Appl. sci. IRes.
Section A, Vol. 8
H E A T T R A N S F E R TO LAMINAR FLOW
B E T W E E N P A R A L L E L PLATES W I T H A
P R E S C R I B E D WALL H E A T F L U X
by R. DI CESS and E. C. S H A F F E R *)
Westinghouse Research Laboratories, Pittsburgh 38, Pa., U.S.A.
Summary
T h e first t h r e e e i g e n v a l u e s a n d c o n s t a n t s , as well as a s y m p t o t i c expressions
for t h e s e q u a n t i t i e s , are p r e s e n t e d for h e a t t r a n s f e r t o l a m i n a r flow b e t w e e n
p a r a l l e l f l a t p l a t e s w i t h a s y m m e t r i c a l l y p r e s c r i b e d wall h e a t flux.
§ 1. Introduction. The problem of heat transfer to fully developed
laminar flow between parallel plates for which the wall heat flux
is prescribed has been considered by S e l l a r s , T r i b u s and K l e i n 1)
through a generalization of constant wall temperature results.
Recently Siegel, S p a r r o w and H a l l m a n 2) have considered
flow through a circular tube directly from the boundary condition
of a prescribed heat flux at the wall.
In the present analysis fully developed laminar flow between
parallel flat plates with a prescribed wall heat flux is considered
directly from the heat flux boundary condition. This is accomplished
by utilizing the transformation employed in 2) for circular tubes.
In addition asymptotic expressions are given for the eigenvalues
and constants appearing in the series solutions.
§ 2. Constant wall heat/lux. The coordinate system for the present
problem is shown in fig. 1. Fluid flows in the x-direction with a fully
developed laminar velocity profile. For x < 0 the flow is isothermal
at temperature to, whereas for x > 0 there is a constant heat flux
qw at both walls. Incompressible flow of a fluid having constant
properties is assumed.
*) Present address, Cornell University, Ithaca, New York.
- - 339 - -
340
R. D, CESS
AND
E. C. SHAFFER
F o r large x a fully developed t e m p e r a t u r e profile te exists of
the form
t~ - - to
4
-
qwa/k
-
I
3
x
39
1
4-~-~2--~4---280
Pe a
'
(1)
where ~ = y/a, k is the t h e r m a l c o n d u c t i v i t y of the fluid, and Pe
the Peclet n u m b e r defined as
Pe = 4Uma/a,
for which c~ and Um are the t h e r m a l diffusivity and m e a n velocity
of the fluid respectively. Following S i e g e l , S p a r r o w and H a l l m a n , a new variable m a y be defined from (1) as
t -- to
qwa/k
O(x, ~) --
t~ - - to
qwa/k
(2)
i'////t//////~//////////iz/////z/11,
/
/
z
/
/
z
/
F
/
/
x:O
qw
Fig. 1. Coordinate system.
I n t e r m s of O(x, ~) the e n e r g y equation becomes
(1 -- ~2)
80
--
8x
8
820
3Pea
8~2
(3)
with the b o u n d a r y conditions
39
3 ~2 4-
o(o,~)- 280
--
8
8~
o(~,
o) =
4
--
8
8~
O(x,
1
T~4'
1) =
o.
B y separation of variables the solution of (3) m a y be w r i t t e n as
oo
0 = Y, cnYn(~) exp
n=l
(
8 ~n2 x)
3
Pe
a
,
(4)
HEAT TRANSFER
BETWEEN
PARALLEL PLATES
341
where fin and Yn(~) are the eigenvalues and eigenfunctions respectively of the equation
v"
+/~2(1
-
~2)y
=
(s)
o
with the b o u n d a r y conditions
Y'(O) = Y'(I) = O.
(6)
Since (5) and (6) constitute a Sturm-Liouville
series coefficients Cn are given b y
/1(~
cn =
system, the
_ ~a ~2 +~1 ~4)( l __ ~2)y n d$
-/'ol (1 -- ~') g n 9' de
(7)
Following G r a e t z a), the integral appearing in the denominator
m a y be written as
and from the b o u n d a r y condition for $ :
1
-
1 this becomes
1 Yn(1)(
-
f0(1 - ~2)Yn2 d~ -- 2fin
\
D2y
/e-2g°
g
(8)
±
The integral appearing in the numerator of (7) m a y also be written
in a more tractable form. U p o n integrating b y parts twice and
making use of (5) there results
L(28o
1
39
_
3 ~2_[_
24 (1 - - ~ 2 ) Y n d ~
-- 1 Yn(1) +
-
/~n~
-- 1 Yn(1)
--
1)
%-
4
3
(1 -- ~ 2 ) V n d ~ =
~1 ( d y n
2fi;4 do d\~--/,
fin 2
such that from the b o u n d a r y conditions for Y(~)
39
280
1)
34 ~2 -b ~_~4
(1 - - ~ 2 ) Y n d ~ - -
-1
fin 2 Yn(1).
(9)
342
R. D. CESS
AND
E. C. SHAFFER
From (7), (8), and (9) the series coefficients become
2
cn =
(10)
The foregoing results m a y be written in terms of the usual heat
transfer parameters. Combining (2) and (4) and setting ~----- 1,
the expression for the local wall temperature tw is
tw--to
qwa/k
4 x + 17
oo
(
Pea
-35 +n=Ix cnYn(1) exp
8 fin2 x )
3 Pe a . (11)
The local Nusselt number is defined as
Nu =
qw
4a
(tw -- tb) k
and from an energy balance the bulk fluid temperature t~ is given by
tb -- to
4
qwa/k
x
Pe a
Thus, from (11)
N u .=
4
17
co ~
3-5 + ~:~
• cnYn(1) exp
(12)
(8fln2x)
3 Pe
a
The eigenvalues fin and constants c~ and Yn(1) are determined
from the solution of (5). Letting~Y(0) = 1, this solution satisfying
the condition Y'(0) = 0 is t h e ~ame as that. for the constant ~waR.
temperature case given by P r i n s , M u l d e r and S c h e n k 4) as
oo
Y(~) = E a2m~2m,
m=0
where
C~0 ~
a2m =
1,
a2
f2
2m (2m -- 1)
----- - - - -
2 '
(a2m-4
--
a2m-2).
HEAT TRANSFER BETWEEN PARALLEL PLATES
343
In order to satisfy the boundary condition Y'(1) = 0, the eigenvalues Bin are the roots of
oo
ma2m = O.
m=0
The first three values of Bin have been determined, and these results
together with the corresponding values of Cn and Yn(1) are listed
in table I.
TABLE I
~n
T
4.287224
/
Cn
0.175024
8.30372 1 --0.051725
12.3114
0.02506
Yn(1)
--1.26970
1.4022
---1.4911
It is of further interest to determine the asymptotic expressions
for fin, cn, and Yn(1). From 1) the asymptotic solution of (5) for
large fi and ~ close to unity is *)
2
~sin (-fi~
~2) j~ (fl3~/-fl
3~-~
~ ~]sl2) -12
)1
, (13)
where ~ = 1 - - ~ . I n order to satisfy the condition d Y / d r t = 0
for ~ = 0, it is required that
Bin = 4n @ ~,l
(14)
which is the asymptotic expression for large u. Correspondingly
Yn(1) = Y n ( ~ = O) --
(-- 1)n0z~*
fin ~ = (-- 1)n 0.97103 fin ~, (15)
whereas from (10)
Cn =
(--1) n+x
211/6JP(~)341a
a/=
f i n - h i 6 = ( - - 1)n+12.4727fln -w".
(16)
These asymptotic expressions can be used to augment the values
given in table I.
*) I n *) the constant appearing in front of (13) is given as ½, whereas the correct
value is }.
344
HEAT TRANSFER BETWEEN PARALLEL PLATES
For the present case of a constant wall heat flux one can define
a thermal entrance length XT as the length for which the Nusselt
number is within 5 percent of the fully developed value 140/17.
From (12) this entrance length is found to be
XT/a = 0.046Pe.
§ 3. Arbitrary wall heat flux. The foregoing results m a y be
extended to include an arbitrary wall heat flux qw(x) by the method
presented in 2). A differential step is considered such that the
heat flux is zero for x < x' and has a value dqw for x > x'. The
wall temperature for x > x' corresponding to this incremental
wall heat flux is given by (11), replacing qw and x by dqw and
( x - x') respectively. The wall temperature distribution for an
arbitrary heat flux qw(X) is obtained through integration by parts
to give
tw -- to
=
IF{ 4
--IX{
S-~cnYn(1)
-- k J0 I.Pe -- 3~=1
Pee exp
-
-
8 Pe
qw(x')dx'. (18)
Received 4th December, 1958.
REFERENCES
I) S e l l a r s , J. R., M. T r i b u s alld J. S. K l e i n , Trans. Amer. Soc. Mech. Engrs 7B
(1956) 444,
2) S i e g e l , R., ~E. M. S p a r r o w and T. M. H a l l m a n , Appl. Sci. Res. A7 (1958) 386.
3) G r a e t z , L., Ann. Phys. u. Chemie ~-5 (1885) 337.
4) P r i n s , J. A., J. M u l d e r and J. S c h e n k , Appl. Sei. Res. A2 (1951) 431.