e n t i a l f lo w .
I n a s m u c h a s the te c h n iq u e
u s e d f o r h e a t t r a n s f e r m e a s u r e m e n t s in t h i s s t u d y a r e n o t
s u ffic ie n tly a c c u r a t e to d e te c t su ch s m a l l in f lu e n c e s ,
trip w ires
o f d i f f e r e n t c o n f i g u r a t i o n s m a y b e u s e d t o d e t e r m i n e if t h e r e
i s a n y a p p r e c i a b l e fl o w in r a d i a l d i r e c t i o n .
T r i p w i r e s (22 g a u g e ) a t 7 d i f f e r e n t c o n f i g u r a t i o n s a r e
f i x e d o n t h e h e a t e d s u r f a c e s of t h e b l a d e a s s h o w n i n f i g u r e (17).
T h e i r i n f l u e n c e on h e a t t r a n s f e r is th e n o b s e r v e d a t v a r i o u s
blade s p e e d s .
T h e a n a l y s i s of t h e r e s u l t s p r o v i d e m o r e d e t a i l
a b o u t t h e f l o w in b o t h r a d i a l a n d t a n g e n t i a l d i r e c t i o n s .
57
B
TYPE
A
NO W IR E
u
3
esc
TYPE
G
TYPE F
TYPE E
TYPE
D
C O NFIG URA TIO N S
TYPE
WIRE
C
TRIP
TYPE
58
H E A T T R A N S F E R M E A S U R E M E N T S W ITH P I T C H E D B L A D E S
In o r d e r to i n v e s t i g a t e the i n f l u e n c e of th e a n g l e o f
i n c i d e n c e a n d s e p a r a t e d f l o w on h e a t t r a n s f e r ,
the b la d e s a r e
p i t c h e d a t v a r i o u s a n g l e s f r o m 0 t o 90 d e g r e e s a n d t h e h e a t
t r a n s f e r m e a s u r e m e n t s ta k en a s b e fo re for s e v e r a l s p e e d s .
TORSION M E A S U R E M E N T S
T o rs io n m e a s u r e m e n t s a r e done a f t e r finishing a ll h e a t
tran sfer m easurem ents.
slip rin g s, a r e re m o v e d
The b r u s h e s ,
in c o n ta c t w ith th e
to e lim in a te the fric tio n a l r e s i s t a n c e
of t h e c o n t a c t s .
T h e b la d e s a r e r o t a t e d at the d e s i r e d s p e e d an d the
s p e e d is m e a s u r e d w ith a s tr o b o ta c .
The stro b o ta c a ls o helps
t o r e a d t h e a n g l e of d i s p l a c e m e n t on g r a d u a t e d r i m of t h e
torsion m e ter.
U nder s te a d y s ta te th is re a d in g g iv e s the
a m o u n t of to r q u e for r o ta t in g the b l a d e s .
The low est sp eed
i s 2 0 0 r . p. m . d u e t o t h e l i m i t a t i o n s o f t h e d r i v i n g m o t o r .
T h e m a x i m u m s p e e d i n t h e c a s e o f b l a d e s o f t y p e 1, u s e d f o r
heat tra n sfe r m e a su re m e n ts,
i s 2 2 0 0 r . p. m .
F o r b l a d e s of
t y p e 2, u s e d f o r t o r s i o n m e a s u r e m e n t s a l o n e , t h e m a x i m u m s p e e d
i s 1000 r . p . m .
R e a d in g s a r e ta k e n both a s s p e e d s a r e i n c r e m e n t e d
and d e c re m e n te d .
In a d d i t i o n to t h e f l a t p l a t e t h e c y l i n d r i c a l p o r t i o n s c o n ­
n e c t in g the b la d e s to th e hub a s well a s th e hub i t s e l f a l s o
re q u ir e so m e to rq u e for rotation.
T herefore,
corrections m ust
be m a d e to d e t e r m i n e t h e a c t u a l t o r q u e n e c e s s a r y to r o t a t e
only th e flat b la d e .
T h i s is done by d e t e r m i n i n g the d i f f e r e n c e
in t o r q u e r e q u i r e d to r o t a t e the e n t i r e b l a d e a s s e m b l y a n d th e
t o r q u e r e q u i r e d to r o t a t e a t the s a m e s p e e d
sh ap e but w ithout the flat blade p o r tio n s .
b o d ie s of i d e n t i c a l
H o w e v e r, the d e s c r i b ­
ing e q u a tio n s a r e not l i n e a r and the b o u n d a r y c o n d itio n s a r o u n d
59
(
F ig u re
18
B LA D ES AND C O U N TER PA R TS
A
A'
B lade
Type 1
C o u n te rp a rt for A
yi =
B
B'
B lade 1
Type 2
C o u n te rp a rt for B
y i =
C
C'
D
D'
7. 7 5 "
y 2 = 19. 375
w = 0. 2 c’"
8. 5"
y 2 = 18"
w = 0. 0 6 2 5
B lade 2
Type 2
C ou n terp art for C
T l = 8. 5 ”
y z ~ 18"
w = 0. 0 6 2 5
B lade 3
Type 2
C o u n te rp a rt for D
Tl =
8. 5 "
y 2 = is"
w = 0. 0 6 2 5
60
the f r e e end of th e b la d e a r e a l s o not the s a m e .
But it c a n be
a s s u m e d th a t t h i s p r o c e d u r e e l i m i n a t e s m o s t of th e h ub and
c y lin d ric a l and effect.
A d i f f e r e n t t y p e o f f l a t b l a d e ( t y p e 2), u s e d f o r t o r s i o n
m easurem ents,
is c o n s t r u c t e d f r o m fla t p i e c e s of a l u m i n u m
in c h in t h i c k n e s s . M e a s u r e m e n t s a r e t a k e n on t h r e e s e t s
1b
of s u c h b l a d e s e a c h h a v i n g d i f f e r e n t c h o r d l e n g t h s ; n a m e l y ,
2” , 4 " a n d 6 " .
F i g u r e (18) i s a p h o t o g r a p h o f t h e t w o t y p e s
of b l a d e s ( t y p e 1 a n d 2) a n d t h e i r c o u n t e r - p a r t w h i c h i s u s e d
for c o r r e c tio n .
R E S U L T S AND DATA R ED U C TIO N S
T h e t h e r m o d y n a m i c p r o p e r t i e s of a i r a r e e v a l u a t e d a t
a n a p p r o x i m a t e l y m e a n t e m p e r a t u r e (1 0 0 ° F ) of th e b la d e
s u r f a c e and the a m b ie n t a t m o s p h e r e .
f r o m J a c o b a n d H a w k i n s ( 6 ).
T h ese v alu es a r e taken
They are;
D ensity
p
= 0 . 0 7 1 0 l b m ft^
Specific h eat
K inem atic v isc o sity
C-,
P
u
= 0.2403 B T u / , L
/ lbm
2
= 0. 6 4 9 ft h r
T h e rm a l conductivity
k
= 0 .0 1 5 5 BTU / .
, o„
/ h r ft
r
F
T h e t e m p e r a t u r e d i f f e r e n c e s a r e m e a s u r e d in m i c r o ­
v o lts with i r o n - c o n s t a n t a n t h e r m o c o u p l e s .
d i f f e r e n c e s a r e u s e d in th e c a l c u l a t i o n s .
the e x p e r im e n t,
i8:
Only t e m p e r a t u r e
O v e r t h e r a n g e of
th e c o n v e r s i o n f a c t o r is c o n s t a n t .
Its v a l u e
o
1 F = 2 9 . 28 m i c r o - v o l t s .
T h e h e a t e r c u r r e n t I is m e a s u r e d in a m p e r e s .
The
p o w e r is c a lc u la te d a s :
P = I ZR
w h e r e R is th e r e s i s t a n c e of th e h e a t e r .
T h e h e a t f l u x i n B T l ^ H r f t Z f r o m t h e s u r f a c e of t h e b l a d e
61
i s g i v e n by
_
Q
3.413 „
=
x
p
•
•
*( 8 2 )
w h e r e A is t h e a r e a of th e h e a t e d s u r f a c e .
T h e a v e r a g e h e a t t r a n s f e r c o e f f i c ie n t h in B T U / H r F t
2 o
F
is d e f i n e d in t e r m s of th e a v e r a g e of th e d i f f e r e n c e s b e t w e e n
the h e a te d b la d e s u r f a c e t e m p e r a t u r e a t th e c e n t e r of the m a in
h e a t e r and the c o r r e s p o n d i n g a d ia b a tic w all t e m p e r a t u r e for
b o t h s i d e s of t h e b l a d e a s ,
2 q"w
(Tw -Taw )
+
top
(Tw -Taw )
bottom
..-{ 8 3 )
I n t r o d u c i n g t h e r e s i s t a n c e of t h e h e a t e r s
H eater 1
in B l a d e
1
16. 9 o h m s
H eater 2
in B l a d e 2
1 6. 7 o h m s
H eater 3
in B l a d e
3
16. 3 o h m s
H eater 4
in B l a d e 4
16. 1 o h m s
e q u a t i o n (83) f o r e a c h b l a d e b e c o m e s
154775
I2
BladeI
ui j
Blade 2
B -d .,
hr
W
*"(84)
e 2
i
152943 I
2 = ~(A T)"
e
V
^
'
• ■' ( 8 5 >
2
e
o, . .
BUde 4
w here
t
4 =
14 7448 I 2
U t T
e
■ ■ ■ (87)
(A T)
r e p r e s e n t s th e s u m of th e t e m p e r a t u r e
e
d i f f e r e n c e s in m i c r o v o l t s b e t w e e n t h e h e a t e d b l a d e s u r f a c e
t e m p e r a t u r e a t the c e n t e r of the m a in h e a t e r and the c o r r e s p o n d i n g
a d i a b a t i c w all t e m p e r a t u r e fo r both s id e s of the b lad e.
In o r d e r t o c o m p a r e t h e r e s u l t s a t a l l l o c a t i o n s t h e c h o r d
C i 8 ta k en a s a c h a r a c t e r i s t i c length.
T h is is ju s tif ie d b e c a u s e
2x ^ y w h ic h i s u s e d to e v a l u a t e t h e a v e r a g e N u s s e l t n u m b e r , is
62
n e a r l y e q u a l to C s in c e y>>G .
T h e r e f o r e the a v e r a g e N u s s e l t
n u m b e r i s g i v e n by
Nu = ^
k
. . . (88)
= 8.0645 h
T h e R e y n o l d s n u m b e r i s g i v e n by
Re = &
v
F r o m e q u a t i o n (58) w h i c h e x p r e s s e s t h e R e y n o l d s a n a l o g y ,
. (89)
the s k i n f r i c t i o n c o e f f i c i e n t i s e v a l u a t e d a s
^
^
P r',!
. . . (90,
2
Re
The c a l i b r a t i o n of the t o r s i o n m e t e r gives a c o n s ta n t
c o n v e r s i o n f a c t o r f o r t h e r a n g e of t h e e x p e r i m e n t .
It i s
1 d i v i s i o n * 40 oz i n c h e s .
T h e t o r q u e i s e s t i m a t e d b y m u l t i p l y i n g t h e d i v i s i o n of d i s p l a c e ­
m e n t by 40 t o g e t t h e t o r q u e i n o z i n c h e s .
63
COMPARISION O F T H E O R E T I C A L AND E X P E R I M E N T A L R ESU LTS
H eat T r a n s f e r M e a s u r e m e n t s On A Flat B lade
T h e d a t a and r e s u l t s of h e a t t r a n s f e r e x p e r i m e n t s on b l a d e s
o f t y p e 1 a r e s h o w n i n T a b l e (2).
As s ta te d e a r l i e r ,
the h e a te r s
a r e a t d i f f e r e n t r a d i a l d i s t a n c e s f r o m t h e a x i s of r o t a t i o n .
They
are:
Blade
1
y r
8. 5"
^ - 11.33
b
B lade 2
y = 1 1. 5"
^ = 15. 3 3
b
Blade
5
y , 14. 5 ”
^ = 19. 33
b
Blade
4
v
17. S"
2. - 22. 33
b
y
F i g u r e (19) s h o w s t h e e x p e r i m e n t a l v a l u e o f Nu a n d R e w i t h —
b
as a p aram eter.
Th e t h e o r e t i c a l a n a l y s i s a l s o p r e d i c t s the
y
N u a s a f u n c t i o n of R e a n d — .
b
A s a l r e a d y s t a t e d , t h e i n f l u e n c e of t h r e e - d i m e n s i o n a l
e f f e c t on Nu i s s m a l l f o r t h e b l a d e s u n d e r t h e p r e s e n t i n v e s t i ­
gation.
Therefore,
t h e e x p e r i m e n t a l r e s u l t s of a l l l o c a t i o n s a r e
c o m p a r e d w i t h t h e r e s u l t s of t w o d i m e n s i o n a l f l o w w h i c h a r e
expressed as,
L a m i n a r flow
Nu -
0. b 4 6 b Re
T u r b u l e n t f l ow
Nu -
0.03601
0. 5
Re '
. . . (91)
...(92)
T h e t h i c k l i n e s i n f i g u r e ( 19) a r e t h e p r e d i c t e d v a l u e s a c c o r d i n g
to l a m i n a r and t u r b u l e n t flow a n a l y s i s .
It i s s e e n t h a t t h e r a t e of i n c r e a s e o f Nu w i t h R e i s
p r a c t i c a l l y t h e s a m e a s p r e d i c t e d by t u r b u l e n t f l ow a n a l y s i s .
I n t h e c a s e o f f l a t b l a d e s t u r b u l e n t f l ow i s g e n e r a l l y o b s e r v e d
a t R e of t h e o r d e r 5 x 1 0 .
B u t i n t h e c a s e of t h e r o t a t i n g b l a d e s
o
-o
on
o
o
o
o
Figu re
o
o
o
fM
19
COMP AR ISION
•4
O
o
nN
OF
oo
c
o
00
o
o
o
EXPERIM ENTAL
RESULTS
64
20
o
00
'■M
CORRELATION
OF
r00
o
o
o
o
o
o
o
EXPERIM ENTAL
-C
F ig ur e
RESULTS
65
66
4
t u r b u l e n t f l o w e x i s t s e v e n a t R e o f t h e o r d e r 1. 5 x 10 .
This
m a y be due to h ig h t u r b u l e n c e in the w ake of th e r o t a t i n g
blades.
F i g u r e (19) a l s o s h o w s t h a t t h e e x p e r i m e n t a l v a l u e s
a r e 25% t o 60% l e s s t h a n t h e p r e d i c t i o n s of t h e t u r b u l e n t f l o w
and,
d e p e n d i n g on R e,
f r o m 60% l e s s t o 40% g r e a t e r t h a n
l a m i n a r flow p r e d i c t i o n s .
T h e r e a s o n s fo r d e v i a ti o n s will
be d i s c u s s e d l a t e r .
The e x p e r i m e n t a l r e s u l t s a l s o c o n f i r m the p r e d i c t i o n s
th a t fo r th e s a m e Re th e Nu d e c r e a s e s a s one m o v e s a w a y f r o m
t h e a x i s of r o t a t i o n .
b l a d e n u m b e r 3.
But b la d e n u m b e r 4 h a s h i g h e r v a l u e s th a n
T h is m a y be due to th e end e ffe c t.
A n o n - d i m e n s i o n a l a n a l y s i s c o r r e l a t e s the e x p e r i m e n t s
a n d i s s h o w n in a p p e n d i x ( B - l ) .
T he r e s u l t is
Nu = C , R e m ( - )n
1
w h e r e C^,
blade.
. . . (93)
y
m a n d n a r e c o n s t a n t s a n d C i s t h e c h o r d of t h e
T he c o n s t a n t s a r e d e t e r m i n e d w ith the e x p e r i m e n t a l
v a lu e a s sh o w n in a p p e n d i x (B -2 ).
T h e r e s u l t is
N u = 0. 0 4 2 3 8 R e ' 87 ( - ) ' 704
y
y
704
In f i g u r e (20) N u (•£:)"
is p lotted a g a i n s t R eynolds n u m b e r.
L/
The r e s u l t s c o r r e l a t e fa irly w ell.
The end effect in c r e a s e d
t h e h e a t t r a n s f e r c o e f f i c i e n t s o f t h e b l a d e n u m b e r 4.
This
i n c r e a s e g iv e s a n i n c r e a s e in th e c o r r e l a t e d v a l u e s .
Hence,
the e x p e r i m e n t a l v a l u e s of th e b la d e n u m b e r 3 lie lo w e r th a n
the c o r r e l a t e d r e s u l t s .
B l a d e s 1, 2, a n d 4 a g r e e f a i r l y w e l l
with the c o r r e l a t i o n s .
H E A T T R A N S F E R M E A S U R E M E N T S W ITH T R I P WIRES
The d a ta a n d r e s u l t s of the h e a t t r a n s f e r e x p e r i m e n t s
with 7 d i f f e r e n t p a t t e r n s of t r i p w i r e s fixed to the s u r f a c e a r e
s h o w n i n T a b l e 3.
T h e r e s u l t s a r e p l o t t e d i n f i g u r e s (21),
(22),
v
t-4
O
a a t x t x c x c x a ^
O
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71
(23) a n d (24) f o r b l a d e s
1,
2,
3, a n d 4 r e s p e c t i v e l y .
T h e r e s u l t s of a ll th e b l a d e s sh o w the s a m e t r e n d .
i n f l u e n c e of e a c h p a t t e r n of t h e t r i p w i r e (fig.
The
17) o n t h e
m e a s u re d heat tra n s fe r coefficients a r e ex am in ed .
w i r e of ty p e A d o e s no t sh o w an y n o tic e a tie in flu e n c e .
The trip
T his
in d ic a te s th at the t r i p w ir e d o e s not c a u s e a p p r e c i a b l e ch a n g e
in th e flow p a t t e r n a n d t r a n s i t i o n f r o m l a m i n a r to t u r b u l e n t
flow .
A lso,
t h e r e a l r e a d y e x i s t s a t u r b u l e n t flow o v e r the
e n t i r e w id th of th e b l a d e a s a r e s u l t o f hi gh t u r b u l e n c e in th e
f r e e s t r e a m a n d t h e b l u n t l e a d i n g e d g e of t h e b l a d e .
The type B trip w ire show s sig n ific a n t influence; h eat
t r a n s f e r c o e f f i c i e n t s a r e i n c r e a s e d a b u t 1 00 % ,
T his in d icates
t h e h i g h i n f l u e n c e of t h e t a n g e n t i a l flow' o n t h e f l o w i n t h e
b o u n d a ry l a y e r of a r o ta tin g b la d e .
T he t r i p w ir e of ty p e C d o e s not sh o w an y n o tic e a b le in flu e n c e .
T h i s i n d i c a t e s th a t th e r a d i a l flow h a s v e r y s m a l l i n f l u e n c e on
t h e flow in t h e b o u n d a r y l a y e r of a r o t a t i n g b l a d e in a g r e e m e n t
with the a n a l y s i s .
T h e t r i p w i r e o f t y p e D,
w h i c h i s a m o d i f i e d t y p e B,
is
d e s i g n e d t o e x a m i n e t h e i n f l u e n c e on t h e f l o w p a t t e r n d u e t o a
s l i g h t c h a n g e in t h e o r i e n t a t i o n of t y p e R.
R esults show that
ty p e D a l s o e x e r t s a h i g h i n f l u e n c e on flow in a m a n n e r s i m i l a r
t o t y p e B.
But the in f lu e n c e v a r i e s s li g h t ly w ith o r ie n t a t io n
w h i c h i s hi e r e a s o n for t h e s c a t t e r
in t h e v a l u e s c o r r e s p o n d i n g
t o t h e t y p e s B a n d D i n f i g u r e s 21 t o 24.
T h e t r i p w i r e of t y p e E is fix ed to th e b l a d e to e x a m i n e t h e
i n f l u e n c e on th e flow p a t t e r n d u e to a c h a n g e in th e o r i e n t a t i o n
of th e ty p e C.
R e s u l t s s h o w th a t a s m a l l c h a n g e in o r i e n t a t i o n
h a s n e g lig ib le influe
jic
c Oil i l l c f l o w p a t t e r n . T h u s t y p e s C a n d E
i n d i c a t e t h a t th e i n f l u e n c e of r a d i a l flow is s m a l l .
72
I
T r i p w i r e s of t h e t y p e
F
and
o u t if t h e r e i s a n y r a d i a l flow a t a ll .
G
a r e u s e d to find
R e s u l t s of t y p e s F
a n d G s h o u ld be e q u a l if t h e r e is no r a d i a l flow .
It is s e e n t h a t th e v a l u e s of t h e m e a s u r e m e n t s w i t h t r i p
w i r e s of ty p e F a r e h i g h e r th a n w ith t r i p w i r e s of ty p e G a n d
both a r e h i g h e r th a n th e m e a s u r e d v a lu e s fo r th e flat b la d e
alone.
T herefore,
does exist.
t h e s e r e s u l t s i n d i c a t e t h a t r a d i a l flow
I n a s m u c h a s th e s e t r i p w i r e s a r e in flu e n c e d by
t a n g e n t i a l flow t h e r e m a y be s e p a r a t i o n w ith a r a d i a l flow
c o m p o n e n t b e in g c r e a t e d by c e n t r i f u g a l a n d c o r i o l i s f o r c e s .
Thus,
th e o r i g i n of th e r a d i a l flow is u n c e r t a i n .
H E A T T R A N S F E R M E A S U R E M E N T S W ITH P IT C H E D B L A D E S
T h e d a t a a n d r e s u l t s of t h e h e a t t r a n s f e r m e a s u r e m e n t s
w i t h d i f f e r e n t p i t c h a r e s h o w n i n T a b l e (4 ).
c o m p a r e the r e s u l t s
F i g u r e 25 t o 28
(Nu a n d Re) of th e f o u r b l a d e s fo r a ll
th e a n g l e s of p itc h .
T h e r e s u l t s of a ll th e b la d e s a r e sh o w n to h a v e p r a c t i c a l l y
the s a m e tr e n d .
degrees,
At s m a l l a n g le s of p itc h ,
n a m e l y 1, 3, a n d 5
the r e s u l ts c a n be g ro u p e d ro u g h ly into 3 r e g io n s
a s follow s:
R egion I
R e < 3 x 10
R e g i o n II
3x 104 < R e < 7 x 10
R e g i o n III
4
4
R e > 7 x 104 .
In r e g i o n I t h e s l o p e o f t h e c u r v e s j o i n i n g t h e p o i n t s i s a b o u t
0. 5,
in r e g i o n II,
w h ich a p p e a r s to be a t r a n s i t i o n r e g io n ,
t h e r e is no d e f in ite v a lu e f o r th e s lo p e ,
a n d i n r e g i o n III t h e s l o p e
i s a b o u t 0. 8 5 .
T h is i n d ic a te s th a t r e g i o n I r e p r e s e n t s the
l a m i n a r flow ,
a n d r e g i o n II r e p r e s e n t s t h e t r a n s i t i o n f r o m
l a m i n a r t o t u r b u l e n t f l o w , a n d r e g i o n III r e p r e s e n t s t h e t u r b u l e n t
flow .
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77
A t h i g h e r a n g l e s of p i t c h , t h e s l o p e f o r e a c h p a r t i c u l a r
angle is constnat.
B u t , a s t h e a n g l e of p i t c h i s i n c r e a s e d t h e
s l o p e of t h e c u r v e d e c r e a s e s .
T h e s l o p e of t h e s e c u r v e s
v a r i e s f r o m 0. 5 a n d 0. 8.
When the b l a d e s a r e p i t c h e d t h e r e is both l a m i n a r and
t u r b u l e n t flow a l o n g t he s i d e f a c i n g the s t r e a m a n d s e p a r a t e d
f l ow a l o n g t h e o p p o s i t e s i d e .
Therefore,
i n c r e a s e d a n g l e s of
p i t c h i n c r e a s e s t h e r e g i o n s of l a m i n a r f l o w a n d t h u s r e d u c i n g t h e
s l o p e of t h e c u r v e .
T h i s a l s o i n d i c a t e s t h a t t h e r a t e of i n c r e a s e of
t h e h e a t t r a n s f e r c o e f f i c i e n t s in a s e p a r a t e d flow is not m o r e t h a n
that
of t u r b u l e n t f l o w.
The p o w e r n e e d e d to r o t a t e the b la de i n c r e a s e s c o n s i d e r a b l y
with i n c r e a s e d p i t ch and t h e r e f o r e ,
t h e r a n g e of Re i s l i m i t e d
l o r l a r ge ' a n g l e ' s of i n c i d e n c e ' .
TORSION M E A SU R E M E N T S
T h e t o t a l d r a g ( p r o f i l e d r a g ) on a b o d y p l a c e d i n a s t r e a m
of f l u i d c o n s i s t s ol s k i n f r i c t i o n d r a g ( e q u a l t o t h e i n t e g r a l of a l l
s h e a r i n g s t r e s s o v e r t h e s u r f a c e of t h e b o d y ) a n d
form
d r a g ( i n t e g r a l of n o r m a l f o r c e s ) .
w h i c h d o e s not
The form drag,
or p re s s u re
e x i s t m f r i c t i o n l e s s flow', i s d u e t o t h e f a c t t h a t t h e p r e s e n c e of t h e
b o u n d a r y l a y e r m o d i f i e s t h e p r e s s u r e d i s t r i b u t i o n on t h e b o d y a s
c o m p a r e d w i t h t h e i d e a l f l ow,
bu t it c o m p u t a t i o n i s d i f f i c u l t .
A s a r e s u l t of v i s c o u s l o s s e s i n t h e b o u n d a r y l a y e r ,
the
p a r t i c l e s c l o s e t o t h e b o d v s u r f a c e 1 d o no t h a v e s u f f i c i e n t
ki n e t i c e n e r g y to o v e r c o m e the a d v e r s e p r e s s u r e g r a d i e n t al ong
t h e r e a r of t h e b o d y .
T h i s r e s u l t s in t h e r e v e r s a l of f low
d i r e c t i o n and s e p a r a t i o n w h ic h c a u s e s a low p r e s s u r e r e g io n
on t h e r e a r s i d e a n d t h e r e f o r e ,
bodies,
drag.
p re ssu re drag.
F o r blunt
t h i s t y p e of d r a g i s m u c h m o r e t h a n t h e s k i n f r i c t i o n
I n e v e r y t u r b u l e n t b o u n d a r y l a y e r a n e x c h a n g e of m a s s
a n d m o m e n t u m ta k e p l a c e a c r o s s the s t r e a m l i n e s .
This
e x c h a n g e r e p r e s e n t s a c o n t i n u o u s m o m e n t u m t r a n s p o r t f r o m the
o u t e r f l o w t o w a r d s t h e s u r f a c e of t h e b o d y .
Thus the r e s u lta n t
l o s s e s a r e h i g h e r than in th e l a m i n a r c a s e .
However,
near
th e s u r f a c e of the body the t u r b u l e n t b o u n d a r y l a y e r c a r r i e s
m u c h m o r e m o m e n t u m than the l a m i n a r b o u n d ary la y e r and
therefore,
th e flow a t t a c h e s to the s u r f a c e for a l o n g e r d i s t a n c e
a n d the d r a g d e c r e a s e s .
A particularly rem arkable phenomenon,
the t r a n s i t i o n f r o m l a m i n a r to t u r b u l e n t flow,
c o n n e c t e d with
o c c u r s in the
c a s e of b l u n t b o d i e s s u c h a s c y l i n d e r s o r s p h e r e s .
The d ra g
coefficient su ffers a sudden and considerable d e c re a s e n ear
5
5
R e y n o l d s n u m b e r 3 x 10 t o 5 x 10 . A s a c o n s e q u e n c e
separation moves dow nstream .
T h i s r e s u l t s in a d e c r e a s e in
th e low p r e s s u r e r e g io n a n d t h e r e f o r e r e d u c e s the p r e s s u r e d r a g .
T h e t h e o r e t i c a l a n a l y s i s p r e d i c t s o n l y t h e t o r q u e d ue to
skin friction drag,
and tu rb u len t flows.
g i v e n b y e q u a t i o n s (57) a n d (79) f o r l a m i n a r
It i s a l s o p o s s i b l e t o p r e d i c t t h e s k i n
f r i c t i o n t o r q u e f r o m h e a t t r a n s f e r m e a s u r e m e n t s by a s s u m i n g
R ey n o ld s analog y as shown below.
The heat t r a n s f e r m e a s u r e m e n t s a r e c o r r e l a t e d as:
Nu :
0. 0 42 38 Re
0. 87
C 0. 704
( -)
y
A c c o r d i n g to Reynolds analogy
Nu =
Hence
Cf
—
J l
2
Re
PrJ/ ?
1 /30,0,704
P r (- )
4
y
T h e to r q u e due to f r ic t io n a l d r a g is giv en a s
y2
=
A A J , , 0 r. - 0 . 1 3
0 . 0 4 2 3 8 Re
. . . (94)
1600
1800
2000
79
1400
OO
1200
OO
OO
200
400
bOO
800
1000
rg
o
o
o
~r-
o
Ln
cv]
o
O
o
(q n n
o
o
l
T.
rsj
—i
•70)
—<
nb
j
o j
o
>r
80
1
/
3 n l . 8 7 ^ 0 . 13c 0 . 5 7 4
= 0.04238 P r
T h e s k i n f r i c t i o n t o r q u e g i v e n by e q u a t i o n s ( 5 7 ) ,
. . .(96)
( 7 9 ) , a n d ( 96)
a n d to t a l t o r q u e d e t e r m i n e d by th e e x p e r i m e n t a l m e a s u r e m e n t s
a r e s h o w n in T a b l e ( 5 ) .
A c o m p a r i s i o n of a l l t h e s e v a l u e s a r e
d o n e in f i g u r e ( 29) .
It i s s e e n t h a t t h e t o r q u e e v a l u a t e d f r o m h e a t t r a n s f e r
m e a s u r e m e n t s a r e l e s s than the t h e o r e t i c a l p r e d i c t i o n s for
t u r b u l e n t flow but h i g h e r than that f o r l a m i n a r flow.
It i s
a l s o s e e n t h a t the p r o f i l e d r a g is m u c h h i g h e r t ha n the s k i n
friction drag.
An a v e r a g e d r a g c o e f f i c i e n t i s d e f i n e d ,
b a s e d on t h e a r e a
n o r m a l to t h e f l o w a s
M
where
M
is the tot al m e a s u r e d t o r q u e f o r a l l b l a d e s .
W i s t h e t h i c k n e s s of t h e b l a d e .
T a b l e ( 6 ) s h o w s t h e v a l u e s of t h e d r a g c o e f f i c i e n t s .
F i g u r e ( 30)
s h o w s t h e v a l u e s of d r a g c o e f f i c i e n t a s a f u n c t i o n of
a v e r a g e R e y n o l d s n u m b e r d e f i n e d by
- Re
(V1 + 72 )
= ------------
Q W
—rr-
. . . (98;
F i g u r e (3 1) s h o w s t h e d r a g c o e f f i c i e n t s f o r t w o d i m e n s i o n a l
fl o w a r o u n d c y l i n d e r s a n d th r e e - d i m en s l o r a l f l o w a r o u n d s p h e r e s
a n d c i r c u l a r d i s c s a s g i v e n in t h e l i t e r a t u r e .
A s s e e n in f i g u r e (3 1) t h e d r a g c o e f f i c i e n t of t h e r o t a t i n g
b l a d e i s l e s s t h a n t h o s e of c y l i n d e r s a n d s p h e r e s a t t h e s a m e
Reynolds num ber and rem a in s fairly constant.
This indicates
t h e e x i s t a n c e of t u r b u l e n t f l o w e v e n a t m u c h s m a l l e r R e y n o l d s
«
i-
I
Figure
n
H)
IZ
Ri- \
AVERAGE
^ u.
ZO
2.)
28
in
Cd
VERSUS A V ERAG E
Re
100
SO
ZO
Rota ting Blade
. 0Z
10
“
i0
1
Re
F'i mi r e U
DRAG C O E F F I C I E N T VERSUS REYNOLDS NUMBER
82
w
D
a
I
cd
0
H
J
O
o
o
<
H
O
H
l \
o
Q
O'
cd
U
o
D
cn
1 ‘
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\
U
2
o
o
oc
Q
<
w
c
o
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r-
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cd
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c
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U
c
(
o
o
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o
o
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X3
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Uh
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Cl
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W
Q
Cd
H
U
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Q
cd
CL
o
o
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TJ
3
13
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fNJ
3
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o
z
Q
1-4
CO
<
Cl
2
O
u
rv)
rO
u
L,
3
QC
( e a q a u i zo) a n b a o x
>*
H
[3
Q
2
UJ
ro
w
nq
d
0
Ui
83
n u m b e r s . T h i s i s p o s s i b l e a s a r e s u l t of t h e h i g h i n t e n s i t y
o f t u r b u l e n c e in t h e w a k e a n d t h e b l u n t l e a d i n g e d g e .
Also,
t h e c e n t r i f u g a l a n d c o r i o l i s f o r c e s i n d u c e r a d i a l f l o w in t h e
wake and in c re ase
the b a c k p r e s s u r e .
T h e m a j o r c o n t r i b u t i o n of t h e p r e s s u r e d r a g o n p r o f i l e
d r a g d i d n o t g i v e a n y c h e c k on t h e t h e o r e t i c a l r e s u l t s of s k i n
friction coefficient.
It is p o s s i b l e to r e d u c e t h e p r e s s u r e d r a g
by r e d u c i n g t h e t h i c k n e s s - t o - c h o r d r a t i o .
Therefore,
three
s e t s of b l a d e s o f t y p e 2 ( s e e f i g u r e 18) a r e u s e d .
T a b l e (7) s h o w s t h e t o r q u e d u e t o s ki , , f r i c t i o n d r a g
a c c o r d i n g to l a m i n a r a n d t u r b u l e n t f l o w t h e o r i e s a s g i v e n by
e q u a t i o n s ( 57) a n d ( 7 9 ) , a n d a l s o t h e m e a s u r e d t o r q u e .
Figure
( 32) s h o w s t h e c o m p a r i s i o n of t h e s e v a l u e s .
It i s s e e n t h a t t h e t o t a l d r a g is h i g h e r t h a n t h e s k i n f r i c t i o :
drag.
In t h e c a s e of b l a d e s w i t h Z" c h o r d t h e m a x i m u m p e r c e n ­
t a g e d e v i a t i o n b a s e d on m e a s u r e d t o t a l t o r q u e i s 32%.
For
4 ” c h o r d b l a d e s it i s 24% a n d f o r 6 " c h o r d b l a d e s it is 13%.
T h e s e d e v i a t i o n s a r e d u e to t h e c o n t r i b u t i o n of t h e p r e s s u r e
drag,
v i b r a t i o n of t h e b l a d e s a t h i g h s p e e d s , a s s u m p t i o n s i n t h e
theoretical analysis,
a n d e r r o r in t h e e x p e r i m e n t s .
Despite
t h i s d i s a g r e e m e n t t h e m e a s u r e m e ’ ts c o n f i r m t h a t the i n f l u e n c e
of t h e p r e s s u r e d r a g on p r o f i l e d r a g d e c r e a s e s a s t h e r a t i o
of t h i c k n e s s t o c h o r d d e c r e a s e s .
E X P E R I M E N T A L ERROR ANALYSIS
N,, p h y s i c a l m e a s u r e m e : t c a n b e k n o w n t o i n d e f i n i t e l y
high p r e c i s i o n .
O n e of t h e p r i m a r y c o n c e r n s of t h e e x p e r i ­
m e n t a l i s t is a d e t e r m i n a t i o n of t h e e r r o r a s s o c i a t e d w i t h h i s
m easurem ents.
F o r th i s the u n c e r t a in i ti e s in e a c h m e a s u r e d
v a ria b le m u s t be known.
84
In t h e p r e s e n t w o r k ,
the following a r e the m e a s u r e d
v a ria b le s and th e ir uncertainties:
Current
+
0. 005 a m p
Voltage
+
0. 05 v o l t s
Tem perature
+
1 microvolts
Speed
+
10 r . p . m .
Torque
+
2. o z .
Resistance
f
0. 1 o h m
inches
T h e s p e c i f i c a t i o n s of t h e i n s t r u m e n t s u s e d t o m e a s u r e t h e s e
v a r i a b l e s a r e g i v e n in A p p e n d i x E.
T h e l i m i t of f r a c t i o n a l e r r o r i n h,
calculated.
Nu,
Re ,
andM
are
T h e t o t a l e r r o r i s a s s u m e d to b e t h e s u m of t he
e r r o r s for e a c h i n d e p e n d e n t v a r i a b l e .
The heat t r a n s f e r
c o e f f i c i e n t h is g i v e n a s ,
I R
h = c o n s t (— )
.. f , ( 9 9 )
w h e r e I is t h e c u r r e n t
R i s the r e s i s t a n c e
(AT)
h
+AT- . ) in m i c r o v o l t s
1
c
2 dl _ f d R _ H A T i 4A_T^)
I
R
(AT
+ AT^)
e
is ( A T
’ •*(
U '
A p p l y i n g t he r u l e s f o r l i m i t of e r r o r c a l c u l a t i o n s
dh ... . “ I
h
I
I dR
R
I d( A T ,
( A T i + A T 2)
nol)
• '' 1 '
T h e u n c e r t a i n t y dR m R is c a u s e d by t h e u n c e r t a i n t y in t h e
m e a s u r e m e n t of t h e r e s i s t a n c e a n d a l s o by t he c h a n g e m t h e
r e s i s t a n c e d u e t o c h a n g e of t e m p e r a t u r e .
An e x a c t e v a l u a t i o n
of t h e l a t t e r is d i f f i c u l t a n d t h e r e f o r e i t s m a x i m u m v a l u e i s
e s t i m a t e d ( a p p e n d i x B - 3 ) a n d a s s u m e d to be c o n s t a n t .
This
i s j u s t i f i e d b e c a u s e v a r i a t i o n s in t h e p o w e r i n p u t t o t h e
h e a t e r s a n d t h e s u r f a c e t e m p e r a t u r e s of t h e b l a d e s a r e s m a l l .
85
With th i s a s s u m p t i o n ,
it i s p o s s i b l e t o e x p r e s s t h e f r a c t i o n a l
d h— i n v,
error —
h as
h
dh_ dg
. . . ( 102)
+ c
h
g
d g d e p e n d s upon the m e a s u r e d i n d e p e n d e n t v a r i a b l e s
where —
g
I and ( A T ) , , and c is a c o n s t a n t d e p e n d i n g upon the r e s i s t a n c e
dh
of t h e h e a t e r .
—— i s e x p r e s s e d a s
d h _ dg
. . . ( 103)
Blade 1
.02592
h _ g
dh
dg
...(104)
.02599
Blade 2
d
” g
dh
Blade 1
dg
h " g
dh
dg
Blade 4
h
02 61 3
. . . ( 105)
02621
. . . ( 106)
g
d^g a s a f u n c t i o n of ( A T ) (> a n d c u r r e n t I.
F i g u r e (3 3) s h o w s —
T h e N u s s e l t n u m b e r is g i v e n a s
hC
Nu
—
k
d Nu
dh
Nu
h
sin ce C and k a r e c o n s t a n t s .
...(8
8)
. . (107)
Thus,
t h e e r r o r in Nu i s t h e s a m e
a s i n h.
T h e R e y n o l d s n u m b e r is
y il C
Rfc
v
dN
d Re
Re
‘ ~N
. . . (89)
. . . (108)
w h e r e N i s s p e e d in r, p. m .
F i g u r e ( 34) g i v e s t he f r a c t i o n a l e r r o r
d Re
ln ^ r a s a f u n c t i o n
o: N .
T h e t o r q u e 19
M -
40 D
. . . ( 109)
dM _
M _
dD
D
. .. (1 1 0 )
w h e r e D i s t h e d i v i s i o n s of t h e d e f l e c t i o n .
F i g u r e ( 35 ) g i v e s t h e
86
. 06
. 04r~
. 02
1500
1000
2 0 00
A T in m i c r o - v o l t s
F i g u r e 33
8
1^ v e r s u s A T
g
2500
87
10
. 08
. 06
dRe
Re
. 04
.
02
4 00
8 00
1200
1600
2000
2400
4 00
4 80
N ( r . p. m . )
F i g u r e 34
dRe
Re
VERSUS N
. 24
dM
. 08
80
F i g u r e 35
160
dM
M
240
320
M ( oz i n c h e s )
VERSUS M
88
fractional e r r o r
i n M a s a f u n c t i o n o f t o r q u e M_
U s i n g e q u a t i o n s (103),
e r r o r i n h,
Nu,
( 108) a n d ( 110) t h e f r a c t i o n a l
R e a n d M e a n be c a l c u l a t e d .
It i s s e e n t h a t t h e
e r r o r in h a n d N u d e c r e a s e s a s t h e h e a t e r c u r r e n t i s i n c r e a s e d
a n d t h e i n f l u e n c e of t o t a l t e m p e r a t u r e d i f f e r e n c e i s v e r y s m a l l .
T h e e r r o r in t o r q u e M a n d Re d e c r e a s e s h y p e r b o l i c a l l y w i t h
i n c r e a s e d M and Re.
F o r t h e p r e s e n t s t u d y t h e e r r o r in h a n d
Nu v a r i e s f r o m 4% t o 8%,
and the e r r o r m
t h e e r r o r i n Re f r o m 1. 5% t o 0. 5%
M f r o m 12% t o 1%.
E X P L A N A T I O N FOR DEVIATION O F E X P E R I M E N T A L
R E S U L T S F R O M T H E O R E T I C A L ANALYSIS
The e x p e r i m e n t a l a p p a r a t u s d o e s not m e e t al l t h e
r e q u i r e m e n t s ol t h e t h e o r e t i c a l m o d e ] .
As a r e s u l t ,
t he
experimental
r e s u l t s do not a g r e e w e l l w i t h t h e t h e o r e t i c a l
predictions.
The r e a s o n for such d e v i a tio n s a re :
1)
E a c h b l a d e r u n s i n t h e w a k e of t h e p r e c e d i n g o n e .
Therefore,
t h e r e l a t i v e v e l o c i t y ol t h e f r e e s t r e a m w i t h r e s p e c t
t o t h e b l a d e i s no t y!2 b u t yl2 - U' w h e r e U 1 i s t h e d e f e c t in t h e
v e l o c i t y in t h e w a k e .
T h i s r e d u c e d v e l o c i t y c a u s e s a dec r e a s e
in t he h e at t r a n s f e r coettic lent.
2)
The heat t r a n s f e r coefficients a r e evaluated using
( T w - T a w ) as the t e m p e r a t u r e d i f f e r e n c e w h e r e Taw is the
adiabatic
w a l l t e m p e r a t u r e w h e n t h e b l a d e s a r e not h e a t e d .
H e a t i n g of t h e h i a d e s me r e a s e s the' t e m p e r a t u r e ot t he tree'
s t r e a m w h i c h i n c r e a s e s in t u r n t h e a d i a b a t i c
Therefore,
wall t e m p e r a t u r e .
the- e v a l u a t e d h e a t t r a n s f e r c o e f f i c i e n t s a r e l e s s
than the a c t u a l v a l u e s .
As a r e s u l t of high h e a t t r a n s f e r
c o e f f i c i e n t s at h i gh s p e e d s the p o w e r input to t he h e a t e r s
m u s t be i n c r e a s e d t o g e t a p p r e c i a b l e t e m p e r a t u r e d i f f e r e n c e s .
89
F o r the a bo ve m e n t i o n e d r e a s o n t h i s c a u s e s a d e c r e a s e in
the heat t r a n s f e r co efficien ts ev a lu a te d .
T h e h e a t i n g in a d d i t i o n
m a y c h a n g e the flow p a t t e r n du e to v i s c o u s e f f e c t s .
Al l t h e s e
i n f l u e n c e s m i g h t be t h e r e a s o n f o r d e c r e a s e i n t h e r a t e of c h a n g e of
N u w i t h R e a t h i g h v a l u e s of R e .
1)
T h e e l e c t r i c a l h e a t e r s m a y no t be in g o o d c o n t a c t w i t h
t h e e n t i r e i n n e r s u r f a c e of t h e b l a d e .
This c a u s e s a d e c r e a s e
i n t h e h e a t t r a n s f e r c o e f f i c i e n t a s a r e s u l t of d e c r e a s e in
effective heat t r a n s f e r s u r f a c e , and c o n t a c t r e s i s t a n c e s .
4)
T h e h e a t t r a n s l e r c o e f f i c i e n t is b a s e d on t h e a v e r a g e of
t he d i i t e r e n c e s b e t w e e n the h e a t e d b l a d e s u r f a c e t e m p e r a t u r e
at t h e c e n t e r ot t h e m a i n h e a t e r a n d t h e c o r r e s p o n d i n g a d i a b a t i c
w a l l t e m p e r a t u r e t o r b ot h s i d e s of t h e b l a d e .
T h i s m a y not
r e p r e s e n t an a v e r a g e t e m p e r a t u r e d i l t e r e i u e to r the e n t i r e
surface.
5)
The a n a l y s i s n e g l e c t s both end and hub e f f e c t s ,
b)
T h e r e i s h e a t t r a n s f e r by r a d i a t i o n a n d n a t u r a l
convection,
but t h i s i s r e d u c e d t o a m i n i m u m by h a v i n g s m a l l
11“r n p e r a t u r e d i f f e r e n c e s .
7)
The s u r i a e e c ha r a < t e r i s 11 c s m a y a l s o i a u s e d e v i a t i o n s .
•S)
[ l i e a n a 1\ s i s e v a l u a t e s o n l y s k i n i r i i t i<>n t • r<] ue .
>
a s a r e s u l t of t h e t i m t e t h i c k n e s s o f t h e b l a d e s ,
d r a g ini r e u s e s t h e t o t a l d r a g .
Bu t ,
the p r e s s u r e
T h i s c a u s e s t he i n c r e a s e in t he
!n e a s u r e r! t ■■r f) u e .
(J)
i l i e d e t e i t i n ve l uc l t v i n t h e w a k e a l s o i m r e a s e s t h e
skin friction coefficient.
90
COMPARISION WITH OTHER HEAT EXCHANGERS
T h e p e r f o r m a n c e of v a r i o u s h e a t e x c h a n g e r s m a y be
c o m p a r e d in s e v e r a l w a y s ,
d e p e n d i n g on the o b j e c t i v e .
If
i n v e s t m e n t c h a r g e s a n d p o w e r c o s t s n e e d e d be c o n s i d e r e d ,
the
c o m p a r i s i o n is m a d e p e r u ni t a r e a of h e a t - t r a n s f e r s u r f a c e .
however,
c o m p a c t n e s s is m o s t im p o rta n t,
If,
c o m p a r is io n should
be m a d e on unit v o l u m e of t h e h e a t e x c h a n g e r .
In t h i s w o r k , t h e c o m p a r i s i o n s a r e m a d e o n u n i t a r e a of
t h e h e a t t r a n s f e r s u r f a c e a s s u g g e s t e d by M c A d a m s (7).
P
i s a p l o t o f H s t d v e r s e s ( - )g t d P
w h e r e ( —) gt d =
(h)
-
s td
The power,
H o r s e p o w e r p e r s q u a r e f oot of h e a t
a r e a at stan d ard conditions.
A verage heat tr a n s f e r coefficient
c onditions.
This
transfer
at s t a n d a r d
n e e d e d by the r o t a t i n g b l a d e s a r e c a l c u l a t e d
f r o m the to r s io n m e a s u r e m e n t s .
P=
2-*-N- M
33000
’
. ( 111)
1
'
whe r e
N i s s p e e d i n r . p. m .
M i s t o r q u e in
pou n d s feet.
P is p o w e r in H. P .
The friction power,
g i v e n by t h e t u r b u l e n t f l ow t h e o r e t i c a l a n a l y s i s
a n d g i v e n by R e y n o l d s a n a l o g y f r o m h e a t t r a n s f e r m e a s u r e m e n t s
a r e also evaluated.
F r o m heat t r a n s f e r and pow er r e q u i r e m e n t
P
d a t a ( — ) , a n d (h)
, a r e c a l c u l a t e d a s follows:
A std
std
^ A
( P / A )S t d
std
/
pe s t d '
C£_
e_
»3{ P s t d v2
'
r( P r ),td
Pr
. . . (1 12)
2/3
...(113)
91
RUFFLED
FINS
0,0776
0.066
LOUVERED
PLATE
FINS
o .o :
CODE:
PLAIN
PLATE
o. i
FINS
It.M
FINNED
FLAT
TUBES
0
INSIDE
CIRCULAR
TUBES
COOC:
18. 3“
»T —I
HEAT
EXCHANGE
FIGURE 3 6
SURFACES
.
COMP ARI S I ON
OF
HEAT
EXCHANGE
SURFACES
92
OG
oo
<]►
"O
o
o
o
oo
o
o
OO
o
o
o
oj
o
pi e
93
The standard values are:
n=
C p = 0. 2 4 8 ,
0.0678,
P r = 0.671
p = 0.0413
P
T a b l e ( 8 ) s h o w s t h e v a l u e s o f ( — )8t(j a n d h g ^ j f o r v a r i o u s
A
speeds.
Several heat tra n sfe r surface configurations a r e selected
from Kays,
L o n d o n a n d J o h n s o n (18).
i n f i g u r e ( 36) .
T h e i r g e o m e tr y is shown
F i g u r e (37) s h o w s t h e c o m p a r i s i o n o f c o m p a c t
e x c h a n g e r s u r f a c e s o n e q u a l a r e a b a s i s a s s h o w n i n M c A d a m s (7).
It i s s e e n t h a t t h e p o w e r r e q u i r e m e n t s o f t h e r o t a t i n g b l a d e
is h i g h e r t h a n the c o m p a c t h e a t e x c h a n g e r s .
P o w e r m e a s u r e d by
to r s io n m e a s u r e m e n t s a r e hig h er than the friction p ow er,
obtained
f r o m t u r b u l e n t flow a n a l y s i s a n d f r o m R e y n o l d s a n a l o g y o n h e a t
transfer m easurem ents.
ments,
A s d i s c u s s e d in t h e t o r s i o n m e a s u r e ­
t h i s i s d u e t o t h e c o n t r i b u t i o n of p r e s s u r e d r a g .
It s h o u l d
be n o t e d t h a t t h e p o w e r r e q u i r e m e n t s for o t h e r c o m p a c t h e a t
exchangers,
c o m p a r e d h e r e do not t a k e into a c c o u n t the p o w e r
l o s s in t he d u c t s ,
h e a d e r s and o t h e r flow p a s s a g e s .
F r o m t h e t o r s i o n m e a s u r e m e n t s it i s s e e n t h a t t h e p r e s s u r e
d r a g c a n be c o n s i d e r a b l y r e d u c e d
n e s s to c h o r d w idth.
by r e d u c i n g the r a t i o of t h i c k ­
T h i s i n d i c a t e s t h a t t h e p r o p e r d e s i g n of t h e
b l a d e s will m a k e the r o t a t i n g b l a d e h e a t e x c h a n g e s m o r e a d v a n t a ­
g e o u s t h a n o t h e r c o n v e n t i o n a l h e a t e x c h a n g e r s b e c a u s e of t h e i r
s i m p l i c i t y in c o n s t r u c t i o n ,
tra n s fe r coefficients.
lo w e r initial c o st and high h e a t
94
IN FLUENCE OF ELECTRICAL FIELD
T h e i n f l u e n c e of the i n t e r a c t i o n s of fluids a n d e l e c t r i c a l
l i e l d s on h e a t t r a n s f e r a r e n o w i n v e s t i g a t e d .
An e l e c t r i c a l
f iel d c a n i n f l u e n c e not only fl ui ds c o n t a i n i n g c h a r g e d p a r t i c l e s
and co nducting fluids,
well.
bu t n e u t r a l n o n - c o n d u c t i n g f l u i d s a s
Thi s w i d e r a n g e of a c t i o n m a y p r o v i d e c o n t r o l l a b l e b o d y
f o r c e w i t h i n t h e f l u i d w i t h o u t h i g h d e g r e e of i o n i s a t i o n ,
generally
u s e d m M a g n e t o - hyd r o dy na mi c s ,
E l e c t r o s t at
11
Influence
1'he s i m p l e s t a c t i o n of a n e l e c t r i c a l f i el d is t h a t of t he
f o r c e e x e r t e d on-sr c h a r g e d p a r t i c l e in t h e f l ui d.
are electrons,
negative ions,
Such p a r ti c le s
c o l l o i d s a n d i m p u r i t i e s in s o l u t i o n s .
E l e c t r i c a l field a l s o p o l a r i s e s and i n d u c e s d i p o l e s in n o n ­
conducting d ie le c tric s.
Bu t s o m e m o l e c u l e s h ave1 p e r m a n e n t
d i p o l e s w h i c h a r e i n d e p e n d e n t of t h e e l e c t r i c a l f i e l d .
The
e l e c t r i c a l d i p o l e s ( i n d u c e d o r p e r m a n e n t ) p r o v i d e a m e a n s of
a p p l y i n g < out r i d l e d l ore e to a n e u t r a l m o l cm ul e .
O b s e r v e d I n t e r a c t i o n s of E l e c t r i c a l F i e l d
A tew d ! t h e o b s e r v e d i n t e r a c t i o n s a r e :
I'1
I hi' m o 1 1 1 1 1 •, ot a cfcndl e ! 1a m e b e t w e e n i b a r g e d h i g h
p o t e n t i a l p l a t e s of a c o n d e n s e r c a u s e d by t h e d r i f t of t h e i o n s
in t h e h i g h t e m p e r a t u r e f l a m e .
2)
E l e c t r o l y t i c a c t i o n i n a s o l u t i o n d u e t o t h e d r i f t of
io ns to the an ode and c a t h o d e .
1)
Electric
wind,
d u e t o t h e l a r g e n u m b e r of i o n s c r e a t e d
at a p o i n t a s a r e s u l t of i n t e n s e e l e c t r i c a l f i e l d a n d d r i f t of
95
»
t h e s e io ns in the e l e c t r i c a l field.
( T h e r e is m o m e n t u m
t r a n s f e r betw ee n the ions and the n e u tra l m o l e c u le s ,
a n o t h e r w i s e s t a t i o n a r y g a s to flow.
causing
A c o r o l l a r y e f f e c t is the
e l e c t r i c a l wind p r e s s u r e g e n e r a t i o n . )
4)
T h e K e r r e f f e c t w h ic h r e f e r s to the c h a n g e in the
r e f r a c t i v e i n d e x of l i q u i d u n d e r t he i n f l u e n c e of e l e c t r i c a l
fields.
5)
T h i s is u s e d f o r h i gh s p e e d l i g h t s h u t t e r s in K e r r c e l l s .
E l e c tr i c a l p ro p u ls io n w h e re the e l e c tr i c fields a c c e l e r a t e
the ions p ro d u cin g v e r y high th ru st.
I N F L U E N C E O F E L E C T R I C A L F I E L D ON F L U I D
P R O P E R T I E S AND T R A N S P O R T P H E N O M E N A
T h e n a t u r e of t h e f l u i d p r o p e r t i e s i s f u n d a m e n t a l t o a n y
i n v e s t i g a t i o n of the flui d b e h a v i o r .
viscosity,
th e rm a l conductivity,
Important properties are
and diffusivity.
C h a n g e of
t he fluid p r o p e r t i e s c h a n g e s t he v e l o c i t y a n d t e m p e r a t u r e
d is tr ib u tio n s and th e r e f o r e ,
tra n s fe r coefficients.
c h a n g e s the d r a g and the h eat
D o b i n s k i ( 8 ) o b s e r v e d i n c r e a s e s in
v i s c o s i t y of p o l a r l i q u i d s w h e n u n i f o r m e l e c t r i c a l field is
a p p l i e d to the flo w in g liquid .
T h is e f fect d e p e n d e d on the
i m p u r i t y a n d v a r i e d w i t h t h e s q u a r e of t h e e l e c t r i c a l f i e l d
strength.
A n d a r d e (1) c o r r o b o r a t e d t h e e a r l i e r f i n d i n g s of
D o b i n s k i (8 ) i n a s e r i e s o f v e r y c a r e f u l e x p e r i m e n t s .
e x p e r i m e n t s w e r e l i m i t e d to u n i f o r m fi el d s .
All
T h e c h a n g e in
v i s c o s i t y w a s e x p l a i n e d a s b e i n g d u e to t h e a c c u m u l a t i o n of
c h a r g e wi t hi n the l i qui d.
liq u id s led to c l u s t e r i n g ,
T h e a c t i o n of t h e c h a r g e s on t h e
polar
a n d a s u b s e q u e n t i n c r e a s e in v i s c o s i t y .
S c h m i d t a n d L e i d e n f r o s t (29) o b s e r v e d s m a l l i n c r e a s e s in t h e r m a l
c o n d u c t i v i t y o f l i q u i d s a n d l a r g e i n c r e a s e in h e a t t r a n s f e r
c o e f f i c i e n t s of n o n - c o n d u c t i n g l i q u i d s .
A s t r o n g i n f l u e n c e is
96
e x p e c t e d on d i f f u s i v i t y d u e t o a m b i p o l a r e f f e c t .
Th e a b o v e d i s c u s s e d i n t e r a c t i o n s can in f l u e n c e the p r o ­
p e r t i e s of t h e b o u n d a r y l a y e r w h i c h e x i s t s b e t w e e n a m o v i n g
r e a l fluid and i t s b o u n d a r y .
T h e d e v e l o p m e n t of t h i s b o u n d a r y
l a y e r depends upon the v is c o s ity ,
conditions,
l o c a l flow v e l o c i t y ,
d e n s it y and m a n y o t h e r p a r a m e t e r s .
surface
The transition
of t h i s b o u n d a r y l a y e r f r o m l a m i n a r t o t u r b u l e n t a l s o d e p e n d s
upon the s u r f a c e c o n d i t i o n s ,
fluid p r o p e r t i e s ,
t h e i n t e n s i t y of
t u r b u l e n c e of t h e e x t e r n a l f l o w a n d p r e s s u r e g r a d i e n t .
Local
c h a n g e s in fluid p r o p e r t i e s a n d a d d i t i o n a l body f o r c e s due to
e l e c t r i c a l field c a n h a v e i n f l u e n c e on s e p a r a t i o n a nd r e a t t a c h ­
ment.
The i n t e r n a l h e a t g e n e r a t i o n due to o h m i c h e a t i n g ,
and
c h a n g e in t h e t e m p e r a t u r e a n d v e l o c i t y d i s t r i b u t i o n s d u e t o
c h a n g e in f l u i d p r o p e r t i e s m o d i f y t h e h e a t t r a n s f e r a n d d r a g
coefficients.
influences.
Only a c c u r a t e e x p e r i m e n t s c a n i n d i c a t e s u ch
A few of t h e e x p e r i m e n t a l r e s u l t s a r e d i s c u s s e d n e x t .
S e n f t l e b e n a n d B r a u n (32) s h o w e d t h a t a n i n c r e a s e i n h e a t
t r a n s f e r r a t e u p t o 50%.
electric gas,
c a n be o b t a i n e d in t h e c a s e of p a r a -
en closed betw een a h o rizo n tal w ire and a c o n ­
centric cylinder.
The effect was negligibly s m a l l when the
gas was d ie le c tric .
T h e a u t h o r s a t n h u t e d t h i s t o t h e c h a n g e in
c i r c u l a t i o n c u r r e n t due to e l e c t r o s t r i c t i v e f o r c e s .
K r o n i g a n d S c h w a r z (19) a l s o s h o w e d a n i n c r e a s e in h e a t
t r a n s f e r rate from a h o riz o n ta l w ir e and a c o n c e n tric
c y l i n d e r due
to e l e c t r o s t r i c t i v e f o r c e s .
and
They used a rg o n ,
oxygen,
ethylc h l o r i d e and i n t r o d u c e d a new c h a r a c t e r i s t i c n u m b e r ,
s i m i l a r to G r a s h o f f n u m b e r to c o r r e l a t e t h e i r r e s u l t s .
I y e y a n d L e e (16) r e p e a t e d t h e e x p e r i m e n t s of S c h w a r z
using m o ist a i r .
e l e c t r i c field.
N o a p p r e c i a b l e c h a n g e w a s p r o d u c e d by t h e
97
A s h m a n n a n d K r o n i g (2), a n d a l s o D e H a n n (7) c o n d u c t e d
s e v e r a l e x p e r i m e n t s with p o l a r liquids ( n - h e p t a n e ,
carbon tetrachloride).
h - hexane,
They concluded that the heat t r a n s f e r
r a t e i n c r e a s e s in t h e s a m e f a s h i o n a s w a s f o u n d i n g a s e s .
L e i d e n f r o s t (21) m e a s u r e d h e a t t r a n s f e r c o e f f i c i e n t a n d
p r e s s u r e d r o p in a c o u n t e r - f l o w type h e a t e x c h a n g e r with w a t e r
at one s id e and t r a n s f o r m e r oil at the o t h e r .
The e le c tr ic a l
f i e l d s t r e n g t h o n t h e oil w a s i n c r e a s e d up t o 70 k. v / c m . a n d
t h e h e a t t r a n s f e r c o e f f i c i e n t i n c r e a s e d by 4 0 0 %, b u t t h e p r e s s u r e
d r o p a l o n g t h e oi l s i d e i n c r e a s e d by 40 %.
The e l e c tr i c a l pow er
l o s s in m a i n t a i n i n g t he f ield w a s a p p r o x i m a t e l y one m i l l i o n t h
ot t h e p u m p i n g p o w e r n e e d e d t o a c h i e v e a n e q u a l i n c r e a s e i n
h e a t t r a n s f e r by i n c r e a s i n g t h e v e l o c i t y of t h e o i l ,
V e l k o f f (38) s t u d i e d t h e e f f e c t of c o r o n a d i s c h a r g e
( e l e c t r i c a l w i n d ) on n a t u r a l c o n v e c t i v e h e a t t r a n s f e r .
his e x p e r i m e n t a l work,
B a s e d on
he c o n c l u d e d t h a t the e l e c t r i c a l field
c a n h a v e s i g n i f i c a n t i n f l u e n c e on f r e e c o n v e c t i o n .
The actions
o b s e r v e d w e r e q u i t e c l e a r l y t h e r e s u l t of c o r o n a w i n d i m p i n g i n g
on t h e h e a t e d p l a t e r a t h e r t h a n d u e t o e l e c t r i c a l f i e l d o r t h e r m a l
gradient.
T h i s c o n c l u s i o n w a s s u p p o r t e d b o t h by e v a l u a t i o n of
t h e e x p e r i m e n t s a n d by t h e c o r r e l a t i o n a c h i e v e d w i t h a n a l y s i s
which a s s u m e d that the a c t i o n was due to c o r o n a wind.
At a t m o s p h e r i c c o n d i t i o n s t h e a i r b e h a v e s l i k e a d i e l e c t r i c
gas.
Hence,
t h e i n f l u e n c e of e l e c t r i c a l f i e l d on p r o p e r t i e s is
not a p p r e c i a b l e .
The p o s s i b i l i t y to io n i s e a i r u n d e r a m b i e n t
c o n d i t i o n s i s l e s s d u e to r e l a t i v e l y h i g h p r e s s u r e a n d l o w
t e m p e r a t u r e and th e r e f o r e ,
c h a n g e in p r o p e r t i e s due to i o n i z a t i o n
c a n be e x p e c t e d to be i n s i g n i f i c a n t .
But c o r o n a wind m a y h a v e s o m e i n f l u e n c e u n d e r t h e s e
conditions.
To study such effect,
corona discharge realtions
98
a r e d e t e r m i n e d in f o r c e d c o n v e c t i o n w h e r e the d r i v i n g f o r c e is
m u c h l a r g e r c o m p a r e d to n a t u r a l c o n v e ctio n .
C H A R A C T E R I S T I C S O F C O R O N A D I S C H A R G E IN A N A I R S T R E A M
T h e e l e c t r i c a l d i s c h a r g e a t p o i n t w a s d e s c r i b e d by
H a r n e y (12) t o t a k e p l a c e i n v a r i o u s d i f f e r e n t p h a s e s a s f o l l o w s :
1)
C o n d u c t i o n b e lo w i o n i z a t i o n by c o l l i s i o n .
F o r low a p p l i e d v o l t a g e the c u r r e n t flow d e p e n d s upo n the
i o n s p r o d u c e d by e x t e r n a l s o u r c e s .
In n o r m a l a t m o s p h e r i c
a i r the m a g n i t u d e of t h i s c u r r e n t is m i n u t e a n d d i s c o n t i n u o u s .
Z)
C o n d u c t i o n w i t h i o n i z a t i o n by c o l l i s i o n .
For h i g h e r a p p lie d v o l ta g e s and therefore.,
h ig h field
s t r e n g t h s the e l e c t r o n s r e s u l t i n g f r o m e x te r n a l io n iz a tio n gain
s u f f i c i e n t e n e r g y in t h e f i el d t o c a u s e f u r t h e r i o n i z a t i o n by
collision.
3)
T h e c u r r e n t flow i n c r e a s e s .
G eiger counter region
F o r still h i g h e r field s t r e n g t h ,
a single e x te rn ally
p r o d u c e d ion r e s u l t s in an a v a l a n c h e in which s ufficie nt r a d ia tio n
is p r o d u c e d to c a u s e f u r t h e r i o n i z a t i o n in the g a s .
A large
c u r r e n t p u l s e b u i l d s up to a point w h e r e the s p a r e c h a r g e
d i s t o r t i o n o f t h e f i e l d c h o k e s t h e p r o c e s s o f i o n i z a t i o n by
c o llis io n and the d i s c h a r g e c e a s e s .
4)
Continuous corona regim e
F or yet
higher
a p p li e d v o l t a g e s the s p a c e c h a r g e d i s ­
t o r t i o n c a n n o t r e d u c e t he field s t r e n g t h to the point of c h o k i n g
and th e re r e s u l ts a s e lf - s u s ta in e d d is c h a r g e .
The visible
c o r o n a now a p p e a r s about the point.
5)
B reakdow n stage
T h e field s t r e n g t h m a y be i n c r e a s e d to the p o i n t w h e r e
the b r e a k d o w n of the gas e x t e n d s a c r o s s the e l e c t r o d e gap.
99
I n i t i a l l y t h i s m a y be in the f o r m of p r e - b r e a k d o w n s t r e a m e r s .
F i n a ll y the c o r o n a d i s c h a r g e c u l m i n a t e s in an a r c d i s c h a r g e .
T h e only p h a s e of d i s c h a r g e w h i c h is c o n s i d e r e d h e r e is
the c o n t i n u o u s one.
ELECTRIC
FIELD -C O R O N A DISCHARGE RELATIONS
In a n a ly z in g the c o r o n a d i s c h a r g e p h e n o m e n o n ,
it is
e s s e n t i a l to o b t a i n r e l a t i o n s h i p s c o v e r i n g the field s t r e n g t h ,
cu rren t distribution,
and the induced p r e s s u r e
rise.
The
following a s s u m p ti o n s a r e made:
1)
O n l y o n e k i n d of i o n i s p r e s e n t o v e r m o s t t he
discharge space,
2)
e x h i b i t i n g on ly one v a l u e of m o b i l i t y .
T h e i o n d e n s i t y is s o h i g h t h a t t h e s p a c e c h a r g e
c o n d i t i o n s d e t e r m i n e f i e l d d i s t r i b u t i o n a n d i on c u r r e n t f l o w .
3)
T h e i o n i z i n g a n d a c c e l e r a t i n g e l e c t r o d e s do not
i n t e r f e r e wi th t h e p r e s s u r e b u i l d up a n d t h e h y d r o d y n a m i c flow.
4)
T u r b u l e n c e in the fluid is n e g l i g i b l e .
5)
Volume ch arg e density P
and m o b ility K a r e s c a l a r .
T h e e l e c t r i c field e q u a tio n s a r e
~
Pc
V. E - —
c
=
F
where a
. . . (1 14)
a Cr E + e c Prt
V
. . . (1 15)
p c E
...(lib)
is e l e c t r i c a l c o n d u c t i v i t y .
T h e total v e lo c i ty V
of t h e c h a r g e
(ions) is e q u a l to
the v e c t o r s u m of th e v e l o c i t y of ions r e l a t i v e to the g a s a n d the
v e l o c i t y of the g a s .
V
=
K E +V
w h e r e V is gas v e lo c ity .
T h e c u r r e n t d e n s i t y is
. . . (1 17)
100
J *
O
c
( KE + V)
. . . (118)
A s a r e s u l t of t h e c h a r g e a n d t h e e l e c t r i c a l fi e l d ,
is an e le c tr ic a l body force.
Consequently,
there
the Navier-Stok.es
e q u a tio n is w r i t t e n as
du
i
P ~ ;—
dt
where
d P
E - ——
=
3 x.
u
V
2
u .+
i
i
A = ^ Ui ,
A
3x
d
——
3
i
an d F = p E
"a—
a Xj
...(119)
c
A c c o r d i n g to the b o u n d a r y l a y e r a s s u m p t i o n s the p r e s s u r e a c r o s s
the b o u n d a ry l a y e r is c o n s ta n t.
Therefore,
the p r e s s u r e
rise
d u e to the e l e c t r i c a l field will o c c u r onl y in the r e g i o n of the
a i r w h e r e e s s e n t i a l l y n o n v i s c o u s p o t e n t i a l flow o c c u r s .
In
th is re g io n the N a v i e r S to e k s eq u a tio n r e d u c e s to
VP=Pc E
...(120)
T h e s e p r e s s u r e f o r c e s a c t on t h e e d g e of t he b o u n d a r y l a y e r an d
r e d u c e its th ic k n e s s and t h e r e b y i n c r e a s e the heat t r a n s f e r
coefficient.
A n e s t i m a t e of the m a x i m u m p o s s i b l e p r e s s u r e ,
due to e l e c t r i c a l f o r c e s ,
wi ll i n d i c a t e t h e m a g n i t u d e of t h e
p o s s i b l e i n f l u e n c e on h e a t t r a n s f e r .
C o m b i n i n g e q u a t i o n s (114) an d (119)
VP
E =■ Tp—
p ^ - fV.
. . . (121 )
In o r d e r t o s i m p i l i y t h e e s t i m a t e
that E and P a r e u n i d ir e c t io n a l .
t
XT —
dE =
E
dy
Integrating between y -
P - P
. . . ( 122)
yields
y ^ a n d y = y,
= - 5L
it is a s s u m e d
E q u a t i o n (121) t h e n r e d u c e s to
d—P
dy
«
o
turther,
r
[ E
2
-E
2,
o
]
. . . ( 123)
10 1
where P
a n d E a r e the v a l u e s of p r e s s u r e a n d e l e c t r i c a l
o
o
f i e l d s t r e n g t h a t y = y . A s s t a t e d by T h o m s o n (34), E = 0
if i t i s a s s u m e d t h a t t h e i o n s o u r c e i s a p l a n e i o n i z e d l a y e r .
Therefore,
the m a x i m u m p r e s s u r e d i f f e r e n c e w hich c a n be
p r o d u c e d due to e l e c t r i c a l field follows as:
(AP)
max
-
-f2
E
2
...(124)
max
The above relation holds true strictly for e le c tro sta tic p r e s s u r e
r i s e in a d i e l e c t r i c b e t w e e n t w o p a r a l l e l p l a t e s .
hold p r e c i s e l y for a c o r o n a d i s c h a r g e .
It d o e s n o t
However,
S t u e t z e r ( 33)
i n d i c a t e s t h a t t h e r e l a t i o n s h i p s of t h e f o r m
(ap) -
42 -
e Zm a x
hold quite well for c o r o n a d i s c h a r g e .
(F) is a c o n s t a n t w h i c h
d e p e n d s on the g e o m e t r y w h e n th e fluid v e l o c i t i e s a r e n e g l i g i b l e .
But the d i s t r i b u t i o n of the body f o r c e and the e l e c t r i c a l field
changes at higher a ir velocities.
Tw o s i m p l e c a s e s with the
e l e c t r i c a l field p a r a l l e l and n o r m a l to the d i r e c t i o n of a i r
f l o w a r e d i s c u s s e d in A p p e n d i x C.
The ( A P)
g i v e n by e q u a t i o n (124) is e v a l u a t e d in
max
a p p e n d i x ( A - 17) a n d f o u n d t o be 0 . 0 1 3 4 i n c h e s o f w a t e r .
This
p r e s s u r e f o r c e is c o n s i d e r a b l y low c o m p a r e d to the p r e s s u r e
f o r c e s i n f o r c e d c o n v e c t i o n b u t s i g n i f i c a n t i n t h e c a s e of
natural
convection.
H E A T T R A N S F E R E X P E R I M E N T S WITH E L E C T R I C A L F I E L D
T h e e x p e r i m e n t a l s e t up u s e d f o r all h e a t t r a n s f e r m e a s u r e ­
m e n t s is m o d ifie d ,
a s s e e n in f i g u r e
(38),
to v e r i f y the
i n f l u e n c e of e l e c t r i c a l fie ld o n h e a t t r a n s f e r in a i r .
The co ro n a
i g u r e 38
MODIFIED EXPERIM EN TA L APPARATUS
103
d i s c h a r g e i s p r o d u c e d a s a r e s u l t of h i g h e l e c t r i c a l field
a r o u n d t he s h a r p e d g e of a c i r c u l a r r i m e l e c t r o d e ,
ab o v e the h e a t e r s in the b la d e,
mounted
by 6 h o l d e r s s o t h a t i t s c e n t e r
c o i n c i d e s w i t h t h e a x i s o f r o t a t i o n of t h e b l a d e s .
The d istance
b e t w e e n t h e e l e c t r o d e a n d t h e b l a d e c a n be v a r i e d by m o v i n g
t h e h o l d e r up a n d d o w n .
detail.
F i g u r e (39) s h o w s t h e h o l d e r s i n
The b la d e is g r o u n d e d and the v a r i a b l e high v o lt a g e is
a p p l i e d to t h e e l e c t r o d e .
F O R C E D C O N V E C T I V E HE A T T R A N S F E R E X P E R I M E N T S
WITH E L E C T R I C A L F I E L D
F o r c e d c o n v e c t i v e heat t r a n s f e r e x p e r i m e n t s a r e done
as e x p l a i n e d e a r l i e r without any e l e c t r i c a l field.
T h e n t he
p o t e n t i a l of the e l e c t r o d e is r a i s e d a n d the b l a d e s u r f a c e
t e m p e r a t u r e s a r e m e a s u r e d without ch an g in g the h e a t e r p o w e r
input.
Th e p r o c e d u r e is r e p e a t e d at i n c r e a s e d high p o t e n t i a l s
until a r c d i s c h a r g e o c c u r s .
T a b l e (9) s h o w s t h e o b s e r v a t i o n s a t d i f f e r e n t s p e e d s a n d
d i s t a n c e b e t w e e n the b l a d e s and th e e l e c t r o d e .
D u r i n g t h e r u n s 1,
2,
and 3 a r c d i s c h a r g e o c c u r e d before
any c o r o n a d i s c h a r g e c u r r e n t flows.
D u r i n g r un 4 t h e r e is a
fl ow of 100 pt a m p c o r o n a d i s c h a r g e c u r r e n t .
The o b s e rv a tio n s
i ndi c a t e t h a t t h e rl< . t r i c a l f i e l d a n d c o r o n a d i s c h a r g e h a v e
ne g! i gi id i' i n f l u e n c e <m f o r c e d c o n v e c t i v e h e a t t r a n s f e r .
NATURAL CONVECTIVE HEAT TRANSFER
E X P E R I M E N T S WITH E L E C T R I C A L F IE L D
E v a l u a t i o n of t h e p r e s s u r e f o r c e d u e t o e l e c t r i c a l f i e l d
i n d i c a t e d t h e p o s s i b i l i t y f o r s t r o n g i r f l a e n c e on n a t u r a l
convective heat tra n s fe r.
Therefore,
natural convective
104
::
— - 2 0 B r a s s Rod
4
( Al l T h r e a d )
1"
1"
— x — Polystyrene
2
2
1" X —
3
Copper Electrode
V
F i g u r e 39
ELECTRO D E HOLDER
105
e x p e r i m e n t s a r e c a r r i e d out with the b l a d e in a s t a t i o n a r y
h o r i z o n t a l p o s iti o n with and w ithou t the e l e c t r i c a l field and
corona discharge.
E x p e r i m e n t s a r e r e p e a t e d by c h a n g i n g t h e
d is ta n c e b etw een the blade and the e le c tr o d e ,
pitching the b la d es.
a n d a l s o by
T a b l e (10) s h o w s t h e o b s e r v a t i o n o f a l l
these experim ents.
No s i g n i f i c a n t c h a n g e i s o b s e r v e d w h e n t h e r e i s n o
corona discharge.
B u t a c o r o n a d i s c h a r g e c u r r e n t of 30
m i c r o a m p e r e s a t 30. k. v. r e d u c e s t h e t e m p e r a t u r e d i f f e r e n c e
betw een the a t m o s p h e r e and the heated blade fro m 39.65 ° F
t o 9. 39
F for the s a m e h e a t e r p o w e r input.
With i n c r e a s e d
d i s t a n c e b e t w e e n the blade and the e l e c t r o d e this in fluence
decreases.
Also,
t h e c h a n g e in t h e t e m p e r a t u r e d i s t r i b u t i o n
a s a r e s u l t of p i t c h i n g d i d no t s h o w a n y n o t i c e a b l e i n f l u e n c e .
T h e s e r e s u l t s c o n f i r m t h e f i n d i n g s o f V e l k o f f (38)
a n d t h e e a r l i e r c o n c l u s i o n s of t h i s s t u d y .
106
CONCLUSIONS
The b oundary la y e r and m o m e n t u m in te g ra l equations for
flow a r o u n d a r o t a t i n g flat b l a d e a r e d e v e l o p e d i n a r o t a t i n g
stream line co-ordinate system .
These equations a r e simplified
a n d s o l v e d by a s s u m i n g p r o p e r v e l o c i t y p r o f i l e s t o g iv e l o c a l
and a v e r a g e skin fric tio n c o e ffi c ie n t s for both l a m i n a r and t u r ­
b u l e n t f l o w.
The heat t r a n s f e r c o efficien ts a r e e s t i m a t e d using
Reynolds analogy.
The th e o re tic a l a n a ly sis shows the ex ista n c e
of r a d i a l a n d t a n g e n t i a l f l o w i n t h e b o u n d a r y l a y e r .
The influence
of t h e r a d i a l f l o w on s k i n f r i c t i o n c o e f f i c i e n t s a n d h e a t t r a n s f e r
c o e f f i c i e n t s a r e found to be s m a l l .
Th e t u r b u l e n t flow a n a l y s i s
p r e d i c t s v e r y l o w i n f l u e n c e of r a d i a l f l o w c o m p a r e d t o l a m i n a r
flow.
T h e a n a l y s i s a l s o p r e d i c t s t h a t t h e i n f l u e n c e of t h e r a d i a l
flow i n c r e a s e s a s one m o v e s t o w a r d s th e a x i s of r o t a t i o n and
a lo n g the t a n g e n tia l d i r e c t i o n f r o m the le a d in g edge.
The heat t r a n s f e r coefficients d e t e r m in e d e x p e r i m e n t a lly
a r e le s s th a n the v a l u e s p r e d i c t e d u s in g R eynolds an ^ l °g y •
The
q u a l ita t iv e e x p e r i m e n t s with t r i p w i r e s i n d i c a t e s the s m a l l
i n f l u e n c e of r a d i a l f l o w.
The experim ental resu lts also indicate
t h a t t h e a v e r a g e v a l u e of h e a t t r a n s f e r c o e f f i c i e n t f o r t h e s a m e
R e y n o l d s n u m b e r i n c r e a s e s a s t h e a x i s of r o t a t i o n i s a p p r o a c h e d .
M e a s u r e m e n t s on flat b l a d e s i n d i c a t e t h e e x i s t a n c e of t u r 4
b u l e n t f l o w e v e n a t R e y n o l d s n u m b e r s o f t h e o r d e r 1. 5x10 . T h i s
is due to high t u r b u l e n c e in th e f r e e s t r e a m an d the blunt le a d in g
e d g e of t h e b l a d e s .
H e a t t r a n s f e r m e a s u r e m e n t s on p i t c h e d b l a d e s i n d i c a t e
t h a t t h e r a t e o f i n c r e a s e of h e a t t r a n s f e r c o e f f i c i e n t s i n t h e
107
I
s e p a r a t e d flow r e g i o n is s m a l l e r t h a n t h a t in t he t u r b u l e n t
p o r t i o n of t h e f l o w ,
region.
an d is g r e a t e r t h a n t h a t in t h e l a m i n a r
When the b l a d e s a r e p i t c h e d l a m i n a r , t u r b u l e n t , and
s e p a r a t e d flows e x i s t s i m u l t a n e o u s l y a r o u n d the bl ad e.
fore,
There­
t h e n e t r a t e of i n c r e a s e of h e a t t r a n s f e r c o e f f i c i e n t s i s
l e s s t h a n t h a t of t u r b u l e n t f l ow.
T o r s i o n m e a s u r e m e n t s on the flat b l a d e s of t y p e 1 u s e d f or
heat t r a n s f e r m e a s u r e m e n t s indicate that the d ra g coefficients
b a s e d on a r e a n o r m a l to flow a r e f a i r l y c o n s t a n t a n d l o w e r t h a n
t h o s e of c y l i n d e r s a n d s p h e r e s a t t h e s a m e R e y n o l d s n u m b e r s .
T h i s m a y be due to t u r b u l e n t flow e v e n at low R e y n o l d s n u m b e r s
a n d r a d i a l flow in t h e w a k e .
T o r s i o n m e a s u r e m e n t s o n b l a d e s of t h e t y p e i w i t h d i f f e r e n t
w
—
( t h i c k n e s s to c h o r d r a t i o ) i n d i c a t e t h a t t h e p r o f i l e d r a g
W
c a n b e r e d u c e d by r e d u c i n g t h e v a l u e of — .
L/
T h e o r e t i c a l and e x p e r i m e n t a l w o r k show that the c e n t r i ­
f u g a l a n d c o r i o l i s f o r c e s h a v e v e r y s m a l l i n f l u e n c e on h e a t
t r a n s f e r c h a r a c t e r i s t i c s of r o t a t i n g f l a t b l a d e s .
It i s a l s o f o u n d
t h a t t h e c o r o n a d i s c h a r g e h a s s i g n i f i c a n t i n f l u e n c e on n a t u r a l
c o n v e c t i v e h e a t t r a n s f e r and n e g l i g i b l e i n f l u e n c e on f o r c e d c o n ­
vective heat tra n s fe r.
In o r d e r t o i n v e s t i g a t e t h e p r a c t i c a l a p p l i c a t i o n of r o t a t i n g
flat b l a d e s a s h ea t e x c h a n g e r s ,
t h e p o w e r r e q u i r e m e n t s of t h e s e
r o t a t i n g b l a d e s u s ed for he at t r a n s f e r m e a s u r e m e n t s a r e c o m p a r e d
w i t h t h e f r i c t i o n p o w e r r e q u i r e m e n t s of c o m p a c t h e a t e x c h a n g e r s
on u nit a r e a b a s i s .
T h e y a r e found to be h i g h e r .
d u e t o t h e c o n t r i b u t i o n of t h e p r e s s u r e d r a g ,
w h i c h c a n be r e d u c e d
c o n s i d e r a b l y t h r o u g h p r o p e r d e s i g n of th e b l a d e s ,
n e g l e c t o f l o s s e s in h e a d e r s ,
compact heat exchangers.
T h i s m a y be
and a l s o to the
i n l e t d u c t a n d o t h e r f l ow p a t h s of
108
A s a r e s u l t of t h e s e s t u d i e s r o t a t i n g b l a d e s a r e found
to have the following adv an tag es:
1)
C o n s tru c tio n is s im p le
2)
Initial co st is s m a ll
3)
T h e r e is no need for duct w o r k an d fan to blow the
coolant o v e r the h e a t t r a n s f e r a r e a .
4)
H i g h f l o w s p e e d s of t h e c o o l a n t a n d h i g h h e a t t r a n s f e r
c o e f f i c i e n t s c a n be a c h i e v e d .
5)
T h e r e is l i t t l e r e s t r i c t i o n on t h e i r a d a p t a b i l i t y .
109
RECOMMENDATIONS
T h e p r e s e n t i n v e s t i g a t i o n e x p l a in s the flow and the
heat t r a n s f e r p h e n o m e n a a r o u n d a ro ta tin g flat blade.
It a l s o
b r i n g s to light s o m e m e r i t s of a r o t a t i n g flat b la d e a s a
heat exchanger.
However,
further investigations are d esirable
in the following a r e a s .
1} I n v e s t i g a t i o n s of h e a t t r a n s f e r c h a r a c t e r i s t i c s of
b l a d e s of d i f f e r e n t c r o s s s e c t i o n s ,
Z)
including c i r c u l a r tubes.
I n v e s t i g a t i o n s of t h e i n f l u e n c e of s i m u l t a n e o u s
r o t a t i o n of t h e t u b u l a r h e a t t r a n s f e r s u r f a c e s a b o u t t h e i r o w n
a x i s a n d a b o u t the p e r p e n d i c u l a r a x i s ,
center.
p a s s i n g th r o u g h the
T h i s m a y c a u s e i n c r e a s e in h e a t t r a n s f e r c o e f f i c i e n t s
due to i n c r e a s e d s u r f a c e s p e e d s and a l s o due to c e n t r i f u g a l
an d c o r i o l i s f o r c e s a c t i n g on t h e b o u n d a r y l a y e r s .
t h e r e is lift p r o d u c e d due to to m a g n u s e ffe c t .
In ad d ition,
This can be
u s e d a s a s t i r r i n g unit for m i x i n g a n d h e a t i n g v i s c o u s l i q u i d s .
1)
I n v e s t i g a t i o n s to u s e c e n t r i f u g a l f o r c e f o r c o n d e n ­
sation heat tra n sfe r.
T he c o n d e n s a t e on the r o t a t i n g cool
s u r f a c e s c a n b e t h r o w n out r a d i a l l y d u e t o c e n t r i f u g a l f o r c e .
T h i s r e s u l t s in h i g h h e a t t r a n s f e r r a t e s a s i s f o u n d in d r o p w i s e
eond en s a t ion.
4)
I n v e s t i g a t i o n s i n t h e a r e a of b o i l i n g h e a t t r a n s f e r
using cen trifug al force.
rotating hollow blade,
If t h e b o i l i n g l i q u i d i s i n s i d e a
v ap o r bubbles which a r e g e n e ra te d during
th e b oi l in g p r o c e s s c a n m o v e t o w a r d s the a x i s of r o t a t i o n .
The
d r i v i n g f o r c e ca n be c o n s i d e r a b l y i n c r e a s e d due to the i n c r e a s e d
centripetal acceleration.
T h u s e f f i c i e n t n u c l e a t e b o i l i n g c a n be
110
achieved.
5)
I n v e s t i g a t i o n s o f t h e p o s s i b i l i t y of d e s i g n i n g t h e b l a d e s
n ot o n ly a s a h e a t t r a n s f e r s u r f a c e but a l s o a s a p u m p to
d i s c h a r g e the h e a t e d o r cooled fluid.
6
) I n v e s t i g a t i o n s of t h e e l e c t r i c f i e l d i n t e r a c t i o n s w i t h
p o l a r l i q u id s in f o r c e d c o n v e c t i v e h e a t t r a n s f e r .
7)
T h e r o t a t i n g s u r f a c e c a n be c o n s t r u c t e d s i m i l a r to
a V a n de G r a a f f g e n e r a t o r an d t h e h e a t t r a n s f e r s u r f a c e c a n
be r a i s e d to v e r y high e l e c t r i c a l p o t e n t i a l s .
This may cause
c h a n g e in f l ui d p r o p e r t i e s a n d i n f l u e n c e t h e h e a t t r a n s f e r .
LIST O F R E F E R E N C E S
111
LIST O F R E F E R E N C E S
1.
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The Royal Society
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2.
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3.
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p.
4.
5.
.
6
7.
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, a n d D o d d , C. , " T h e E f f e c t o f A n
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C o n v e c t i v e H e a t T r a n s f e r In L i q u i d s I I " . A p p l i e d S c i e n c e
R e s . , V o l . A3 p. 8 5 - 8 8 . 1951.
.
D o b i n s k i , Von S . ,
" O b e r Den E i n f l u s s E i n e s E l e k t r i s c h e r
F e l d e s A u f Di e V i s k o s i t a t Von FI u s s i g k e i t e n . "
P h y s i k a l i s c h e Z e i t s c h r i f t , V o l . 3 6. 193 5.
9.
F o g a r t y , L. E. , " T h e L a m i n a r B o u n d a r y L a y e r O n A
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V o l . 18, No. 4, p. 2 4 7 - 2 5 2 , A p r i l 1951.
8
10.
G r a h a m , M. E . , " C a l c u l a t i o n o f L a m i n a r B o u n d a r y L a y e r F l o w
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11.
G r u s c h w i t z , E. , " T u r b u l e n t e R e i b u n g s s c h i c h t e n M i t
S e k u n d a r s t r o m u n g " . I n g e n i e u r - A c h i v , Bd . VI , 1935,
S. 3 5 5 - 3 6 5 .
12.
H a r n e y , J . D. , " A n A e r o d y n a m i c S t u d y O f T h e E l e c t r i c W i n d " .
T h e s i s , Calif. I n s titu te of T e c h n o lo g y . P a s a d e n a , Calif.
( A D - 1 3 4 4 0 0 ) 1957.
13.
H a y e s , W. D. , " T h e T h r e e D i m e n s i o n a l B o u n d a r y L a y e r " ,
N A V O R D R e p . 1313, N O T S 3 8 4 , U . S . N a v a l O r d a n c e T e s t
S t a t i o n ( I n h o k e r n ) , M a y 9, 1951.
14.
H o w a r l t , L. , " T h e B o u n d a r y L a y e r In T h r e e - d i m e n s i o n a l
Flow - P a r t I". P h i l o s o p h i c a l M a g . , Vol. XLII, pp. 2 3 9 - 2 4 3 ,
M a r c h 1951.
15.
H o w a r t h , L. , " N o t e On T h e B o u n d a r y L a y e r On A R o t a t i n g
Sphere".
P h i l o s o p h i c a l M a g . , S e r . 7, V o l . X L I I , p p 130 8- 1315
N o v e m b e r 1951.
16.
I v e y , H. J . a n d L e e , F . 1956 B . S c .
D epartm ent. Bristol University.
17.
Joh n sto n , J. P. , "On The T h r e e - d i m e n s i o n a l T u rb u le n t B o u n d ary
L a y e r G e n e r a t e d By S e c o n d a r y F l o w " . J . Of B a s i c
E n g i n e e r i n g , S e r i e s D, T r a n s A S M E , V o l . 8 2 , p. 2 3 3 , I 9 6 0 .
18.
K a y s , W. M. , L o n d o n , A . L. , J o h n s o n , D. W. , " G a s T u r b i n e
P l a n t H e a t E x c h a n g e r s " A m e r i c a n S o c i e t y of M e c h a n i c a l
E n g i n e e r s , N e w Y o r k 1951.
19.
K r o n i n g , R. a n d S c h w a r z , N. , " O n T h e T h e o r y O f H e a t
T r a n s f e r F r o m A W i r e In A n E l e c t r i c F i e l d " . A p p l i e d
S c i e n c e R e s . , V o l . A 1. pp. 3 5 - 4 6 , 1949.
20.
K u e t h e , A, M. , M c K e e , P . F3. , C u r r y , W. H. ,
M easurements
In T h e B o u n d a r y L a y e r Of A Y a w e d W i n g " .
N A C A T N 1946,
1949.
Thesis.
Engineering
21.
L e i d e n f r o s t , W. , " R o t a t i n g H e a t E x c h a n g e r s A n d T h e i r
Technical Feasibility" L ec tu re p re se n te d at R e s e a r c h
Institute, Huntsville, A labam a.
22.
M a g e r , A . , " G e n e r a l i z a t i o n Of B o u n d a r y L a y e r M o m e n t u m
Integral Equations To T h re e -d im e n s io n a l Flows Including
T h o s e Of R o t a t i n g S y s t e m s " .
N A C A T R 1067, 1952.
113
23.
M ager, A. , " T h re e -d im e n s io n a l Boundary L a m in a r Boundary
L a y e r W i t h S m a l l C r o s s F l o w " . J . A e r o . S c i . , 21,
p p . 8 3 5 - 8 4 5 (1954).
24.
M o o r e , F . K. , T h r e e - d i m e n s i o n a l C o m p r e s s i b l e L a m i n a r
B o u n d a r y L a y e r F l o w " , N A C A T N No . 2 2 7 9 , 1951.
25.
M o o r e , F . K. , " T h r e e - d i m e n s i o n a l B o u n d a r y L a y e r F l o w " ,
J . A e r o . S c i . , Vo l . 2 0 , No. 8 , p p 5 2 5 - 5 3 4 , A u g u s t 1953.
26.
M o o r e , F . K. , " T h r e e - d i m e n s i o n a l B o u n d a r y L a y e r T h e o r y " .
A d v a n c e s i n A p p l i e d M e c h a n i c s , 4, p p 1 5 9 - 2 2 8 , (1956) .
27.
M o o r e , R. W. , a n d R i c h a r d s o n , D. L. , " S k e w e d B o u n d a r y
L a y e r F l o w N e a r The End W al ls Of A C o m p r e s s o r C a s c a d e " .
T r a n s . A S M E , Vo l . 79, 1957, pp 1 7 8 9 - 1 8 0 0 .
28.
P r a n d t l , L. , " O n B o u n d a r y L a y e r s In T h r e e - d i m e n s i o n a l
F l o w " , R e p o r t s a n d T r a n s a c t i o n s , No. 6 4 , B r i t i s h M . A . P . ,
M a y 1946.
29.
S c h m i d t , E. , a n d L e i d e n f r o s t , W. , " D e r E i n f l u s s
E l e k t r i s c h e r F e l d e r A u f D e n W a r m e t r a n s p o r t In F l u s s i g e n
E l e c t r i s c h e n Nichtleite r n " . F o r s c h u n g A uf Den G ebiete
D e s I n g e n i e w r w e g e n g . N r . 3. 1963.
30.
S e a r s , W. R. , " B o u n d a r y L a y e r s In T h r e e - d i m e n s i o n a l F l o w " .
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51.
S e a r s , W. R. , " T h e B o u n d a r y L a y e r Of Y a w e d C y l i n d e r s " ,
J . A e r o . S c i . , Vol . l b, No. 1, pp 4 1 - 4 5 , J a n u a r y , 1949.
32.
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F e l d e r A u f D e n W a r m e s t r o m In G a s e n " . Z . P h y s i c s 102,
pp 4 8 0 - 5 0 6 , 1936.
~
3 3.
S t u e t z e r , O. M. , " I o n D r a g P r e s s u r e G e n e r a t i o n " .
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3-1.
T a n , H. S. , " O n L a m i n a r B o u n d a r y L a y e r F l o w O v e r A
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36.
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GENERAL REFERENCES
1.
E c k e r t , E. R. G. a n d D r a k e , R. M. , H e a t a n d M a s s T r a n s f e r .
M c G r a w - H i l l Book C o m p a n y , I n c . , New Y o r k . (1959).
2.
H i n z e , J . O. , T u r b u l e n c e ,
N e w "Yo r k ( 1 9 5 9 ) .
3.
I l o e r n e r , S. F . , F l u i d D y n a m i c D r a g ; P r a c t i c a l I n f o r m a t i o n on
A ero d y n a m ic D rag and H y d ro d y n am ic R e s is ta n c e .
Midland
P a r k , N. J . , 1958.
4.
K r e i t h , F . , P r i n c i p l e s of h e a t t r a n s f e r .
T e x t b o o k C o . , 1958.
5.
J a c o b , M. , H e a t T r a n s f e r , V o l u m e 1 a n d 2, J o h n W i l e y a n d
S o n s , Inc. , N e w Y o r k (1949) a n d (1957).
6
.
7.
M c G r a w - H ill Book C om pany,
Scranton,
Inc.,
International
J a c o b ,, M . , a n d H a w k i n s , G . A . , E l e m e n t s of H e a t T r a n s f e r
and I n s u l a t i o n J o h n Wiley and Sons, Inc. , New Y o r k ,
T h i r d E d i t i o n . (1957).
M c A d a m s , W. H. , H e a t T r a n s m i s s i o n .
M c G r a w - H i l l Book
C o m p a n y , I n c . , N e w Y o r k , T h i r d E d i t i o n , (1954) .
.
P a i , S . , V i s c o u s F l o w T h e o r y , V o l u m e 1, D. V a n N o s t r a n d
C o m p a n y , I n c . , P r i n c e t o n , N e w J e r s e y , (1956).
9.
S c h i i c h t i n g , H. , B o u n d a r y L a y e r T h e o r y , M c G r a w - H i l l B o o k
C o m p a n y , I nc . , N e w Y o r k ,
F o u r t h E d i t i o n ( I 960) .
8
APPENDICES
APPENDIX A
DERIVATIONS FOR THEORETICAL INVESTIGATION
1.
D e r i v a t i o n of m o m e n t u m i n t e g r a l e q u a t i o n s .
The m o m e n t u m e q u a t i o n i n t h e x d i r e c t i o n is
a 2u
u du
()u
du
uv
— —— + v —— + w —— + ---- - 2 12v
y d x
dy
dz
y
( 12)
I n t e g r a t i n g w i t h r e s p e c t to z f r o m 0 t o 6
<5
I
7 ^ 7
-j
dz
6
du .
C
\
dy d z + J
I U v dz * y
v -2—
6
9u
f* u v
w —— d z f \----- d z
dz
J y
(7 ^ )
dz
(a)
I n t e g r a t i n g t h e t h i r d t e r m by p a r t 3
du
W —— (17 ” Wll
n7
[ 7 - 0
F r o m c o n t i n u i t y e q u a t i o n ( 14)
/
*
’
-
(\
7 ~ ;> J
U
.
dw
(iz
1
0 7
117
Integ ratin g the a b o v e equation a c r o s s the b o u n d a r y l a y e r with
r e s p e c t to z
<5
f
6
9 w
J
C
J a l dz = J
'
O
, 1
9u
V
{7
a^ + 7 +
*
- ~\
/ I
7 - r>
Substituting equations
f
dz
, 1
V
du
P
V — dz - U \
dy
J
I du
V
dv
( - — t - + — ) dz
y dx
y dy
6
i3u v
3v
u (— — + - + — )
y dx y
dy
o
dz +
uv
\ — dz
J y
p
o
6
y
2 f i v d z = y
o
v
o
6
dz
' 8z
5
^
6
( t ~)
dy
d z -U
J
o
6
ill
J
dz
6
(—) d z -U\ —{— ) d z
J y dx
o
1
y
(5
V d z :
I
- i
^
o
o
- -
P OZ
( r
zx
) d z
It i s r e w r i t t e n a s
^
(f)
( uv) dz + Jo 2(T 1 dl
6
-U C
J
o
)
6
j"o 7 al (u‘)4,; +1 o
—
(e)
6
6
-2
3 v
(b) , ( c ) , a n d ( e ) i n t o e q u a t i o n ( a ) g i v e s
6
r
+ \
J
(d >
( - T— -t- — + -r— ) d z
9 y'
'y ox
'y
Jr\
6
('
u du
P
\ _
dz + \
J
y Ox
J
.
6
, , / T
-y
6
3 v
( U u - u ^ d z + ^ y
(uv) dz
(g)
118
6
6
+— \
y Jo
( u v ) dz
6
V
- U
v dz
-
— \ v dz
yJo
9y J o
6
- 2 £2 ^
vdz
=
-
■—^
o
(h)
In t h e f i n a l f o r m it i s w r i t t e n a s
— — y
( f t y u - u 2) dz _ ~
y
X O
y
6
~ ~
^
y
(uv)
9z
o
6
(uv) d z - ily
—— ^
6
v d z f 3 Si ^
v dz
o
r
ox
( 15)
T h e m o m e n t u m e q u a t i o n in t h e y d i r e c t i o n i s
+
+
T
+ 2Q
u
- nV
( 13)
I n t e g r a t i n g w i t h r e s p e c t to z f r o m o to fi
6
I
J
6
dv
6
i'
v 5 7 dI ’
JO
9v
j
('
w 5 7 d7 *
6
u 9v
JO 7^7
('
JO
u l
y
dz
6
+y
2 f2 u
dz - ^
f 2 *y dz = ^ p ^ —7
o
( i)
119
In te g ra tin g
t h e s e c o n d t e r m by p a r t s
Z
Substituting (c),
J
v| x
-
z = o
6
- Io
9w
V7 7 d "
0)
(j) , i n t o e q u a t i o n (i)
dz + J
v (i
y
9u
v
9v
f
9 x + -y + “9y > d z + J
u 9v
7 ^
dz
<M
( v 2 ) dz +
9y
1 77
) d z + \ ( -----) d z
y
< f > d- I
- C — (u 2 + ft 2 y 2 - 2 Q y u ) d z =
J y
%
P
(l)
In t h e f i n a l f o r m it is w r i t t e n a s
6
(*
9y
J
6
V2
o
dz +—
y
~
dx
f
J
( uv) d z + — C
y J
o
o
- Qy — ^ ( i 2 y - u ) d z + — ^
2.
6
V2 dz
( 12y - u ) u d z = - 2 1
p
( 16)
E v a l u a t i o n of l a m i n a r t a n g e n t i a l v e l o c i t y p r o f i l e ,
u
U
w h e r e 17
*
= •—
0
G( tj)
= a Q + a 1 rj + a 2 rj2 f a 3 T]3
(38)
120
Applying b oundary conditions
a t
tj
=
o
G(rj)
=
o
a t
tj =
o
G " ( tj) =
at
tj
=
1
at
tj
=
1
t h e r e f o r e
ao
o
t h e r e f o r e
*2
G ' ( tj) =
o
t h e r e f o r e
a l
G ( tj)
=
1
t h e r e f o r e
a 1
3_
2’
a l
Solvi' g f o r c o n s t a n t s
a 2
_1
2
~~
T h e p r o f i l e is
G( r j )
3.
=
(39)
E v a l u a t i o n of l a m i n a r r a d i a l v e l o c i t y p r o f i l e
« G(rj) g(ij)
yU
= t
3
1
( - t j - - tj
3
) ( b o + b jT) f b ? T? )
(41)
Applying b oundary conditions
a t T) =
o
g(Tj)
at
1
G ( tj) g(rj)
tj
=
=
1
therefore
= o therefore
+ G(rj) g ' ( r j )
l +b j + b ^
=o
=
o
t h e r e f o r e b j ! 2 b 2- o
Solving for co n stan ts
bQ = 1 ,
bj
=
-2,
b2 =
1
(1
17) 2
T h e p r o f i l e is
« G (tj) g(r})
4.
= «
(
~
Relation between
tj
-
1
T]3 )
-
and 6 for la m in a r profiles
6
o
I
*f / 3
"
4 J
1\
' i ”
- 7
”
, 3
)<
'
- I "
1\
+ ~ "
)
1
. f r
3
O
3
9
-
5.
2
7 ” ■ 4 ”
11
280
1
3
• 2 ”
3
+ 2 "
4
1 6
4 "
a
E v a l u a t i o n of (5 a nd 0 * f o r l a m i n a r p r o f i l e s
-1 ill
dx
R e w r i t i r 1g a s,
{ 280
6
'
"
-I
2
(
—
y l26)
122
Integrating and applying boundary conditions
at x
=
- x
at x
=
x
=
sin
w h e r e xn
°
5
6
.
0
6 = 6
- 1
b
(—)
y
=
y (x 4 x )
1/ 2
4. 6 4 [ ------ - — — ]
~
4.64 Rex
°x
=
6
o
=
6
^
~
y(x 4 x )
0.646 Rex
^ Z y(x 4 x0 )
( 45)
(46)
E v a l u a t i o n of t h e p a r a m e t e r e f o r l a m i n a r p r o f i l e ,
S o l u t i o n of t h i s e q u a t i o n is
X + X
, , H ~ ^ 2 (* + x 0 )
. /
r,
1 ,
K’J
,1 /2[1+ 1 T 5 !
I
(> ‘
Boundary c o 1dition
t
=
Therefore
c
o
=
atx
H- 1
( ——- )
=
-x Q
2 ( * + x o)
------------- ---—
(3 +
)
'
K-J
( 49)
123
7.
E v a l u a t i o n of t h e r a t i o s of i n t e g r a l t h i c k n e s s e s f o r l a m i n a r
velocity profiles.
The following in te g r a ls a r e e v a lu a te d first.
6
^
1
[ 1 - G{tj) ] d z
=
y
6
[ 1 - G(tj) ]
d?] w h e r e 17 ~
1
6 y
T)
+
= 0 .37^
6
( 1 -
~
~
Jj)
dr?
1
\ 1 - G( 17) ] G(rj)
J
dz
= <5^ ( 1 - -p? + - j v) ( -j V - ~ 17)
6
= 0. 139286
J
G( tj)
g(ij)
dz
=
6
~ TJ - J TJ 3) ( 1 + TJ2 - 2 17}
= 0. 1 16667
6
\
hj
drj
drj
6
1
[ 1 - G(rj) ] G(tj)
g(r))ds
6 \
O
f 1 - - T] * j r] S ( ~ 1 - y 77 ^)
i-'
—
«-
( 1 4- n Z - 2 tj)
= 0.05496
6
drj
124
y
[ G(r, ) g( f j ) ] 2
dz
o
1
= 0.019156
^
H
=
( 1 - G)
6
dz
=
o
2.6923
( 1 - G) G d z
o
6
( 1 - G) G g dz
J
= — r --------------------------o
y
= 0.3946
( 1 - G) G d z
O
K
y
=
G g dz
=
o
y
0.8376
( 1 - G) G d z
o
6
(G g ) Z dz
\
L.
=
-_2--------------------------=
y
0. 1376
( 1 - G ) G dz
o
8
.
E v a l u a t i o n of a v e r a g e s k i n f r i c t i o n c o e f f i c i e n t a t a c o n s t a n t
r a d i a l d i s t a n c e for l a m i n a r flow
125
C f»x
=
0. 3233
1
Re x ‘
1 + [ 1. 4 5 3 ( x + x 0 ) ] ‘
1/ 2
•
y
S u b s t i t u t i n g R e, x
(55)
(x-t-xQ)
in t h e a b o v e e q u a t i o n a n d
y
e x p a n d i n g in b i n o m i a l s e r i e s
C
fl X = n0. 3» 233
»
-—
1 +
\
+2
v l/ 2
[ !• 4 ^ 3 ( x + x 0 )
/( x +. x f )* ' ^ 2 y “ 1 ofi -
}Z
-
i I- 4 R. 3 ( x + x l>) ]b -
2
2
j 1. 4 5 3 ( x + x 0 ) ] 4
1. 4 5 3 ( x +x ()) ] 8 +
I n t e g r a t i n g a t c o n s t a n t y f r o m x = * x Q t o x = x q a n d di v i d i n g b y
2 x Q,
the a v e r a g e s k i n f r i c t i o n c o e f f i c i e n t is g iv e n a s ,
-2
= 0 . 6466 R
'
x 11 Z
o
1+0. 1 (2. 9 0 6 x J
- 0 . 0 1388 ( 2 . 9 0 6 x o ) 4 + 0 . 0 0 4 8 0 8 ( 2 . 9 0 6 x ) b
- 0. 0 0 2 2 9 8 ( 2. 9 0 6 x q )
where R
and
x
fi 2 x c
e, x
=
"pT7~
s i n - 1 (/ -b x)
y
+
(56)
126
9-
E v a l u a t i o n of t o r q u e d u e to s k i n f r i c t i o n d r a g f o r l a m i n a r f l o w .
=
plyQ )2
0.3233
[----
- ---------sr
fi ( x f x ) y 2
T o r q u e d u e t o a n e l e m e n t of a r e a y d x d y is
dM
=
r QX y
2
dx dy
T o t a l t o r q u e is e s t i m a t e d b y i n t e g r a t i n g t h e a b o v e e q u a t i o n
y2
- f
yi
x 2
r
xi
y2
J
xo
[ J
yp
p (fi y ) 2 (
5y ^ x - t x 0 )~Q’
”xo
y2
0 . 6 4 6 6 p fi3 / 2 , 1 / 2 V
y 3 {2 s m ' 1 2 ) ° ‘ 5d y
yi
10.
y
E v a l u a t i o n of t u r b u l e n t r a d i a l f l o w v e l o c i t y p r o f i l e
=
where
e
G( n)
G(»j) =
g(*?)
tj
= T
g(n)
^ ^
= co + c
z
V
dx]dy
] t? +
c z -nz
(57)
127
Applying b o u n d ary conditions
a t rj =
o
g(ij)
= 1
at
1
G ( tj)
g ( 7])= o
tj
=
a t tj
=
therefore
therefore
C0
=
1 + Cj
1
+ C^ = o
G 1 ( tj ) g ( tj ) hG ( TJ) g * ( tj ) = o t h e r e f o r e C j + Z C ^
1
Solving for c o n s ta n ts
c 0
= 1 .
The velocity
-2
=
,
C2
=
1
p r o f i l e is
=
11.
C,
c
(1 " rj) ^
Relation between
O
tj 1 / 7
( 65)
and 6 m tu rbulent profiT
6
G»
'
.f
o
f n
"
6
- 7 tr>
dz
1/7
(17)
1/7
1
=
12.
dz
6
< b
6
>
S o l u t i o n of m o m e n t u m i n t e g r a l e q u a t i o n f o r t u r b u l e n t f l o w ,
i
y
±
dx
( o x)
x
.
C). 0 1 2 5 5 { —
y lit)
T h e a b o v e e q u a t i o n is r e w r i t t e n a s
e ' Z5 d e
v
= 0.01255 ( ~ y
W
25
y * 7 5 dx
<6 8 >
=0
128
I n t e g r a t i n g the a b o v e equation an d ap ply in g the b o u n d a r y conditions
at
x
*
-xQ
at
x
= x
0
*
=
o
e x = ex
0X = °* 036 0 1 y (X + Xo) R e x 0
5
13.
:
y
Ox
=
0- 3 7 0 4
(69)
y ( x + x Q) R e x ° ' 2
(70)
E v a l u a t i o n of t h e p a r a m e t e r t f o r t u r b u l e n t f l o w .
S o l u t i o n of t h e a b o v e e q u a t i o n is
HO
K-J
(x +
5Jx t j j a )
4 ( 2 . 2 5 +-
XQ )
K-J
The boundary conditions
at x
=
- x.0
Therefore
HO
K-J '
14.
5 Q. J ; » p ) ,
4( 2. 24 *— ■ >
I\ ~J
'
"
E v a l u a t i o n of t h e r a t i o s of i n t e g r a l t h i c k n e s s e s f o r t u r b u l e n t
velocity profiles.
The following in t e g r a ls a r e ev alu ated f irs t
129
1
j" [1-G(>,)]d„ =
[ 1 - a(n) ] dr, = 6 y
= 0. 125
i/
(l-„
7) d»
<5
i
J
[ 1 - G(rj) ] G ( tj) d z
= 6
J
( tj ^ 7 -
tj
^ 7 ) drj
o
= 0.097222
6
1
j
G ( n ) g(rj) d z
= 6 j
rj / 7 ( 1 f TjZ - 2 rj) drj
= 0. 2 5 9 8 4 8
^
f 1 - G(tj) ] G(tj) g(rj) d z
6
= 5 J
= 0.0527228
j)
[G(f))g{rj)]Z dz
=
0
6
[T71 / , 7 ( l + fjZ - 2 f j ) ] Z drj
. 100 7 2 0 ^
<5
6
J
H
( 1 - G) d z
c
=
=
0
j
( 1 - G) G d z
( 1 - rj ^ 7 ) tj ^ 7 ( 1 + tj2 - 2 tj) drj
1. 2 8 5 7
130
6
j*
J
=
( 1 - G) G g d z
—2
o
= o. 5423
^
< I - G) G d z
o
6
G g dz
1
K =
—?
=
2. 6 7 2 7
( 1 - G) G d z
O
J
L
=
(G g)
dz
= 1. 1286
o
{ 1 - G) G dz.
15.
E v a l u a t i o n of a v e r a g e s k i n f r i c t i o n c o e f f i c i e n t a t a c o n s t a n t
r a d i a l d i s t a n c e for t u r b u l e n t flow
Cr
n z f
’
2
=
Substituting
0.02881
R Cf X =
R
x
■j" 1 + [ 0. 061 >4( x + x C)) ]
I
L
2
——^ y.( x.—
>1
^ ^
!■
I
(77)
J.
^ e above equation and
e x p a n d i n g in b i n o m i a l s e r i e s
C f, *
—j —~
=
„
0.02881 v
° - Z
f. ~ 0 . 2
il
y
-0. 4
-0.2
( x + x Q)
1 -t — [ 0 . 06 164<x + Xq ) ] 2 - - ^ - [ 0 . 0 6 164 (x f x 0 ) ]*
f IT
^°* 0 6 1 6 4 <X + Xo) ] 6 “ 1 7 8 ^°* ° 6 1 6 4 <x + *o> I 8 +
131
In tegrating the above at c o n s ta n t y f r o m x = -x
to x = x n
and
d i v i d i n g b y 2 x Q, t h e a v e r a g e s k i n f r i c t i o n c o e f f i c i e n t i s g i v e n a s
Cf
—
=
....................
- 0 ' 2 I ..............................
2
0.0360 1 R e, X
1 1 + 0 . 1429 (. 1 2 3 3 x n )
0 . 0 2 0 8 3 (• 1 2 3 3 x q ) 4
-
+ 0 . 0 0 7 3 5 3 (• 1 2 3 3 x Q) 6
0. 0 0 3 ^5 1 (•
' 123 3x I) )f 8 t . . . .
( 78)
where
R
e, x
2
Y n 2x0
------v
o
and
xQ
16.
= Sin
1
, b
( y )
E v a l u a t i o n of t o r q u e d u e t o s k i n f r i c t i o n d r a g f o r t u r b u l e n t f l o w
T° X z
pftrn)
T o r q u e due
to
dM
-
. 0 2 8 8 1 R (e , x - ° - 1
an e l e m e n t
=
t qx
of
y
2
area
dx
of
y dx dy i s
dy
T o t a l t o r q u e is e s t i m a t e d by i n t e g r a t i n g t h e a b o v e e q u a t i o n
132
y
r 2
M = J
*
c 2
2
J
y i
Tox y dxdy
xi
Y j u -*0
( x + x o)
y2
o
= n0 . 0n 3, 6A 0n . 1 pQ
h
8
v
°- 2
f\
y3
* 6 /( ?2cS-i n - 1 b—)
. 0. 8 d
Jy
yl
................................
(79)
17.
E s t i m a t e of m a x i m u m p r e s s u r e r i s e d u e to e l e c t r o s t a t i c
i nflue
nc e
The following values a r e a s s u m e d for a i r
E
max
«
(A P )
=
=
91500 v o l t s / f t
2. 7 x 10
max
= 2. 7 x 10
=
coul/ volt-feet
4 (E
)2
2
max
x 9. 15 x 9. 15 x 10
= 0 . 0 1 3 4 i n c h e s of w a t e r .
c o u l v o l t / ft
APPENDIX B
DERIVATIONS FOR E X P E R IM E N T A L INVESTIGATION
1.
Non-dimensional analysis
Let
h
-
h ( P , y , C , p , k, p , C ,
W)
then
„ ^
h = C , f 2
a
y
1
b
d e f e
i
4*
C p k P KC J W
P
A p p l y i n g t h e d i m e n s i o n of e a c h v a r i a b l e in t h e M L T H ©
sy s t e m
H T ' V ' 2 ©’ 1 = C 1 ( T ‘ 1 ) a ( L ) b ( L ) d ( M L " 1 T " 1) e
(HT_ Y ~ 1 ©" V ( M L ' V (HM- 1 0“ V (L)^
E q u a t i n g t h e e x p o n e n t s of l i k e t e r m s ,
gives!
H
1 =
f + j
M
o
e + g - j
=
T
- 1 =
I
-2
=
b+ d -
O
- 1
=
- e - g
-a
- e - f
e-
Solving the above equations
f
= 1 - j
e = j - g
*
*
g
d
=
2g
- b - j f - l
f-3g
Jt
+■
1 34
Therefore
h
= C j ( f i ) g ( y ) b ( C ) 2 g r b " i ' 1 (M ) J " g (k) 1 _J( p ) 8 ( C p ) J ( W ) '
R ear ranging
k
=
c . ( i £ £ £ . ) g , f i £ B ) j ( £ ) b "e ( ^ , ■*
i •
n
k
C
' C
which can be w r itte n as
Nu
=
C ( Re 8
t
(P r )J ( i )n A
C '
'c
F o r a g i v e n b l a d e o p e r a t i n g in a f l u i d w h o s e p r o p e r t y d o e s n o t
c ha n g e
Nu
2.
=
C2
R em ( ^ ) n
C o r r e l a t i o n of e x p e r i m e n t a l r e s u l t s .
T he N u and R e at any location a r e
c o r r e l a t e d by
ni
Nu
=
ci
Re
w ith l e a s t m e a n s q u a r e e r r o r fit u s i n g
IBM 7094
The results are
Blade
1
Nu
=
0.0528 R e ° " 733
Blade
2
Nu
=
0.0042 Re ° - ^ 7
Nu
=
d
it
0.0038 R e° ’ 9 2 4
_
0. 8 6 4
0 . 0090 Re
Blade 3
Blade 4
T h e a v e r a g e v a l u e of t h e e x p o n e n t is t a k e n a n d t h e c o r r e s ­
ponding c o n s t a n t s
are
d e t e r m i n e d f o r e a c h b la d e by l e a s t
m ean square e r r o r method.
Blade
0. 87
1
N „u
= 0 . 0 126 1
Blade 2
Nu
= 0 . 0 109 1 R e
e
0, 87
135
0 87
Blade
3
N U. =
0. 0 0 6 9 9 R e
Blade
4
Nu
0. 0 0 8 4 7 R e °* 8 7
=
Theseconstants a re assu m ed
t o b e t h e f u n c t i o n s of
radial distance as
c,
= C
Vi n
« _ )
T h e l e a s t m e a n s q u a r e e r r o r fit g i v e s
c.
=
0.04238
1
c
( - )
y
0,704
T h u s t h e f i n a l c o r r e l a t i o n is
N LI =
3.
0.04238 R C '
87
C ,7°4
( -y )
D e t e r m i n a t i o n of t h e c h a n g e in h e a t e r r e s i s t a n c e d u e to
c h a n g e i1 t e m p e r a t u r e .
MICA
STAINLESS
STEE L . 03^
AI R G A P
. 007
H E A T E R W I R E . 008
F i g u r e 4 0 , CROSS SE C T IO N Of* B L A D E A N D H E A T E R
136
Fig.
( 4 0 ) i s t h e c r o s s s e c t i o n a l d r a w i n g of t h e b l a d e .
The
f o l l o w i n g a s s u m p t i o n s a r e m a d e t o d e t e r m i n e t h e d i f f e r e n c e in
t h e s u r f a c e t e m p e r a t u r e of t h e h e a t e r w i r e a n d t h e b l a d e .
1)
H e a t e r is s y m m e t r i c a l l y p l a c e d i n s i d e the b l a d e a n d
e q u a l a m o u n t of h e a t f l o w s t h r o u g h b o t h s i d e s of t h e b l a d e .
2)
The surface te m p e ra tu re s a r e uniform.
3)
T e m p e r a t u r e o f s t a i n l e s s s t e e l w a l l is u n i f o r m .
M aximum heater current
=
M aximum heater resistance
0.725 am p s.
=
16.9 o h m s .
_ _ n
n , . ,,
0 * 7 . 0 * 7. 16*9. 3 * 4 13 .
2. 3- 143
H e a t f l u x on e i t h e r s i d e o f b l a d e
=
647* 5
...
144
BTU/ Hr F t 2
T h e r m a l R e s i s t a n c e du e to a i r ga p
0*033
l c ,_
" 12. 0* 0 1 7 8 "
Therm al Resistance
H r F t2 °F
BTU
du e to m i c a
0* 0 0 7
H rF t2 °F
t 0 0 0 13412. 0* 4351
BTU
T em perature drop across surfaces
= 6 4 7 - 5 ( 0 * 0 0 1 3 4 + 0* 1545)
= 100* 9 ° F
M a x i m u m i n c r e a s e in w i r e s u r f a c e t e m p e r a t u r e
= 170°F
T h e a s s u m p t i o n of t h i s a n a l y s i s i s
pitched.
Also,
=
37* 7 6 ° C.
ot v a l i d w h e n t he b l a d e s a r e
t h e r e is c o n t a c t r e s i s t a n c e b e t w e e n e a c h
e l e m e n t s a n d t h e h e a t e r s m a y not b e e x a c t l y l o c a t e d a t t h e
c e n t e r d u e to d i f f i c u l t i e s n c o n s t r u c t i o n .
To account for these
t h e m a x i m u m i n c r e a s e in w i r e s u r f a c e i s a s s u m e d to b e 5 0 ° C .
A s s u m i n g t h e c o e f f i c i e n t o f t h e r m a l r e s i s t i v i t y of
n i c h r o m e to b e 0 * 0 0 0 4 o h m s / ° C ,
the m a x i m u m
c h a n g e in
h e a t e r r e s i s t a n c e i s ( * 0 0 0 4 x 5 0 ) R o h m s w h e r e R is t h e r e s i s ­
t a n c e of t h e h e a t e r in o h m s .
^5
R
=
R
Therefore
+ 0.02
137
APPENDIX C
E le c tric a l F ie ld - Corona D isch arg e Relation
Two sim p le m o d e ls with flow p e r p e n d i c u la r and p a r a l le l
to t h e e l e c t r i c a l f i e l d a r e a n a l y s e d t o d e t e r m i n e t h e f i e l d s t r e n g t h
and body f o r c e d is tr ib u tio n .
V. E
=
T he e le c tr ic field equations a r e
it-
. . .
(C-l)
€
With the
J
=
<7C E +
F
=
Pc E
P c
Vt
. . .
(C-2)
. . .
(C-3)
a s s u m p t i o n s m a d e in t h e s e c i o n " I n f l u e n c e of E l e c t r i c a l
F i e l d " ( P a g e 94 )
V
where
KE
+ V
. . .
( C ~4 )
. . .
(C-S)
. . .
(C-6)
V is the a i r v e l o c i t y
1
where J
*
=
Pc
(KE
+
V)
is the c u r r e n t d e n s i t y .
V P
=
p
c
E
138
1.
F l o w p e r p e n d i c u l a r to the e l e c t r i c a l f iel d.
C o lle c to rs plane
L
Y
Space charge
free region
E m i t t e r plane
F i g u r e 41
M O D E L FOR F L O W P E R P E N D IC U L A R TO
ELECTRICAL FIELD
T h e e m i t t e r is l o c a t e d a t y = o.
( f i g u r e 41) .
Bipolar
c o n d u c t i o n i s a s s u m e d t o b e p r e s e n t on i y i n a l a y e r y Q t h i c k .
Ins id e th is l a y e r p o s i t i v e a n d n e g a t i v e io ns m o v e in o p p o s i t e
d i r e c t i o n s a n d t h e i r c o n t r i b u t i o n t o t h e d r a g f o r c e is t h e r e f o r e
neglected.
In t h i s m o d e l
E =
j
V =
w h e r e i, j,
k a r e unit
E
. . .
( C - 7)
i U
. . .
(C-8)
vectors
i n x, y,
z directions respectively.
F r o m eq u atio n (C-5)
J" =
1J
x f j Jy
=
i P c U + j pc K E
. . .
(C-9)
139
M a g n i t u d e of J
is
0. 5
J
= p c ( U2 + K2 E 2 )
Therefore
P c
=
g- g -
. . .
(C-10)
. . .
(C -ll)
[U2 + (K E )2 ]
Combining equations (C -l) and (C-10)
dE
1
dy
€
[U2 + (KE) 2 ]0 , 5
w h i c h is r e w r i t t e n a s
0. 5
[ U 2 + (KE) 2 ]
j
dE = -
dy
I n t e g r a t i n g t h e a b o v e e q u a t i o n f r o m y Q to y
7
( Y ~Yo)
+
=7k { ke
J
(ke) * + u2
U 2 In ( K E + J ( K E ) 2 + U 2 )
- KEq
y (KEq) 2 + U2
-
In ( K E q
U2
+
y ( K E q ) 2 + U 2l J
w h e r e E 0 is t h e e l e c t r i c a l f i e l d s t r e n g t h a t y = y 0 .
i s a s s u m e d to b e a p l a n e i o n i z e d l a y e r ,
Thomson
(35)
J
7
.
Then equation (C-12)
.
<y- y <>)
U2
' 7k
KE
1 ~
. .
( C - 12)
If t h e i o n s o u r c e
t h e n E Q = o a s s t a t e d by
r e d u c e s to
/ ' KE. ,
/
K
T h e a b o v e e q u a t i o n i s m u l t i p l i e d by '" y
~
, KE
/ KE 2 , ,
M +ln(~
V ( — ' +1)l
to y i e l d the d i m e n s i o n l e s s
e q u a t i o n ( C - 13)
“
£
' I I f
. . .
( C - 13)
T h e b o d y f o r c e is o b t a i n e d b y c o m b i n i n g ( C - 6 ) a n d ( C - 1 0 )
F
=
_______ 3 ______O R
[U2 + (K E )2 ] '
( C- 14)
. . .
E lim in atin g J fro m (C-14) using (C-13) and r e a r r a n g i n g
TT*‘
,
-> '
U
KE
)
( C - 15)
E q u a tio n ( C -1 3 ) a n d (C - 1 5 ) give the n o n - d i m e n s i o n a l d i s t r i b u t i o n
KE
of t h e f i e l d s t r e n g t h a n d b o d y f o r c e a s a f u n c t i o n of ——
f o r flow
p e r p e n d i c u l a r to e l e c t r i c a l f i e l d .
2.
F l o w p a r a l l e l to t h e e l e c t r i c a l f i e l d .
I
I
Collector
plane
J
- f
y=
v
F i g u r e 42
MODEL FOR FLOW P A R A L L E L TO
ELECTRICAL FIELD
T h e a s s u m p t i o n s of t h e p r e v i o u s m o d e l h o l d t r u e f o r
this m o d e l a l s o .
In t h i s m o d e l
141
Therefore
E
=j
E
. . .
( C - 16)
V
=j
U
. . .
( C - 17)
. . .
( C - 18)
‘ * *
( C " 19)
J
= j
(U + KE)
. .
a n d t h e m a g n i t u d e of J i s
=p
J
c
(U +K E )
a nd
Pc
=
u T KE
Combining equations (C - l) and (C-10)
dE
dy ~
1
€
J
( K E + U)
* • ‘
( C - 20)
w h i c h is r e w r i t t e n a s
( K E + U)
dE
=
- dy
€
I n t e g r a t i n g the a b o v e f r o m y 0 to y
= -_L
[ ( K E + U) 2 - ( K E q + U) 1 ]
. . .
(C-21)
w h e r e E q i s t h e f i e l d s t r e n g t h a t y = y Q a n d it i s a s s u m e d t o b e e q u a l
to z e r o .
T he equation (C-21)
i s s i m p l i f i e d a n d r e a r r a n g e d in a
■■on-dimensional f o r m as,
ifcajK.inJ™
«u*
z
...
<c-zz)
u
T h e b o d y f o r c e is o b t a i n e d b y c o m b i n i n g ( C - 6 ) a n d ( C - 1 9 )
-
JE
( K E + U)
Elim inating J from (C-23)
( ^
(C-23)
using (C-22) and re a rra n g in g
)
(
* * ‘
U
— r
+ '
! ( i r f + *>* - 1'
• • •
( c - 24)
142
Equations (C-22) and (C-24) give the n o n - d im e n s io n a l d i s t r i b u KE
t i o n of t h e f i e l d s t r e n g t h a n d b o d y f o r c e a s a f u n c t i o n o f
f o r flow
p a r a l l e l to t h e e l e c t r i c a l f i e l d .
E q u a t i o n s ( C - 13) , ( C - 1 5 ) ,
(C-22) an d (C-24) a r e plotted
a s u n i v e r s a l c u r v e s a s s h o w n i n fig.
sional p a r a m e te r s
43
in t e r m s of n o n - d i m e n ­
and
F i g u r e s (43) a n d (44) s h o w t h a t f o r a s p e c i f i e d c u r r e n t
flow and a i r velocity the body fo rce,
the e l e c t r i c a l f ie ld a n d the
in d u c ed p r e s s u r e i n c r e a s e s a s the d i s t a n c e f r o m the e m i t t e r
source increases.
143
60
J(y-y
E l e c t r i c a l field n o r m a l to flow
E l e c t r i c a l f i e l d p a r a l l e l to flow
)k
40
20
10
KE
U
F i g u r e 43
J ( y - y Q)k
VERSUS
U2 (
12
KE
U
E l e c t r i c a l f i e l d n o r m a l t o flow
E l e c t r i c a l f i e l d p a r a l l e l t o flow
F(y-y
)k
2
4
F i g u r e 44
6
KE
F *y - y 0 *
eJ T -
~
8
10
12
VERSUS
U
144
APPENDIX D
T A B U L A T I O N O F DATA AND R E S U L T S
TABLE
1
T o r s io n C a lib r a tio n Data
M eter Reading
(division g)
Torque
(oz in c h)
M eter Reading
(divisions)
Torque
(oz in c h)
0
. 3^
15
6.35
255
0
. 9^
39
6
. 95
279
55
63
7.5 5
303
2. 15
87
8
. 15
327
2. 75
111
8.
3. 35
135
9 . 30
375
3.95
159
9. 85
399
4. 5 5
183
10. 45
423
5, 15
207
10. 95
447
5. 75
23 1
1 1. 35
47 1
1.
75
35 1
145
TABLE 2
Dat a a n d R e s u l t s f or Meat T r a n s f e r
Blade
Numbe r
1
Speed
r . p. m .
200
300
<.00
500
600
800
1000
1200
1*00
1600
1800
2000
2200
Current
amps
0 . 3*0
0.350
0. 375
0. 375
0 . 380
0. * 35
0. ** 5
0. *85
0.**0
0 . *7 5
0. * 8 0
0. *70
0 . * 70
2 (T w - T a w )
a*, v o l t s
30*9.0
250*.0
2316.0
2008.0
1652.0
1851.0
156*.0
17*3.0
1236.0
1*15.0
1189.0
1151.0
1122.0
F. xpe r i m e n t s
- 0 degree
11TU
H r FT1" ° F
5.87
7.57
9 . *0
10.8*
13.53
15.82
19.60
20.89
2*.2*
2*.68
29.99
2 9 . 70
30.*7
Nu
Re
*7.3
61.1
75.8
87.*
109.1
127.6
158.0
168.*
195.5
199.0
2*1.9
2 39.6
2*5.7
10291
15*36
20581
25726
30872
*1162
51*53
617*3
7203*
8232*
92615
102906
113196
I
200
300
*00
500
600
800
1000
1200
1*00
1600
1800
2000
2200
0 . 330
0 . 330
0.335
0 . 350
0.*00
0. *70
0. *05
0.535
0. *90
0. 530
0. 5*0
0. 5*0
0. 550
2987.0
2*1*.0
205*.0
1860.0
1873.0
1888.0
1559.0
1609.0
1 109.0
12**.0
1081.0
1035.0
1039.0
5.58
6.90
8 . 36
10.07
13.07
I 7.89
2 * . 0*
2 7.21
33.11
3*.5*
*1.26
*3.09
**.53
*5.0
55.6
67.*
81.2
105.*
1**. 3
193.9
2 19.*
267.0
2 78.5
332. 7
3*7.5
359. 1
13923
2088*
278*5
3*806
*1768
55690
69613
83535
97*58
111380
125303
139225
1531*8
3
200
300
*00
500
600
8 00
1000
1200
1*00
1600
1 BOO
2000
2 2 00
0.325
0.325
0. 330
0. 3*0
0 . 380
0. * 2 0
0 . *2 5
0. *65
0. *25
0. *50
0. *60
0. *55
0 . 505
3697.0
2819.0
2*18.0
22*3.0
17*7.0
1710.0
1*22.0
1563.0
11*9.0
1205.0
1039.0
966.0
1033.0
*.26
5.59
6.72
7.69
12.3*
15.*0
18.96
20.65
23.*7
25.09
30.*0
31 . 9 9
36.85
3*.*
*5.1
5*.2
62.0
99.5
I 2*.2
152.9
166.5
189.3
202.3
2*5.2
258.0
297.2
1755*
26332
35109
*3886
52663
70218
87772
106327
122881
1*0*36
157990
1755*5
193099
200
300
*00
500
600
800
1000
1200
1*00
1600
18 0 0
2000
2200
0 . 3*0
0.335
0 . 3 35
0. 3*0
0 . 360
0. *00
0. * I 5
O.*70
0 . * 30
0 . *75
0.510
0. 500
0. 500
25 3 3 . 0
2001.0
1553.0
1326.0
1237.0
1183.0
961 . 0
1129.0
819.0
931 . 0
912.0
8 92 . 0
886.0
6.73
8.27
10.66
12.85
15.*5
19.9*
26.*2
28.85
33.29
3 5 . 73
*2.05
*1.33
*1.61
5*.3
66. r
85.9
10 3 . 7
12*.6
160.8
213.1
2 32.7
268.5
288.2
339. 1
333.3
335.5
2 1186
3 1780
*237 3
52966
63559
8*7*6
105932
127119
1*8305
169*92
190678
21186*
233051
146
TABLE 3
Data and R e su lts for Heat T r a n s f e r K x p e r im e n t s - T r i p W ire Type A
BTU
Hr Ft* ° F
Nu
3497.0
2871.0
2341.0
2204.0
2722.0
2476.0
2204.0
2011.0
2316.0
2237.0
5.90
7.38
9.05
10.68
12.03
15.01
17.56
20.02
22.48
24.08
47.6
59.5
73.0
86. 1
97.0
121.0
141.6
161.4
181.3
194.2
10291
15436
20581
25726
30872
41162
51453
61743
72034
82324
0. 340
0 . 345
0. 360
0 . 360
0.425
0.435
0. 460
0,460
0. 540
0. 540
3127.0
2662.0
2305.0
2064.0
2327.0
1858.0
1526.0
1327.0
1477.0
1407.0
5.65
6.84
8.60
9.60
11.87
15.58
21.21
2 4 . 39
30.20
31.70
45.6
55.1
69. 3
77.4
95.7
125.6
171.0
196.7
243.5
255.6
13923
20884
27845
34806
41768
55690
69613
83535
97458
111380
200
300
400
500
600
800
1000
1200
1400
1600
0. 320
0 . 325
0 . 325
0. 330
0.440
0.470
0.470
0.470
0.500
0.500
3549.0
2895.0
2384.0
2148.0
3471.0
2988.0
2250.0
1925.0
1760.0
1679.0
4.31
5.45
6.61
7.57
8.33
11.04
14.66
17.13
21.20
22.23
34. 7
43.9
53.3
61.0
67.1
89.0
118.2
138.1
171.0
179.3
1 7554
26332
35109
43886
52663
70218
87772
105327
122881
140436
200
300
400
500
600
800
1000
1200
1400
1600
0. 340
0.350
0. 360
0 . 375
0. 450
0.460
0.460
0. 470
0. 540
0.540
2522.0
2167.0
186 3 . 0
1686.0
2112.0
1729.0
1381.0
12 3 0 . 0
1385.0
1279.0
6.76
8.34
10.26
12.30
14.14
18.05
22.59
26.48
31 . 0 4
33.62
54.5
67.2
82.7
99. 2
114.0
145.5
182.2
213.6
250. 4
271.1
21186
31780
42373
52966
63559
84746
105932
127119
148305
169492
Blade
Number
Speed
r , p. m .
Current
amps
1
200
300
<>00
500
600
800
1000
1200
1400
1600
0. 365
0 . 370
0 . 370
0.390
0.460
0.490
0.500
0.510
0. 580
0. 590
200
300
400
500
600
800
1000
1200
1400
1600
3
X(Tw-Taw>
volts
Re
147
T A B L E 3 (Continued)
Data a n d R e s u l t s for H eat T r a n s f e r E x p e r i m e n t s - T r i p W ire T ype B
Blade
Numbe r
1
Speed
r. p. m .
200
300
*00
500
600
auo
1000
1200
1*00
1600
200
300
t
*00
5 00
600
BOO
1000
1200
1*00
1600
3
200
300
*00
500
600
800
1000
1200
1*00
1600
200
300
*00
500
600
800
1000
1200
1*00
1600
BTU
Current
amps
Z(Tw -Taw )
XL VOltS
0 . 380
0.*10
0.*10
0 . * 30
0.525
0.530
0.610
0.630
0.695
0 . 700
2219.0
19*7.0
1583.0
1563.0
2036.0
1756.0
1931.0
1877.0
2067.0
1879.0
10.07
13.36
16.**
18.31
20.95
2 * . 76
29.82
3 2 . 73
3 6 . 17
* 0 . 36
81.2
137.8
I 32.5
1*7.7
169.0
199.7
2*0.5
263.9
291.7
325.5
10291
15*36
20581
25726
30872
*1162
51*53
617*3
7203*
8232*
0.370
0. 390
0.380
2027.0
1626.0
1323.0
1122.0
139*.0
1295.0
1379.0
1380.0
1539.0
1*61.0
10.33
I * . 29
16.69
1 9 . 17
27.*3
3*.**
*1.95
* 5. * 0
50.80
55.02
83.3
115.2
13*.6
15*.6
221.2
277. 7
338. 3
366. 1
*09.7
**3.7
13923
2088*
278*5
3*806
*1768
55690
69613
83535
97*58
11 1 3 8 0
2618.0
2028.0
17 3 5 . 0
1*28.0
159*.0
1192.0
1119.0
1055.0
1235.0
1322.0
9.59
12.98
16.28
18.**
21.58
27.66
33.35
35.37
39.27
* 1 . 33
77.3
10*. 7
131.3
1*8. 7
17*.0
223. 1
269.0
285.3
316.7
333.3
1 755*
26332
35109
*3886
52663
70218
87772
105327
122881
1*0*36
1870.0
1398.0
1268.0
1216.0
1261.0
1192.0
1319.0
1208.0
1358.0
1297.0
12.62
17.73
22.00
26.22
26.9*
3 3 . *5
38.91
*3.9*
*8.7*
52.57
101.7
1*3.0
177.*
211.*
217.3
269.7
313.8
35*.*
39 3 . 1
*23.9
21186
31780
*2373
52966
63559
8*7*6
105932
127119
1*8305
169*92
0.375
0.500
0.5*0
0.615
0.6*0
0.715
0. 725
0.*10
0. *20
0. *35
0. *20
0. *80
0. *70
0.500
0.500
0.570
0.605
0.*00
0.*10
0. *35
0. *65
0. *80
0.520
0. 590
0.600
0.670
0.680
Hr F t * 6 F
Nu
Re
148
T A B L E 3 (Continued)
D a t a a n d R e s u l t s f o r H e a t T r a n s f e r E x p e r i n i e n t 9- T r i p W i r e T y p e C
Blade
Number
Speed
r . p. m .
Current
amps
X(Tw-Taw)
V0lt8
BTU
Hr Ft1 b F
Nu
Re
10291.
15*36 .
20581.
25726.
30872.
61162.
51453.
61743.
72034.
8 2 324 o
1
200
300
400
500
600
800
1000
1200
1400
1600
0.405
0.410
0.405
0.405
0.510
0.520
0.600
0.600
0. 620
0.625
4021.0
3256.0
2637.0
2239.0
3106.q
2668.0
2953.0
2726.0
2563.0
2400.0
6 . 31
7.99
9.63
11.34
12.96
15.69
18.87
20.44
23.21
25.19
50.9
64.4
77.6
91.4
104.5
126.5
152.2
164.8
187.2
203.2
2
200
300
400
500
600
800
1000
1200
1400
1600
0 . 365
0.365
0 . 365
0 . 370
0 . 550
0 . 560
0.630
0.640
0. 640
0.650
3514.0
2772.0
2331.0
1940.0
3327.0
2659.0
2599.0
2394.0
2084.0
1975.0
5.80
7 . 35
8 . 74
1 0 . 79
1 3.91
18.04
23.36
26.17
30.06
32.72
46.8
59. 3
70.5
87.0
112.1
145.5
188.4
211.0
242.4
263.9
13923.
20884.
27845.
34806.
41768.
55690.
69613.
83535.
97458.
111380.
3
200
300
400
500
600
800
1000
1200
1400
1600
0.345
0 . 350
0 . 355
0.350
0.450
0.440
0.495
0.500
0.500
0. 500
4061.0
3264.0
2749.0
2360.0
3146.0
2350.0
2167.0
1913.0
1628.0
1509.0
4.38
5.60
6.84
7.75
9.61
12.30
16.88
19.51
22.92
2 4 . 73
35.3
45.2
55.2
62.5
77.5
99.2
136.1
157.3
184.9
199.4
17554.
26332.
35109.
43886.
52663.
70218.
87772.
105327.
122881.
140436.
200
300
400
5 00
600
8 00
1000
1200
1400
1600
0.370
0 . 360
0 . 380
0. 390
0.500
0.510
0. 600
0 . 600
0. 605
0.615
2887.0
2132.0
1916.0
1644.0
2420.0
1905.0
2146.0
1863.0
1668.0
1566.0
6.99
8.96
11.11
1 3.64
15.23
20. 13
24.74
28.49
32.36
35.61
56.4
72.3
89.6
I 10.0
122.8
162.4
199.5
229.8
260.9
287.2
2 1186.
31780.
42373.
52966.
6 3559.
84746.
105932.
12T119.
148305.
169492.
149
T A B L E 3 (Continued)
Data and Results for Heat T r a n s f e r E x p e r im e n ts - T r i p W ire Type D
Blade
Number
Speed
r . p. m .
Current
amps
Tw-Tawj
yu. v o l t s
BTU
Hr F ^ ° F
Nu
Re
1
200
300
400
500
600
800
1000
1200
1400
1600
0.430
0.435
0.445
0. 440
0.560
0.560
0.620
0. 630
0.695
0.695
2655.0
2004.0
1738.0
1440.0
2376.0
2005.0
2074.0
1923.0
2076.0
1941.0
10.78
14.61
17.63
20.81
20.43
24.21
28.69
31.94
36.01
38.52
86.9
117.9
142.2
167.8
164.7
195.2
231.3
257.6
290.4
310.6
10291
15436
20581
25726
30872
41162
51453
61743
72034
82324
I
200
300
400
500
600
800
1000
1200
1400
1600
0. 405
0.410
0.420
0. 4 20
0.560
0. 570
0.64C
0.645
0 . 720
0 . 725
2102.0
1609.0
1372.0
1138.0
2065.0
1737.0
1753.0
16 3 6 . 0
1778.0
1679.0
11.93
15.98
19.66
23.71
23.23
28.61
35.74
38.89
44.59
47.88
96. 2
128.9
158.6
191.2
187.3
230. 7
288.2
313.6
359.6
386. 1
13923
20884
27845
34806
41768
55690
69613
83535
97458
11 1 3 8 0
3
200
300
400
500
600
800
1000
1200
1400
1600
0.425
0.425
0.430
0.435
0.435
0.420
0.465
0.590
0. 595
0. 590
2808.0
2060.0
1793.0
1541.0
1590.0
1222.0
1236.0
1740.0
1560.0
1409.0
9.60
I 3.09
15.39
18.33
17.77
21.55
26.11
29.86
33.88
36.88
77.4
105.6
124. 1
147.8
143.3
173.8
210.6
240.8
2 73.2
297.4
17554
26332
35109
43886
52663
70218
87772
105327
122881
140436
4
200
300
4 00
500
600
800
1000
12 0 0
1400
1600
0.425
0.430
0.430
0.430
0.550
0.570
0.635
0.640
0 . 720
0.720
2005.0
1409.0
1323.0
1115.0
1834.0
1588.0
1724.0
1560.0
1720.0
1622.0
13.28
19.35
20.61
24.45
24.32
3 0 . 17
34.49
38.71
44.44
4 7.13
107.1
156.0
166. 2
197.2
196. 1
243.3
2 78.1
312.2
358.4
380.0
21186
31780
42373
52966
63559
84746
105932
127119
140305
169492
150
T A B L E 3 (Continued)
D ata a n d R e s u l t s lo r Meat T r a n s f e r E x p e ri m e a t s_T ri p W i r e T y p e E
Blade
Numb* r
1
3
BTU
H r Ftr u K
Current
amps
^ ( T w -Taw)
volt s
200
300
<.00
500
600
800
1000
1200
1400
1600
0.400
0.415
0.420
0.420
0.485
0.485
0.490
0.495
0 .615
0.620
4006.0
3363.0
2876.0
2469.0
2976.0
2388.0
2052.0
1881.0
2565.0
2389.0
6.18
7.93
9.49
1 1.06
12.23
15.25
18.11
20.16
22.82
24.90
49.9
63 .9
76.6
89.2
98. 7
122.9
1 46.0
162.6
184. 1
200.8
10291
15436
20581
25726
30872
41162
51453
81743
72034
82324
200
300
400
500
600
800
1000
1200
1400
1600
0.395
0.390
0.410
0 .400
0.555
0.560
0.565
0.575
0. 710
0.715
4112.0
3302.0
2896.0
2308.0
3909,0
2930.0
2186.0
2016.0
2622.0
2442.0
5.80
7.05
8.88
10.60
12.05
16.37
22.33
25.08
29.40
32.02
46.8
56.8
71.6
85.5
97.2
1 32.0
I 80. 1
202. 3
2 37.1
258.2
13923
20864
2 7845
34806
4 1 768
55690
69613
83535
97458
111380
200
3o0
400
500
600
800
1000
1200
1400
1600
0. 355
0.355
0. 350
J . 350
0. 460
0 .450
0.455
0.455
0.565
0 . 555
4179.0
3279.0
2712.0
2282.0
3493.0
2590.0
1944.0
1645.0
2143.0
1989.0
4.5 0
5 . 74
6 . 74
8.01
9 . 04
1 1 .67
15.90
18.79
22.24
23.12
36. 3
46.3
54.4
64.6
72.9
94. 1
128.2
151.5
179. 3
186.4
1 7554
26332
35109
43886
52663
70218
87772
105327
122881
140436
200
300
400
500
600
800
1000
1200
t <.00
1600
0.395
0.400
0 .410
0.410
0.530
0.535
0.535
0.540
0.685
0.690
3337.0
2645.0
2312.0
1898.0
2902.0
2164.0
1/78.0
1543.0
2140.0
1990.0
6.89
8.92
1 0 . 72
1 3 .06
14.27
1 9 . 50
2 3 . 74
2 7.87
32.33
35.28
55.6
7 1.9
86. 5
105.3
115.1
157.3
191.4
224 . 7
760. 7
284.5
21186
31780
42373
52966
63559
84746
105932
12 7 1 1 9
148305
169492
Speed
r . p. n i .
Nu
Re
151
T A B L E 3 (Continued)
it a and R e s u l t s toi
Blade
Numbe r
Speed
r . p. m .
200
300
<.00
1
600
600
800
1000
1200
1600
1600
200
300
600
6 00
600
800
1000
1200
1600
1600
3
200
300
600
500
600
e o o
1000
1200
1600
1600
200
300
600
600
600
800
1000
I 200
16 00
16 00
Cu r r e n t
amps
Heat
Transfer
j( T w -Taw)
y x volts
Is \ p e r i m e r i t s -T r ip W i r e
Type K
13T U
Hr n * ~ ° K
Nu
Re
I 3.95
16.37
16.69
19.62
21.90
2 6.3 1
2 7.36
28.77
67.2
90. 6
112.5
132.0
133.0
156.6
I 76. 6
196.0
220. 7
2 32.0
10291
15636
20581
25726
30872
61162
5 1653
81763
72036
82326
2 769.0
2083.0
1787.0
1666.0
2666.0
2156. 0
1811.0
1677.0
1625.0
1661.0
9. 5 1
1 2 . 36
16.76
18.26
18.13
22.67
26.68
29.63
33.16
36.66
76. 7
99. 5
118.9
167.1
I6fe.2
182.8
2 13.6
2 39.0
2 67.6
2 93.9
1 3923
20886
27865
36806
6 1760
55690
6961 3
8 3535
9 7658
111380
0 . 355
0 .3 /0
0. 3 /0
0. 3/0
0. 665
0.655
0.660
0. 660
0.665
0.660
2591.0
2271.0
1971.0
1659.0
2626.0
1913.0
1623.0
1656.0
1355.0
1363.0
7.26
9. 00
10.37
1 2 . 32
13.31
1 6 . 16
19.66
21.72
23.82
2 3.52
58.6
72.6
83.6
99. 3
137.3
1 30. 3
157.0
175.2
192.1
189.7
17556
26332
35109
63886
52663
70218
87772
135327
12288 1
160636
0. 600
0.600
0.600
0 .605
0.550
0 . 566
0.660
0.856
0.566
0 .570
2 321 . 0
1836.0
1558.0
1367.0
2191.0
1765.0
1661.0
1687.0
1629.0
1390.0
10.16
12.85
1 5 . 16
1 7.95
2 0 . 36
26.8 1
2 7.86
30.56
32.96
36.66
82.0
133.6
122. 1
1 66.8
166.2
230. 1
22 6 . 5
266,3
265.6
27 7 . 9
21186
3 1780
62373
52966
6 3559
86766
135932
127119
168305
1 59<»92
0.500
0.510
0.525
0.525
0.525
0 . 5 36
3122.0
2210.0
1865.0
1589.0
2367.0
2073.0
1968.0
1755.0
1559.0
1660.0
0.615
0.610
0.615
0.616
0 . 560
0. 565
0. 560
0 .5 /0
0.576
0. 590
0.610
0.600
0.610
0.6 10
8.33
11.21
152
T A B L E 3 (Continued
Data and R e s u lts for Heat T r a n s f e r E x p e r i m e n t s - T r i p W ire T yp e G
Blade
Numbe r
1
2
3
j(T w -T aw )
y i. volts
BTU
H r Ft* ° F
Speed
r, p. m .
C urrent
amps
100
200
300
*00
500
600
800
1000
1200
1*00
1600
0. 330
0 . 330
0 .350
0.355
0. *20
0. *20
0 .525
0 .530
0.535
0. 5*0
0 . 5*0
3605.0
2329.0
1972.0
1673.0
2 087.0
1908.0
2*62.0
2 178.0
2015.0
1821.0
1707.0
*.68
7.2*
9.61
11.66
13.08
1 * . 31
17.33
19.96
21.99
2 * . 78
26.**
37.7
58.*
77.5
9* . 0
105.5
115.*
139. 7
161.0
177.3
199.9
2 13.2
51*5.
10291.
15*36.
20581.
25726.
30872.
*1162.
51*53.
617*3.
7203*.
8232*.
100
200
300
*00
500
600
800
1000
1200
1*00
1600
0. 3*0
0. 3*0
0.350
0 .350
0. *35
0 . **0
0. 600
0.595
0. 600
0 .600
0.600
3563.0
2385.0
1913.0
1658.0
2070.0
192*.0
2576.0
2 10*.0
191*.0
1715.0
1605.0
*.96
7. * I
9 . 79
11.30
13.98
1 5 . 39
21.37
25.73
28.77
32.10
3*.30
*0.0
59.8
79.0
91.1
112.7
12*. 1
172.*
207.5
2 32.0
258.9
276. 7
6961.
13923.
2088*.
278*5.
3*806.
*1768.
55690.
69613.
83535.
97*58.
111380.
100
200
300
*00
500
600
800
1000
1200
1*00
1600
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
325
330
335
3*0
390
390
*90
*90
*95
500
500
*183.0
2660.0
2096.0
1726.0
1 988.0
1838.0
2295.0
1930.0
1870.0
I 789.0
I 861.0
3.77
6. 11
7.99
10.00
1 1.*2
1 2 . 35
15.62
18.57
19.56
20.86
20.05
30.*
*9. 3
6*. 5
80.6
92. 1
99.6
125.9
1*9.8
157.7
168.2
161.7
8777.
1755*.
26332.
35109.
*3886.
52663.
70218.
87772.
105327.
1228 8 1.
1*0*36.
100
200
300
*00
500
600
H00
1000
1200
1*00
1600
0 . 350
0 . 3*5
0. 3*0
0 . 3*0
0. *00
0. * 15
0 . 5 70
0.575
0.575
0. 580
0 . 5 70
3171.0
2139.0
1657.0
1385.0
1630.0
1532.0
229*.0
1993.0
I 791.0
1620.0
1*63.0
5.70
8.20
10.29
12.31
1* . * 7
16.58
20.88
2*.*6
2 7.22
30.62
32.7*
*5.9
66. 2
8 3.0
99. 2
116.7
13 3 . 7
168.*
19 7 . 3
219.5
2*6.9
26*. 1
10593.
2 1166.
31780.
*2373.
52966.
63559.
8*7*6.
105932.
12 7 1 1 9 .
1*8305.
169*92.
Nu
Re
153
TABLE 4
D a ta a n d R e s u l t s Io r H e a t T r a n s f e r E x p e r i m e n t s - 1 d e g r e e
B lade
Num ber
Speed
r . p. m .
C urrent
amps
j(T w -T aw )
yt LVOl l S
BTU
Hr F t* 6F
Nu
Re
1
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0.370
0 .370
0.390
0.410
0.350
0 .350
0 .390
0.395
0.440
0.445
0.445
0 .445
0.440
3742.0
2870.0
2567.0
2 370.0
1523.0
1271.0
1336.0
1221.0
1330.0
1250.0
1157.0
1032.0
996.0
5.66
7.38
9.17
10.98
12.45
14.92
17.62
19.78
22.53
24.52
26.49
29.70
30.08
45.7
59.5
74.0
88.5
100.4
120.3
142. 1
159.5
181.7
197.7
*L 3.6
2 39.5
2 42.6
10291.
15436.
20581.
25726.
30872.
41162.
51453.
61743.
72034.
82324.
92615.
102906.
113196.
2
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0.360
0.360
0.360
0.375
0.380
0 .395
0 .465
0.475
0.525
0 .520
0.525
0.525
0 .500
3642.0
2827.0
2445.0
2132.0
1683.0
1393.0
1468.0
1239.0
1384.0
1227.0
11 3 4 . 0
983.0
8 86.0
5.44
7.01
n . 11
10.09
13.12
17.13
22.53
2 7.85
30.46
33.70
37.17
42.88
4 3 . 16
43 .9
5 6.5
6 5.4
8 1.4
105.8
138. 1
161.7
224.6
2 45.6
271.6
299.8
345.8
348.0
13923.
20884.
27845.
34806.
41768.
55690.
69613.
83535.
97458.
1 11380.
125303.
139225.
153148.
1
200
3C0
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0 .350
0 .340
0 .350
0 .350
0.355
0 .355
0 .410
0 .410
0 .455
0 .455
0 .455
0 .450
0.445
4213.0
3124.0
2852.0
2 306.0
1637.0
1322.0
1337.0
1027.0
1131.0
1027.0
9 92.0
1070.0
1041.0
4.34
5.52
6.41
7.93
11.49
14.23
18.77
24.43
2 7 . 33
30.09
31.15
28.25
28.40
35.0
44. 5
51.7
64 .0
92 .7
114.8
151.4
197.0
220.4
242. 7
251.2
227.8
229.0
17554.
26332.
35109.
43886.
52663.
70218.
87772.
105327.
122881.
140436.
157990.
175545.
193099.
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0. 370
0 . 365
0 . 385
0. 380
0 . 370
0 . 390
0 .445
0 .445
0 .505
0 .505
0 .500
0 .505
0 .495
3041.0
2281.0
2050.0
1708.0
1390.0
1170.0
1263.0
1095.0
1218.0
1096.0
1042.0
1014.0
957.0
6.64
8.61
10.66
12.47
14.52
19.17
2 3 . 12
2 6.67
30.87
3 4 . 31
35.38
37.08
3 7 . 75
53.5
69. 5
86. 0
100. 5
117.1
154.6
186. 4
2 15.0
249.0
276. 7
285. 3
299. 1
304.4
2 1186.
31780.
42373.
52966.
63559.
84746.
105932.
127119.
148305.
169492.
190678.
211864.
233051.
1 54
T A B L E 4 (Continued)
D a t a a n d R e s u l t s f o r H e a t H - a r i Bf e r
B la d e
Num ber
Speed
r_ . p_ . m .
C urrent
a. .m. . p. . s.
I(T w -T iw )
, ,
yU
. volts
experim ents
BTU
iHt r. 5rtY
FtX o
°F
- 3 degrees
INU
Ke
1
200
100
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0 . 380
0.405
0 .405
0.415
0 .345
0.370
0 .445
0 .445
0.450
0 .500
0.520
0.565
0 .560
3864.0
3390.0
2883.0
2494.0
1655.0
1443.0
1698.0
1566.0
1418.0
1592.0
1441.0
1690.0
1627.0
5.78
7.49
8.81
10.69
11.13
14.68
18.05
19.57
2 2.10
2 4.31
2 9.04
2 9 .24
2 9.83
4 6.6
60.4
71 .0
86.2
8 9.8
118.4
1 45.6
1 57.8
178.2
196.0
2 34.2
2 35.8
2 40.6
10291.
15436.
20581.
25726.
30872.
41162.
51453.
61743.
72034.
82324.
92615.
102906.
113196.
2
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0.365
0.365
0 .370
0.360
0 .400
0 .400
0.510
0 .510
0 .510
0 .570
0 .560
0 .610
0.600
3 544.0
2 880.0
2535.0
2109.0
2109.0
1525.0
1758.0
1561.0
1351.0
1534.0
1273.0
1479.0
1346.0
5 .7 5
7.0 7
8 .26
9.40
11.60
16.05
22.63
2 5.48
2 9 .45
3 2.39
37.68
3 8.48
40.91
4 6.4
57. 1
66.6
75.8
9 3.6
129.4
182.5
2 05.5
2 37.5
2 61.2
303.8
310. 3
3 29.9
13923.
20884.
27845.
34806.
41768.
55690.
69613.
83535.
97458.
111380.
125303.
139225.
153148.
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0 .350
0 .350
0 .360
0.360
0.365
0 .365
0.425
0.410
0 .410
0 .450
0 .435
0.4T 5
0.470
4 028.0
3223.0
2980.0
2599.0
2390.0
1821.0
1791.0
1455.0
1228.0
1367.0
1189.0
1620.0
1610.0
4.5 4
5 .67
6.49
7.44
8.3 2
10.92
15.06
I 7.25
2 0.43
22.11
2 3.76
2 0 .7 9
2 0.48
36.6
4 5.8
52 .4
6 0.0
67. 1
88. 1
121.4
139. 1
164.8
178. 3
1 91.6
167. 7
165.2
17554.
26332.
35109.
43886.
52663.
70218.
87772.
105327.
122881.
1404 3 6 .
157990.
175545.
143099.
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0 . 360
0 . 365
0 . 365
0.365
0 .395
0 .400
0 .485
0 .475
0 .465
0.540
0.540
0.570
0 .550
2870.0
2 36 3 . 0
2037.0
1773.0
1875.0
1533.0
1815.0
1508.0
1225.0
1273.0
1091.0
1191.0
1061.0
6 .66
8.31
9 .64
1 1.08
1 2.27
1 5 . 39
19.11
2 2.06
2 6.03
3 3.78
39.41
4 0.22
4 2.04
53.7
67.0
77.8
89. 3
9 8 .9
124. 1
154. 1
1 77.9
209.9
2 72.4
317.8
324.4
339.0
21 1 8 6 .
31780.
42373.
52966.
63559.
84746.
105932.
127119.
148305.
169492.
190*78.
21 1 8 6 4 .
233051.
155
T A B L E 4 (Continued)
Data and R e » r ’t i fo r H eat T r a n s f e r E x p e r i m e n t * - 5 d e g r e e s
B la d e
Num ber
Speed
r. p. m .
C urrent
amps
(Tw -Taw)
volts
BTU
Hr FT* °F
Nu
Re
1
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0.380
0 .385
0 . 385
0 .405
0 .385
0 .410
0 .420
0 .475
0.475
0 .530
0.52 0
0.575
0.585
3945.0
3017.0
25O 4.0
2225.0
1964.0
1741.0
1533.0
1719.0
1521.0
1764.0
1525.0
1773.0
1737.0
5.67
7.60
8.95
11.41
11.68
14.94
17.01
20.31
22.96
2 4 .65
27.44
2 8.86
30.49
4 5.7
6 1.3
72.2
9 2.0
9 4.2
120.5
143.6
163.8
185.2
198.8
221.3
232.8
2 45.9
10291.
15436.
20581.
25726.
30872.
41162.
51453.
61743.
72034.
82324.
92615.
192906.
113196.
2
2 00
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0 . 360
0. 360
0 .360
0.365
0 .400
0.440
0.445
0 .495
0 .490
0.545
0 .545
0 .580
0 .570
3578.0
2824.0
2577.0
2164.0
2256.0
1983.0
1561.0
1644.0
1363.0
1578.0
1376.0
1547.0
1422.0
5.54
7.02
7.69
9.42
10.85
14.93
19.40
2 2 . 79
2 6.94
28.79
33.01
33.26
34.94
44.7
56.6
6 2.0
75.9
8 7.5
120.4
156.5
183.8
217.3
2 32.2
2 66.2
268.2
281.8
13923.
20604.
27845.
34806.
41768.
55690.
69613.
83535.
97458.
111380.
125303.
139225.
153148.
3
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0.330
0.325
0 .330
0 .330
0 .365
0 .370
0 .370
0.455
0 .435
0.530
0.535
0 .560
0.96 0
3822.0
2856.0
2562.0
2278.0
2485.0
2103.0
1799.0
2039.0
1792.0
2121.0
1935.0
2321.0
2285.0
4.25
5.52
6 . 35
7.14
8.00
9.72
11.36
1 5 . 16
17,25
19.77
22.00
2 0.17
2 0.49
34. 3
44 .5
51.2
57 .6
64 .5
78.4
9 1.6
122.2
139. 1
159.4
178. 1
162. 7
165.2
17554.
26332.
35109.
43886.
52663.
70218.
87772.
105327.
122081 .
140436.
157990.
175545.
19 3 0 9 9 .
200
300
400
500
600
800
1000
1200
1400
1600
1800
2000
2200
0 . 305
0 .385
0 .390
0. 390
0 .375
0.390
0 .420
0.480
0 .485
0 .530
0.325
0.580
0.580
3459.0
2713.0
2341.0
1972.0
1471.0
1257.0
1154.0
1286.0
1141.0
1229.0
1101.0
1231.0
1103.0
6 . 32
8.06
9.5 0
1 1 , 37
12.41
17.04
22.54
26.42
30.40
33.70
36.91
40.29
41.93
51.0
65.0
77.3
9 1.7
100. 1
143.9
181.8
213.0
245. 1
271.8
297. 7
324.9
338. 1
21106.
31700.
42373.
52966.
63559.
04746.
105932.
127119.
140305.
169492.
190678.
211864.
233051.
156
T A B L E 4 (Continued)
D a t a a n d R e s u l t s f u r H e a t T r a n s f e r K x p e r i n i e n t s - 10 d e g r e e s
Blade
Numbe r
Speed
r . p. m
200
300
400
500
600
1
SOO
1000
1200
1400
1600
1800
200
i
300
400
500
600
800
1000
1200
1400
1600
1600
200
>00
3
400
500
600
800
1000
1200
1400
1600
1800
200
300
400
500
600
800
1000
1200
1400
1600
1800
C urrent
amps
j
(T w -T .iw )
«.l v o l t s
/ L
ivru
H r KtT ' J K
Nu
Re
45.4
10291.
15436.
20581.
25726.
3C972.
41162.
51453.
61743.
72034.
82324.
92615.
2155.0
1964.0
2157.0
1947.0
2040.0
1938.0
1698.0
5.63
7.69
9.46
11.35
1 3.59
17.04
20.16
21.91
25.52
26.87
30.66
0.360
0 . 390
0 . 390
0 .400
0.420
0.420
0 .490
0.495
0 . 540
0.540
0.540
3244.0
2460.0
2007.0
1 t50.0
1660.0
1426.0
1597.0
1543.0
1627.0
1567.0
1477.0
6.11
9 . 46
11.59
1 3.98
16.25
18.92
22.99
24.29
27.41
28.10
30.20
131.1
152.6
185.4
195.9
221.1
226.6
243.5
0. 330
0.330
0 .345
0 .350
0.410
0 .400
0.440
0 .460
0.560
0.575
0 .570
3379.0
2327.0
1996.0
1713.0
2063.0
1591.0
1626.0
4 .81
6.99
8.89
38.8
56.3
71.7
10.68
86. 1
2187.0
2066.0
12.16
15.01
1 7 . 77
19.40
22.60
22.57
23.48
0.415
0 .420
0.430
0.430
0.420
0.430
0.490
0.500
0.555
0.555
0.550
2606.0
1991.0
1 763.0
1540.0
1404.0
1241.0
1391.0
1332.0
1490.0
1442.0
1424.0
9 . 74
1 3.06
15.46
17.70
18.53
2 1.97
25.45
2 7.67
30.48
31.50
31.32
0 . 380
0 . 380
0 . 380
0 . 385
0.435
0 .465
0 . 530
0.525
0.580
0.580
0.580
3970.0
2905.0
2363.0
2021.0
11628.0
2222.0
62.0
76.3
91.5
139.6
137.4
162.5
176.7
205.8
216.7
247.3
49.3
76. 3
93.5
112.8
13923.
20884.
27845.
34806.
41768.
55690.
69613.
83535.
97458.
111380.
125303.
143.3
156.5
182.3
182.0
189.3
I 7554.
26332.
35109.
43886.
52663.
70218.
87772.
105327.
122881.
140436.
15 7 9 9 0 .
78.6
105.4
124. 7
142.8
149.4
177.2
205. 2
223.2
245.8
254.0
252.6
21186.
31780.
42373.
52966.
6 3559.
84746.
105932.
127119,
148305.
169492.
190678.
98.1
121.1
157
T A B L E 4 (Continued)
Data and R e s u lts for Heat T r a n s f e r
Blade
Numbe r
i
3
E x p e r i m e n t s - 20 d e g r e e s
B T If
—
H r F t * 0 !'
M
Current
amps
J(Tw -T aw )
^U. v o lt s
200
300
*00
500
600
BOO
1000
1200
1*00
1600
0.390
0 .*20
0 .* 1 5
0.*15
0. *80
0 .500
0.560
0.570
0 .6*0
0.635
2727.0
2280.0
19711.0
16*0.0
2126.0
1879.0
2075.0
1980.0
2301.0
2101.0
8.63
11.97
13.*8
16.25
16.77
20.59
23.39
25.*0
27.55
29.70
69.6
96.6
108.7
131.1
135.3
166. 1
188.6
2 0*.a
222.2
239.6
10291.
15*36.
20581.
25726.
30872.
*1162.
51*53.
617*3.
7203*.
8232*.
200
300
*00
500
600
800
1000
1200
1*00
1600
0 . 390
0 .*20
0 . * 30
0. *30
0.*30
0.*50
0.515
0.515
0.570
0.570
2 7 39.0
2397.0
2193.0
16 3 8 . 0
1612.0
1*66.0
1637.0
1517.0
1706.0
1606.0
8.*9
1 1.26
12.90
15.39
1 7.5*
21.13
2 * . 78
2 6 . 7*
29.09
30.9*
68.5
90.8
10*.0
12*. 1
1*1.5
1 70.*
199.8
215.6
23*.6
2*9.5
13923.
2088*.
278*5.
3*806.
*1768.
55690.
69613.
83535.
97*58.
111380.
200
300
*00
500
600
600
1000
1200
1*00
1600
0 .330
0.3*0
0 . 335
0.335
o.**o
0 .**5
0.500
0.510
0.565
0 . 580
2822.0
220*.0
1857.0
1561.0
2379.0
19*1.0
20*9.0
1 950.0
2170.0
2190.0
5.76
7.83
9 .02
10.73
12.15
15.23
18.21
19.91
21.96
22.93
*6.5
63. 1
72.8
86.5
98.0
122.8
1*6.9
160.6
177.1
18*.9
1755*.
26332.
35109.
*3886.
52663.
70218.
87772.
105327.
122881.
1*0*36.
200
300
*00
500
600
800
1000
1200
1*00
1600
0 . *25
0 . *2 5
0. *20
0 . *2 5
0 . * 30
0 . * 50
0.510
0. 525
0.590
0.580
31*0.0
2*73.0
2023.0
1 739.0
1663.0
1*70.0
16*5.0
1513.0
1 7 7* . 0
1751.0
8.*8
10.77
12.86
1 5 . 32
16.39
20.31
23.31
26.86
2 8.9)
28.3)
68.*
86.9
103.7
123.5
1)2.2
163.8
188.0
2 16.6
2 )3 .)
228.*
21186.
31780.
*2373.
52966.
6)559.
8*7*6.
105932.
127119.
1*8305.
169*92.
Speed
r . p, m
Nu
Re
158
T A B L E 4 (Continued)
D a t a a n d R e s u l t s f o r H e a t T r a n s f e r E x p e r i m e n t s - JO d e g r e e s
Blade
Number
1
3
Speed
r. p. m .
C urrent
amps
j(T w -T iw )
V O ltS
BTU
Hr Ft1 F
Nu
Re
200
300
400
500
600
800
1000
1200
1400
0.410
0 .420
0 .430
0 .430
0 .470
0 .470
0 .580
0.580
0.600
2863.0
2248.0
1896.0
1668.0
1 897.0
1538.0
2002.0
1741.0
1547.0
9.09
12.15
15.09
17.16
18.02
22.23
26.01
29.91
36.02
73.3
97.9
121.7
138.4
145.3
179.3
2 09.7
241.2
290.5
10291
15436
20581
25726
30872
41162
51453
61743
72034
200
300
400
500
600
800
1000
1200
1400
0.390
0.415
0.430
0.430
0.450
0.455
0.540
0.550
0.555
2998.0
2567.0
2310.0
1907.0
1881.0
1575.0
1983.0
1830.0
1768.0
7.76
10.26
12.24
14.83
16.47
20.10
22.49
25.28
26.65
62.6
82.8
98. 7
1 19.6
1 32.8
162. i
181.4
203.9
214.9
13923
20884
27845
34806
41768
55690
69613
83535
97458
200
300
400
500
600
800
1000
1200
1400
0.325
0. 330
0 . 350
0.335
0.460
0 .465
0.580
0.580
0.580
2981.0
2183.0
1867.0
1533.0
2931.0
2317.0
2941.0
2637.0
2356.0
5.29
7.45
9.79
10.93
10.78
13.93
1 7.08
19.04
21.31
42. T
60. 1
79.0
88.1
8 6.9
1 12.3
137.7
153.6
171.9
17554
26332
35109
43886
52663
70218
87772
105327
122881
200
300
400
5 C0
600
800
1000
1200
1400
0.400
0.415
0.420
0.420
0.470
0 .480
0.600
0.600
0.600
3101.0
2565.0
2 196.0
1838.0
2 126.0
1806.0
2420.0
2116.0
1972.0
7.61
9.90
11.84
14.15
15.32
18.81
21.93
25.09
26.92
61.4
7 9.8
95.5
114.1
123.6
151.7
1 76.9
202.3
217.1
21186
31780
42373
52966
63559
84746
105932
127119
148305
159
T A B L E 4 (Continued)
Data and R e su lts tor Heat T r a n s f e r e x p e r i m e n t s
Blade
Numbe r
Speed
r . p. m.
C urrent
amps
X(T w - T a w )
y u v o 11 s
BTU
H r Ft * 0 F
- 45 d e g r e e s
Nu
Re
1
200
300
*00
500
600
800
1000
0.420
0.410
0.415
0.420
0.445
0 .450
0.510
2823.0
2012.0
1 750.0
1590.0
1633.0
1378.0
1503.0
9.67
12.93
15.23
17.17
18.77
22.74
26.78
78.0
104. 3
122.8
13B .5
151.4
183.4
2 16.0
10291
15536
20581
25726
30872
41162
51453
I
200
300
400
500
600
800
1000
0.420
0.410
0.415
0 .420
0.450
0.455
0.510
2798.0
2045.0
1787.0
1570.0
1646.0
1410.0
1574.0
9.64
12.57
14.74
17.18
18.82
2 2 . 4o
25.27
77.8
101.4
118.9
138.b
151.7
181.1
203.8
13923
20884
27845
34806
41 768
55690
69613
3
200
300
400
500
600
eoo
1000
0. 340
0 . 340
0 . 340
0.345
0.470
0.470
0.530
2560.0
1940.0
1648.0
1467.0
2 534.0
2 009.0
2 182.0
6.74
8.90
10.47
12.11
13.01
16.41
19.22
54.4
71.7
04.4
97.7
104.9
132.4
155.0
17554
26332
35109
43886
52663
70218
87772
200
300
400
500
600
800
1000
0.385
0.390
0.390
0.400
0.450
0.450
0.510
2832.0
2213.0
1865.0
1663.0
1995.0
1656.0
1818.0
7 . 72
1 0 . 13
12.03
14.19
14.97
18.03
2 1.10
62.2
81.7
9 7.0
114.4
120.7
145.4
170.1
21186
31780
42373
52966
63559
84746
105932
160
T A B L E 4 (C ontinued)
Data and R e s u lts for H eat T r a n s f e r E x p e r i m e n t s
Blade
Number
Speed
r.p .m .
C urrent
amps
J£(Tw-Taw)
volts
200
300
500
500
600
800
0.525
0.525
0.520
0.530
0.555
0.520
29 5 7 .0
2275.0
1810.0
1660.0
1755.0
1839.0
1
200
0.510
0.510
0.500
0.510
0.560
0.520
2571.0
1925.0
1539.0
1502.0
1557.0
1718.0
300
5 00
500
600
800
0.530
0.520
0.520
0.520
0.570
0.530
3675.0
2811.0
2299.0
2063.0
2379.0
2513.0
200
300
500
500
600
800
0.375
0 .380
0.375
0.375
0.550
0.500
300
500
500
600
800
200
}
2557.0
2027.0
1666.0
1383.0
1875.0
1890.0
BT U
H r BT ° r
60 d e g r e e s
Nu
Re
76.2
99.1
121.6
139.0
158.1
183.5
10291.
15536.
20581.
25726.
30872.
51162.
10.50
13.36
I 5.90
18.35
20.92
25.07
8 3.9
107.8
128.2
157.9
168.7
195. 1
13923.
20885.
27855.
35806.
51768.
55690,
7.51
9 . 37
I 1.55
12.76
1 3.86
1 7 . 38
60.6
75.5
92.5
102.9
150. 1
17555.
26332.
35109.
53886.
52663.
70218.
65.7
85.7
100.5
120.9
128.5
157.3
21186.
31780.
52373.
52966.
63559.
85756.
9.55
12.29
15.08
17.25
18.36
2 2.76
8.15
10.50
12.55
15.99
15.92
19.50
111.8
161
T A B L E 4 (Continued)
D a t a a n d R e s u l t s f o r H e a t T r a n s f e r E x p e r i m e n t s - 90 d e g r e e s
Blade
Number
1
2
SpeedC u rren t
X(Tw-Taw)
r . p. m .
amps
yu.volts
i
10.44
1 3 . 75
16.10
1 8 . 12
2 0 . 80
2 3 . 74
84.2
1 10.9
129.8
146. 1
167.8
191.4
13923.
20884.
27845.
34806.
41768.
55690.
3082.0
2225.0
1896.0
1541.0
1397.0
1231.0
7.37
9.69
11.07
13.26
14.63
I 7.97
59.4
78. 1
89.3
106.9
1 18.0
145.0
17554.
26332.
35109.
43886.
52663.
70218.
2273.0
1634.0
1413.0
1242.0
9.37
12.35
14.6'.
1 7.14
19.14
22.42
75.5
99.6
118.3
138.2
154.3
180.8
21186.
31780.
42373.
52966.
63559.
84746.
200
0.400
0 . 385
0 . 390
0 . 380
0 . 395
0.400
2343.0
1649.0
1445.0
1219.0
1147.0
1031.0
0 . 390
0 . 380
0.375
0 . 370
0 . 370
0.385
0.380
0 . 370
0 . 375
0.380
0.395
0 . 395
300
400
500
600
800
200
300
400
500
600
800
Re
10291.
15436.
20581.
25726.
30872.
41162.
0.405
0.39*
0.405
0 .405
0.420
0.420
200
Nu
85.1
1 10.0
130.2
148.7
164.1
187.4
200
300
400
500
600
800
300
400
500
600
800
BTU
H r Ft
F
2407.0
1771.0
1573.0
1377.0
1342.0
1175.0
1202.0
1026.0
10.55
13.64
16.14
18.44
20.34
23.24
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FIE L D
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1 70
A PPEN D IX E
IN ST R U M EN TA TIO N AND M A T E R IA L S
T h i s a p p e n d i x is i n c l u d e d in o r d e r to s u p p l e m e n t t h e t e x t
w ith d e t a ile d in f o r m a tio n r e g a r d i n g the s p e c i f i c a t i o n s a n d s o u r c e s
of t h e i n s t r u m e n t a t i o n a n d i m p o r t a n t m a t e r i a l s u s e d in t h e f a b r i c a ­
t i o n a n d o p e r a t i o n of t h i s a p p a r a t u s .
TH ER M O C O U PLE SYSTEM
P o ten tio m eter
Ranoe:
M fg : :
T y p e K-3
0 - 1 . 5 volts
L eeds and N o rth ru p
P hiladelphia, P a.
1648565
S e r la l:
E l e c t r o n i c Null d e t e c t o r
S e n s lti v it y :
R es ista n ce:
M fg .:
S erial:
T herm ocouple w ire
Type:
2 m i c r o volts p e r div isio r
25000 o h m s .
Leeds and N orthrup Com pany
P hiladelphia, P a.
N 12-966
B i S G A 28
ISA T y p e J
C . L a u d S. G o r d o n C o m p a n y
C hicago
M fg.:
T h e rm o c o u p le s lip -rin g and
Type:
' Rlip- r i ngt
Brush:
Mfg. :
b r u s h unit
S tandard
AC 2 6 3 -2 4
AC 2 6 2 - 2 4
P o ly S cientific C o r p o r a tio
B la ck sb u rg , V irginia
H EA TER SYSTEM
A m m e t e r D. C .
Rang e:
M fg .:
M odel
No.
C om pany
0-1. 5 m p
W atson
901
364
17 1
V o l t m e t e r D. C .
Range:
Mfg. :
M odel:
No.
R heostats
R esistance:
M fg.:
0-15 volts.
W atson
90 1
658
0-10 ohm s,
Ohm ite
0-50 ohm s
Heater s l i p - r i n g and b ru sh
M fg.:
D e s ig n e d by the a u t h o r
a n d c o n s t r u c t e d in M . E .
H eater w ire
Size:
R esistance:
M fg.:
Ba t t e r y .
Output:
M fg.:
Sh o p
. 008 inch dia n i c h r o m e
4 .6 3 o h m s p e r foot
D r iv e r H a r r i s Co.
2 v o l t s D. C .
800 A. H . a t 8 H r . r a t e
The e l e c tr ic s to r a g e b a tte r y Co.
P hiladelphia, P a.
TORSION M E A S U R E M E N T
T o rsio n M eter
M fg.:
T o r s io n S pring
Type:
D e s i g n e d by t h e a u t h o r
a n d c o n s t r u c t e d in M . E . S h o p
F l a t w i r e t o r s i o n type
. 0 5 1 5 " x . 5 0 3 " A IS I 1070
S teel 5 1 / 8 coil f r e e
1, 1 2 5 " a r b o r d i a .
3 3. 5 " d e v e l o p e d l e n g t h
SPEED MEASUREMENT
Strobotac
Range:
M fg.:
Range:
M fg.:
6 0 0 - 3 2 0 0 r . p. m .
G e n e ra l Radio C om p an y
1 10 - 6 9 0 r . p. m .
G e n e ra l Radio C om pany
172
F ollow er
M fg.:
M otor
M fg.:
Speed C ontrol
R heostats:
G e n e ra l Radio C om pany
S ta r K inble
D . C . , 5 K . W.
A m p s . 20
R .P .M .
1750
S tandard p o rtab le rh e o s ta ts
from P u rd u e U niversity
E le c tr ic a l E n g in e erin g School
173
Appendix F
NOMENCLATURE
A
surface area
p a r a m e t e r in v e l o c i t y m o d e l
a.
constant
b
constant
1
1
b
S e m i - c h o r d width
of t h e b l a d e
C
c h o r d w i d t h of t h e
blade
C,
1. x
j r
local skin frictioncoefficient
A v e r a g e d skin friction coefficient
T
A v e ra g e d ra g coefficient
C
Specific heat
P
c.
l
D
constant
F
function
f
function
G
function
g
function
H
P a r a m e te r relating integral thicknessses
h
A v e r a g e heat t r a n s f e r c o e f f i c i e n t ,
i
iurrent
J
p a ra m e te r relating integral thicknesses
K
p a r a m e te r relating integral thicknesses
K
m o b i l i t y of i o n s
k
T h e rm a l conductivity
L
p a r a m e te r relating integral thicknesses
M
Torque
R e a d i n g of T o r s i o n m e t e r
function
174
N
Speed
N
A veraged Nusselt num ber
u
Nu,
x
N
P
lo c a l N u89elt n u m b e r
N u s s e l t n u m b e r b a s e d on c h o r d w i d t h and a v e r a g e
heat tra n s fe r coefficients
Pressure
E le c tric a l power
H o r s e - powe r
Pr
Prandtl number
q"
H e a t f l ux
Re,x
local Reynolds n u m b e r
Re, x
R e y n o l d s n u m b e r b a s e d on t o t a l l e n g t h of
Re
R e y n o l d s n u m b e r b a s e d on c h o r d
Re
A v erag e Reynolds n u m b e r
T
Tem perature
flow p a t h
of t h e b l a d e
t
time
U
F re e s t r e a m velocity
u
X com ponent velocity
v
Y com ponent velocity
W
T h i c k n e s s of t h e b l a d e
w
Z com ponent velocity
X
x ^
Tangential coordinate direction
-1 y
S i n ( — ) d e f i n e s l e a d i n g a n d t r a i l i n g e d g e s of b l a d e
V
R a d i a l c o o r d i n a t e d i r e c 11 o n
d i s t a n c e n o r m a l to the e l e c t r o d e
Z
Axial co o rd in a te d ire c tio n
ft
angular velocity
<5
Boundary layer thickness
wall s h e a r s t r e s s p a r a m e t e r
c
Pe rm itivity
r;
sim ilarity variable
dynam ic viscosity
kinem atic viscosity
d ensity
V olum e c h arg e density
E le c tr ic a l conductivity
S h e a r St r e s s
176
VITA
A b r a h a m E b e n e z e r M u t h u n a y a g a m w a s b o r n on
J a n u a r y 11, 1939 in I d a i y a n g u d i , I n d i a .
He g r a d u a t e d f r o m C. C. M.
1954.
He i s a c i t i z e n of I n d i a .
High School, Idaiya ngudi,
He a t t e n d e d St . J o h n s C o l l e g e ,
Tinnevelly,
in J u n e
f r o m 1954 t o
1956, a n d t h e n A l a g a p p a C h e t t i a r C o l l e g e of E n g i n e e r i n g a n d
Technology,
Karaikude.
He g r a d u a t e d in J u n e I 9 6 0 ,
t h e d e g r e e of B a c h e l o r of E n g i n e e r i n g ,
receiving
Mechanical Branch,
in f i r s t c l a s s w i t h H o n o r s f r o m t h e U n i v e r s i t y of M a d r a s .
He
was ranked second am ong seventy s u c c e s s f u l can did ates.
He
t h e n j o i n e d t h e I n d i a n I n s t i t u t e of S c i e n c e ,
B a n g a l o r e for his
g r a d u a t e s t u d y a n d r e c e i v e d t h e d e g r e e of M a s t e r o f E n g i n e e r i n g ,
P o w e r Engineering (Mechanical),
w i t h D i s t i n c t i o n i n 1962.
was r an k e d f i r s t a m o n g ten s u c c e s s f u l c a n d i d a te s .
He
In S e p t e m b e r
1962 h e a r r i v e d i n t h e U n i t e d S t a t e s a n d j o i n e d P u r d u e U n i v e r s i t y
as a graduate student.
At t h i s t i m e h e w a s e m p l o y e d at
T herm ophysical P r o p e r tie s R e s e a rc h Center as a Graduate
R esearch Assistant.
He w a s g i v e n a P u r d u e R e s e a r c h F o u n d a t i o n
g r a n t f o r 1963 - 1965 e n a b l i n g h i m t o w o r k t o w a r d s t h e P h . D.
d e g r e e in t h e S c h o o l
Mr.
oi
Mechanical Engineering.
M u t h u n a y a g a m is a s t u d e n t m e m b e r ol A . S . M . E . ,
a s s o c i a t e m e m b e r of S i g m a Xi , a n d m e m b e r of Pi T a u S i g m a .