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Perturbation solutions for the effect of prandtl number on heat transfer from a non-isothermal rotating disk
LETTERS IN HEAT AND MASS TRANSFER Volume 4, Pages 53-62, Pergamon P r e s s , 1977. Printed in Gt. Britaln.
PERTURBATION SOLUTIONS FOR THE EFFECT OF PRANDTL NUMBER ON HEAT TRANSFER FROM A NON-ISOTHERMAL ROTATING DISK
Sun-Yuan Tsay and Yen-Ping Shih Department of Chemical Engineering National Cheng Kung University Tainan, Taiwan, China
(C~l,,unicated by Y. Mori)
ABSTRACT A general pertubatlon method for the analysis of the effect of Prandtl number (Pr) on the steady state heat transfer across laminar incompressible boundary layer is proposed. The method is applied to the heat transfer of a rotating disk with step discontinuity of temperature. The method is a combination of coordinate pertubatlon to d e a l l ~ t h the step discontinuity of temperature and parameter (Pr- " ) perturbation to study the effect of Prandtl number on heat transfer. The zero-order solution is the Lighthill solution. The coefficients are universal functions. The convergence is satisfactory for Pr ~ i.
Introduction
This paper s t u d i e s
the s t o a d y o s t a t e hoot t r a n s f e r s c r e e n l ~ s i n a r
inoosrpromsible o o n s t a n t p r o p e r t y boundary l a y e r o f a r o t a t i n K d i s k with a a t o p disHsontinuity o f temperature in a l a r g e body o f qaencont f l u i d . The extreme e a s e o f t h i s problem with i n f i n i t e s t u d y by Smyrl and No~mn [ 1 ] , solution Shill
P r a n d t l number has been
and i s known as L i g h t h i l l
[ 2 ] ° R e c e n t l y , Chso [ ~ ] ,
a p p r o x i m a t i o n t o the s t u d y o f h e a t t r a n s f e r
umiak a two term r e p r e s e n t a t i o n o f the v e l o o i t y step discontinuity
approximation
and Chao and G r i e f [~] extended LiKhfrom r o t a t i n K body
field.
For ~ d g e
flow with
o f s u r f a c e temperature, Chao and Cheean [~3 used a
c o o r d i n a t e p e r t u r b a t i o n t e c h n i q u e t o a n a l y s e the s t e a d y s t a t e h o s t t r a n s f e r s However, t h e P r a n d t l number e f f e c t
on t h e temperature d i s t r i b u t i o n
l o c a l h e a t t r a n s f e r r a t e was not f i l l y
and
investiKatod.
a c o m b i n a t i o n o f the c o o r d i n a t e p a r t u r h a t i o n o f Chao and Chasms [~3
53
54
S.Y. T s a y a n d Y . P .
and p a r a m e t e r p e r t u b a t i o n p a p e r . The c o e f f i c i e n t m ate perturbation
heat transfer
Vol. 4, No. I
w i t h P r "~ a s p a r a m e t e r i n p r o p o s e d i n t h i s
a r e shown t o be u n i T s r s a l
taken care of the discontinuity
whereas the parameter perturbation on h e a t t r a n s f e r ,
Shih
functions. of surface
deals with the Prandtl
The c o o r d i n temperature,
nusber effect
T h i s t e c h n i q u e would be a p p l i e d t o o t h e r b o u n d a r y l a y e r
problems, The Governing Equations
Under t h e 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 t h e e n e r g y e q u a t i o n and boundary oonditions as referred
~T
4. W
U - -
~T - -
to Fig.
k
1
|~"s
~tT
(z)
=
T(r,O)
= T~
r ~
T(r,O)
= Tw
r ~r
re
(2)
e
(3)
T(r, oo) ..T~
The s o l u t i o n
of the Telocity
u = r~
~C7) -
r~
field
is
[6]
(-,mC~/)) 2 (6) Z
w
FIG. i Rotating Disk
VOI. 4, NO. i
HEAT TRANSFER FROM A R3T~TING DISK
55
whore (7)
H(~?) . - . 9 , ,
a ~ 6 ÷ 000
(8)
with
a = 0.510, Introduo~
b ,, - 0 . 6 1 6
of
(9)
g = Tw--T.
r
.
[1-
tol
i.
¢~,].,t-
~. ,-, ,,~ -~.,
(1o)
,. ~,~
gqo. ( 1 - 1 0 ) b e o o w |
~
¢1-1,~21A pr- -I- ~-H,(~z) j
[ R ,,¢
,~" (11)
._ A,I
~¢1-R,)
H,('~)
m
e(R,O) O(R,~)
~O.o "~ R
1
.
.
(12)
0
(15)
"
Perturbation 1~t
i,,o
j,,o
Solutlon
56
S.Y. Tsay and Y.P. Shih
Vol. 4, No. 1
Inserting Zq. (14) i n t o Eqs. (if-I}) and oquatiug tho © o e f f i c i n e t ~ of equal powerB of Ri P r ' ~ yield ~;',,
+ 3 ~'~,o
e~',, + 3 ~'W,,
(3.9
" o
-
~
i A. ~ , W . . ~e,. = ~-
O~',, + 3 ~"Qi', " 6~'Q,,,
1
= T A,
(16)
~',Q,,,, + T
b
A,
~V,, (17)
eJ',, + 3 ~'e~,, - 9 y ~ ,, " T
b A, ~-'Q, ,, A, ~-,~, ,, - A, ~.,Q, ,, - ~..
+ ~
e:,,,
+ )~"w,.,
Q;"
- 12 ~,,,,,
=-~A,.
~-,Q.,,.
- )~.,~, ,,,
¢la)
(19)
~ t4
Zb A' ~,O, ,, " ' i ~ 3
with ,o ( o )
~,o(m)
= ",,
t , ,,, ( o )
. e,,,,,(~)
= ~ ,,,, ( o )
= e,,,,,(~)
. . . . .
.....
o
o
(21)
(2z)
Vol. 4, NO. 1
HEAT TRANSFER F R O M A R C E A T I N G
Oi, j a r e u n i v e r s a l
functions.
~,o
is the Lighthill
DISK
57
approximation nolution
a s g i v e n by S m ' y l and Newnan [ 1 ] . N o t i c e t h a t t h e nonvaninhinK c o e f f i c i e n t s are 0i, i, i l O, i, 2, ... , and 0i+3, i , i = I, 2, 3, . . . . Eq.
Therefore,
(l~) gives
O(~'|~l)
• 01, |
"~" ( 0 | | |
"1" 0 1 ) , ,
i~|)])
]pll-- ]
..1- ° ° .
i ÷ ( G i , i + e l + 3 , i R ) ) R i P r ' ~ - ÷ ...
The a n a l y t i c a l
solution
(23)
o f ~ ,o and • I ,, arc
(2k)
(25) lo
o t h e r e l , j a r e o b t a i n e d by n u m e r i c a l i n t e & T a t i o n o F i g . 2 show8 t h e l ~ m a l t s . Typi©al t e m p e r a t u r e d i s t r i b u t i o n s the series
truncated at i >~0
a r e shown i n F i g . 3 u s i n g Eq. (2}) w i t h
The c o n v e r p n e e
o f Eqo (2~) i n s a t i s f a c t o r y
for Pr~l.
Rate of Heat Transfer
The l o c a l h e a t t r a n n f e r
q.
rate,
q, in
- k ( ~~T ' ~ ) S l O = h(T w
T. )
(26)
and t h e l o c a l N u s s e l t number, Nu, i s
lu.
-~
--
~.o
(27)
58
S.Y. Tsay and Y.P. Shih
I'0
,
,
,
,
,
Vol. 4, No. 1
,
,
,
,
,
,
0"9 0"8
0"7 0"6 0-5 0-4 0"3 0'2
t
0,.,(,jf-~---~
O0
(~4
0'1 0'0
(>8
1"2
2"4
2"0
1~5
¢
0"6
0'4
0"2
-
Oa s(¢)
Oi,j(¢) 0'0
-0'2
"0"40-0
~__~ I
I
0"4
J
I
0-8
I
I
1"2
I
L
1"6
06,31~'1XIO
i
I
2-0
FIG. 2 Universal functions of rotating disk with step discontinuity of surface temperature
Z
i
2"4
V o l . 4, NO. 1 Substitutlns
HEAT TRANSFERFRC~ARDTATINGDISK Eq.
(2~)
i n t o Eq. ( ~ )
~u = - A"~ R"~ P ~
59
one o b t a i n s
Q'(O,R)
(28)
with
o,(o,~) . e,,,,(o) + [ e . . ( o ) + ...
+ e~,,(o)~, ] ~ ~r'~
+ {@I,i(O) + Q[+},i(O)R) ] Ri pr'~ + . . .
whops
@~,. (0) = -
1.1198%
~',4
e;,,(o) .
o.3539~
@~ , t ( 0 )
0~ ,e (0) .
0.0695615
,5 ( o ) . - o . o o 6 ~ 2
e~ ,, (o) .
o.0181006
,~ (o)
e~,, (o) =
0.00000o26
In the calculation @4,,
~,~
---
of ~(0,R),
.
.
= 0.005?965
. o.oo~2o3
soles
the coefficients
.
.
o f R~ i n t h e b r a o k e t l ,
f o r m i l l 1 1 R.
t s a n be n o s l s c t e d I'0
(o) . o.oo8747o6
.
.
.
.
.
.
.
r./r'07"~t r oJJo (>9
~%,
005 ,_
O, ~,
r./r-oo6
e (>~
\~'
(>4
"'-."
-
~'
~
,,,~
-.. ~ ~____~ ~ ~
"~"
o,
...
o., • (>0 0
~.~_'~ 0-4
, Oe
"--.~.-.-'y~._~ 1.2
. . . . 1"6
20
2~
FIG. 3
Temperature distribution of rotating disk with step discontinuity of surface temperature
60
S.Y. Tsay and Y.P. Shih
Vol. 4, No. i
Conclusion
The perturbation method proposed in this paper is a general method which can be applied to other forced flow boundary layer heat transfer problems. The coordinate perturbation in terms of R deals with step change of surface temperature whereas the parameter perturbation in Pr -V3 shows the effect of Prandtl number.
The coefficients are universal functions.
Acknowledgment
The authors appreciate the financial support of National Science Council.
Nomenclature
a, b
flow c o n s t a n t s d e f i n e d by Eq. (8)
A
(~)}-- defined by ~ . (10)
Cp
Specific heat
F, H h
d i m e n s i o n l e s s v e l o c i t y f u n c t i o n s d e f i n e d by Eqs. ( 5 - 6 ) local heat transfer coefficient at r
k Nu
thermal conductivity
Pr
P r a n d t l number,
l o c a l Nmsselt number a t r NPC~k
radial
coordinate
radial
c o o r d i n a t e a t whioh the t e m p e r a t u r e has a d i s c o n t i n u e d
s t e p change R
t r a n s f o r m e d d i m e n s i o n l e s s cOOrdinate d e f i n e d by Eq. (10)
T
temperature
m~
v
v e l o c i t y components i n r-and z - d i r e c t i o n s ,
respectively
F(n)
coordinate norlal gamma f u n c t i o n =
~(n,x)
incomplete gamma function =
r
t r a n s f o r m u d d i m e n s i o n l e s s c o o r d i n a t e d e f i n e d by Eq° (10)
s
to disk surface -~tn-le'tdt ~x
tn-le-tdt
dimensionless coordinate dsfinedbyEq.
(7)
G
d i m e n s i o n l e s s t e m p e r a t u r e d e f i n e d by ER. (9)
p
kinematic viscosity
P
density azimuthal coordinate a n g u l a r v e l o c i t y o f d i s k in
J-direction
Vol. 4, NO. 1
HEAT T R A N S F E R F R O M A R O r A T I N G D I S K
61
References
1. W. H. SQyrl and J . Newman, R i n g - d i s k and S e c t i o n e d d i s k e l e c t r o d e s , J . E l e c t r o c h e m , Soc. ~l~, 212-219 ( 1 9 7 2 ) . 2. M. J .
Lighthill,
C o n t r i b u t i o n s t o the t h e o r y o f h e a t t r a n s f e r
a l a m i n a r boundary l a y e r ,
P r o c . Roy. S o c . , London, A 20_2, 359-~77 (19~0).
~. B. T. Chao, An improved L i g h t h i l l ' s boundary l a y e r s ,
through
a n a l y s i s of heat t r a n s f e r
I n t . J . Neat Mass T r a n s f e r , ~ ,
through
907-920 ( 1 9 7 2 ) .
4. B. T. Chao and R. G r e i f , Laminar f o r c e d c o n v e c t i o n over r o t a t i n g T r a n s . ASME J . Heat t r a n s f e r
bodies,
C, 96, 46~-466 ( 1 9 7 4 ) .
~. B. T. Chao and L. S. Cheema, F o r c e d c o n v e c t i o n in wedge flow with nonisothealal
surfaces,
I n t . J . Heat Mass T r a n s f e r , ~4, 1363-1375 ( 1 9 7 1 ) .
6. W. G. Cochran, The flow due t o a r o t a t i n g
~., ~5-375 (19~).
d i s c , P r o c . Camb. P h i l ,
Soc.
PERTURBATION SOLUTIONS FOR THE EFFECT OF PRANDTL NUMBER ON HEAT TRANSFER FROM A NON-ISOTHERMAL ROTATING DISK
Sun-Yuan Tsay and Yen-Ping Shih Department of Chemical Engineering National Cheng Kung University Tainan, Taiwan, China
(C~l,,unicated by Y. Mori)
ABSTRACT A general pertubatlon method for the analysis of the effect of Prandtl number (Pr) on the steady state heat transfer across laminar incompressible boundary layer is proposed. The method is applied to the heat transfer of a rotating disk with step discontinuity of temperature. The method is a combination of coordinate pertubatlon to d e a l l ~ t h the step discontinuity of temperature and parameter (Pr- " ) perturbation to study the effect of Prandtl number on heat transfer. The zero-order solution is the Lighthill solution. The coefficients are universal functions. The convergence is satisfactory for Pr ~ i.
Introduction
This paper s t u d i e s
the s t o a d y o s t a t e hoot t r a n s f e r s c r e e n l ~ s i n a r
inoosrpromsible o o n s t a n t p r o p e r t y boundary l a y e r o f a r o t a t i n K d i s k with a a t o p disHsontinuity o f temperature in a l a r g e body o f qaencont f l u i d . The extreme e a s e o f t h i s problem with i n f i n i t e s t u d y by Smyrl and No~mn [ 1 ] , solution Shill
P r a n d t l number has been
and i s known as L i g h t h i l l
[ 2 ] ° R e c e n t l y , Chso [ ~ ] ,
a p p r o x i m a t i o n t o the s t u d y o f h e a t t r a n s f e r
umiak a two term r e p r e s e n t a t i o n o f the v e l o o i t y step discontinuity
approximation
and Chao and G r i e f [~] extended LiKhfrom r o t a t i n K body
field.
For ~ d g e
flow with
o f s u r f a c e temperature, Chao and Cheean [~3 used a
c o o r d i n a t e p e r t u r b a t i o n t e c h n i q u e t o a n a l y s e the s t e a d y s t a t e h o s t t r a n s f e r s However, t h e P r a n d t l number e f f e c t
on t h e temperature d i s t r i b u t i o n
l o c a l h e a t t r a n s f e r r a t e was not f i l l y
and
investiKatod.
a c o m b i n a t i o n o f the c o o r d i n a t e p a r t u r h a t i o n o f Chao and Chasms [~3
53
54
S.Y. T s a y a n d Y . P .
and p a r a m e t e r p e r t u b a t i o n p a p e r . The c o e f f i c i e n t m ate perturbation
heat transfer
Vol. 4, No. I
w i t h P r "~ a s p a r a m e t e r i n p r o p o s e d i n t h i s
a r e shown t o be u n i T s r s a l
taken care of the discontinuity
whereas the parameter perturbation on h e a t t r a n s f e r ,
Shih
functions. of surface
deals with the Prandtl
The c o o r d i n temperature,
nusber effect
T h i s t e c h n i q u e would be a p p l i e d t o o t h e r b o u n d a r y l a y e r
problems, The Governing Equations
Under t h e 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 t h e e n e r g y e q u a t i o n and boundary oonditions as referred
~T
4. W
U - -
~T - -
to Fig.
k
1
|~"s
~tT
(z)
=
T(r,O)
= T~
r ~
T(r,O)
= Tw
r ~r
re
(2)
e
(3)
T(r, oo) ..T~
The s o l u t i o n
of the Telocity
u = r~
~C7) -
r~
field
is
[6]
(-,mC~/)) 2 (6) Z
w
FIG. i Rotating Disk
VOI. 4, NO. i
HEAT TRANSFER FROM A R3T~TING DISK
55
whore (7)
H(~?) . - . 9 , ,
a ~ 6 ÷ 000
(8)
with
a = 0.510, Introduo~
b ,, - 0 . 6 1 6
of
(9)
g = Tw--T.
r
.
[1-
tol
i.
¢~,].,t-
~. ,-, ,,~ -~.,
(1o)
,. ~,~
gqo. ( 1 - 1 0 ) b e o o w |
~
¢1-1,~21A pr- -I- ~-H,(~z) j
[ R ,,¢
,~" (11)
._ A,I
~¢1-R,)
H,('~)
m
e(R,O) O(R,~)
~O.o "~ R
1
.
.
(12)
0
(15)
"
Perturbation 1~t
i,,o
j,,o
Solutlon
56
S.Y. Tsay and Y.P. Shih
Vol. 4, No. 1
Inserting Zq. (14) i n t o Eqs. (if-I}) and oquatiug tho © o e f f i c i n e t ~ of equal powerB of Ri P r ' ~ yield ~;',,
+ 3 ~'~,o
e~',, + 3 ~'W,,
(3.9
" o
-
~
i A. ~ , W . . ~e,. = ~-
O~',, + 3 ~"Qi', " 6~'Q,,,
1
= T A,
(16)
~',Q,,,, + T
b
A,
~V,, (17)
eJ',, + 3 ~'e~,, - 9 y ~ ,, " T
b A, ~-'Q, ,, A, ~-,~, ,, - A, ~.,Q, ,, - ~..
+ ~
e:,,,
+ )~"w,.,
Q;"
- 12 ~,,,,,
=-~A,.
~-,Q.,,.
- )~.,~, ,,,
¢la)
(19)
~ t4
Zb A' ~,O, ,, " ' i ~ 3
with ,o ( o )
~,o(m)
= ",,
t , ,,, ( o )
. e,,,,,(~)
= ~ ,,,, ( o )
= e,,,,,(~)
. . . . .
.....
o
o
(21)
(2z)
Vol. 4, NO. 1
HEAT TRANSFER F R O M A R C E A T I N G
Oi, j a r e u n i v e r s a l
functions.
~,o
is the Lighthill
DISK
57
approximation nolution
a s g i v e n by S m ' y l and Newnan [ 1 ] . N o t i c e t h a t t h e nonvaninhinK c o e f f i c i e n t s are 0i, i, i l O, i, 2, ... , and 0i+3, i , i = I, 2, 3, . . . . Eq.
Therefore,
(l~) gives
O(~'|~l)
• 01, |
"~" ( 0 | | |
"1" 0 1 ) , ,
i~|)])
]pll-- ]
..1- ° ° .
i ÷ ( G i , i + e l + 3 , i R ) ) R i P r ' ~ - ÷ ...
The a n a l y t i c a l
solution
(23)
o f ~ ,o and • I ,, arc
(2k)
(25) lo
o t h e r e l , j a r e o b t a i n e d by n u m e r i c a l i n t e & T a t i o n o F i g . 2 show8 t h e l ~ m a l t s . Typi©al t e m p e r a t u r e d i s t r i b u t i o n s the series
truncated at i >~0
a r e shown i n F i g . 3 u s i n g Eq. (2}) w i t h
The c o n v e r p n e e
o f Eqo (2~) i n s a t i s f a c t o r y
for Pr~l.
Rate of Heat Transfer
The l o c a l h e a t t r a n n f e r
q.
rate,
q, in
- k ( ~~T ' ~ ) S l O = h(T w
T. )
(26)
and t h e l o c a l N u s s e l t number, Nu, i s
lu.
-~
--
~.o
(27)
58
S.Y. Tsay and Y.P. Shih
I'0
,
,
,
,
,
Vol. 4, No. 1
,
,
,
,
,
,
0"9 0"8
0"7 0"6 0-5 0-4 0"3 0'2
t
0,.,(,jf-~---~
O0
(~4
0'1 0'0
(>8
1"2
2"4
2"0
1~5
¢
0"6
0'4
0"2
-
Oa s(¢)
Oi,j(¢) 0'0
-0'2
"0"40-0
~__~ I
I
0"4
J
I
0-8
I
I
1"2
I
L
1"6
06,31~'1XIO
i
I
2-0
FIG. 2 Universal functions of rotating disk with step discontinuity of surface temperature
Z
i
2"4
V o l . 4, NO. 1 Substitutlns
HEAT TRANSFERFRC~ARDTATINGDISK Eq.
(2~)
i n t o Eq. ( ~ )
~u = - A"~ R"~ P ~
59
one o b t a i n s
Q'(O,R)
(28)
with
o,(o,~) . e,,,,(o) + [ e . . ( o ) + ...
+ e~,,(o)~, ] ~ ~r'~
+ {@I,i(O) + Q[+},i(O)R) ] Ri pr'~ + . . .
whops
@~,. (0) = -
1.1198%
~',4
e;,,(o) .
o.3539~
@~ , t ( 0 )
0~ ,e (0) .
0.0695615
,5 ( o ) . - o . o o 6 ~ 2
e~ ,, (o) .
o.0181006
,~ (o)
e~,, (o) =
0.00000o26
In the calculation @4,,
~,~
---
of ~(0,R),
.
.
= 0.005?965
. o.oo~2o3
soles
the coefficients
.
.
o f R~ i n t h e b r a o k e t l ,
f o r m i l l 1 1 R.
t s a n be n o s l s c t e d I'0
(o) . o.oo8747o6
.
.
.
.
.
.
.
r./r'07"~t r oJJo (>9
~%,
005 ,_
O, ~,
r./r-oo6
e (>~
\~'
(>4
"'-."
-
~'
~
,,,~
-.. ~ ~____~ ~ ~
"~"
o,
...
o., • (>0 0
~.~_'~ 0-4
, Oe
"--.~.-.-'y~._~ 1.2
. . . . 1"6
20
2~
FIG. 3
Temperature distribution of rotating disk with step discontinuity of surface temperature
60
S.Y. Tsay and Y.P. Shih
Vol. 4, No. i
Conclusion
The perturbation method proposed in this paper is a general method which can be applied to other forced flow boundary layer heat transfer problems. The coordinate perturbation in terms of R deals with step change of surface temperature whereas the parameter perturbation in Pr -V3 shows the effect of Prandtl number.
The coefficients are universal functions.
Acknowledgment
The authors appreciate the financial support of National Science Council.
Nomenclature
a, b
flow c o n s t a n t s d e f i n e d by Eq. (8)
A
(~)}-- defined by ~ . (10)
Cp
Specific heat
F, H h
d i m e n s i o n l e s s v e l o c i t y f u n c t i o n s d e f i n e d by Eqs. ( 5 - 6 ) local heat transfer coefficient at r
k Nu
thermal conductivity
Pr
P r a n d t l number,
l o c a l Nmsselt number a t r NPC~k
radial
coordinate
radial
c o o r d i n a t e a t whioh the t e m p e r a t u r e has a d i s c o n t i n u e d
s t e p change R
t r a n s f o r m e d d i m e n s i o n l e s s cOOrdinate d e f i n e d by Eq. (10)
T
temperature
m~
v
v e l o c i t y components i n r-and z - d i r e c t i o n s ,
respectively
F(n)
coordinate norlal gamma f u n c t i o n =
~(n,x)
incomplete gamma function =
r
t r a n s f o r m u d d i m e n s i o n l e s s c o o r d i n a t e d e f i n e d by Eq° (10)
s
to disk surface -~tn-le'tdt ~x
tn-le-tdt
dimensionless coordinate dsfinedbyEq.
(7)
G
d i m e n s i o n l e s s t e m p e r a t u r e d e f i n e d by ER. (9)
p
kinematic viscosity
P
density azimuthal coordinate a n g u l a r v e l o c i t y o f d i s k in
J-direction
Vol. 4, NO. 1
HEAT T R A N S F E R F R O M A R O r A T I N G D I S K
61
References
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