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A study on laminar motion of a viscous incompressible fluid between two parallel eccentric cylinders
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0093-6413/92
A
STUDY
$8.00 + .00
ON
Vol. 1 9
(1) ,48-49,
LAMINAR MOTION OF A VISCOUS BETWEEN TWO PARALLEL ECCENTRIC O.EI-F.Badr,M.N.Farah
D e p t . o F Math.., F a c u i t y o F
1992
CopyPlght (c) 1991
and
Science
P P i n t e d In the U S A
Pe~.mon
INCOMPRESSIBLE CYLINDERS
Press plc
FLUID
A.Ei-M.Badran
,M a n s o u r e
Uni v e r s i t ~.,E g y p t
(Received 6 June 1991; accepted for print 26 AuEust 1991) SUMMARY The flow o~ incompressible viscous fluid between two eccentric cylinders under the action of a difference between the pressures imposed at the ends of the cylinders is analyzed. Using bipolar coordinates,the resulting boundary value problem is s o l v e d analvtically,and the average ~lux velocity is calculated for various values of the geometric parameters of the problem. INTRODUCTION In m a n y branches o~ nuclear research [12],the exchange oheat inside the reactor o~ the heavy water demands, beside a specific heat eneroy ,an exact chec"~:, o n t h e limited values o~ the decrease ir~ 0 r e s s u r e inside t h e r e a c t o r . It a l s o demands exact information about the volume o,~ t h e w a n t e d heavy water which must be a= little as oossible .With regard to the last demand,speciai technical constructions o E t h e h e a t e~-:changer a p p a r a t u s shou.ld be needed. Such apparatus may be composeO of arranged cor, c e n t r i c or eccen+~,-ic c y l i n d e r s ~ s o that the inner space between them is very smmil to allow flow of little heavy' w a t e r . The simple idea upon which this apparatus is d e s i g n e d is t o i n s e r t a cylindrical body inside .÷ p i p e t o 6 o r m a c o n c e n t r i c space between the body and the oip_~ However,in prar+ice ~t m a y n o t b e D o s s i b i e to a-,,o~d the ~ormation of an eccentric space in-tea.u oE the idea! ,-.'-,nren+r~.~ one.
i~_ i s w e ! l - k n o w n rM, ~ . ±.:,J _ that such ec,..-.entrici~v_, r a u s e s _ a d e c r e a s e in pressure o.~ the fluid,so that it should be taker into _ a ~~m e n t . consideration ~or an exact ~u~-~ The probiem of the lamin_~r ~!ow oE an incompressiDie ~;!t~id b e t w e e r ~ t,~o e...-_centric c y l i n d e r s has been studied in many earlier work,.Mac Donai0 [8] developed the method of the torsional strength to -a.icuiate the volume flux and the frictional forces of the inner and outer cylinders. Some authors [6~9,!3] have used con,~ormal mappino t.o t r a n s f o r m the problem to a oeom.e~rv o~ c~ncent~ic annu!us,Gor which solution was a!read,,, known. Others oer#ormed calculation usinq_ cylindrical ~~-..,~5 ] o;5iu--'_ _ _~rr,n r ~,.~_n a r e s [ 4 , 7 . I0. i ! ] . However,computations based or, these ~.~,r~:'~ m a v produce as much as 20 % error [6,~3]. in t h e o r e s e n t wo.~'i,:,an._=.ivtica.', s c , ! u t i o q ~ n r t h e ~,robie~T. i~ four, d by. usino., b ~ p o l _ ~ r r o n.r d i.n a.t . e.~ _ . a .n d . ~ o n ~ v S n q ~ . t h e ,,~-~~--'~,_,,_,s c.~ -flDu~--er a n a i , / s i s. The advantaoe o-F t h i s solution is these c a]culat!ons can be carried out to a higher decree oE accuracy .Fo~- a l l value.=_ c,E the ._Teometrir_ param~+~_.__.-= . . .~ . . ~ . ' ~ , - p n * r i c i t v ).~ aod 5 iratio o E radii~,. 45
46
M. FARAH and A. B A D R A N
Q. EL-FAROUK,
MATHEMATICAL
FOP.MULATIONS
We considerthe problem of steady laminar ~low of a n incompressible viscous Eluid passing through the interspace between two eccentric parallel cylinders with radii r and r
I
(r < r
i
2
Z
).
Using bipolar coordinates between the axes of the two fol lows: a r
=
arid
i
t-
(~.C,z) the radii and cylinders-fig. (1)-can
[I
2
e=a(coth
[z
- ¢oth
J
,
sLr, h
~z
(1)
~rI )
('2
}'
)
Fig. (I) G e o m e t r y ~ is ~ h e d i s t a n c e ~rom the equation
between
the
x
o F the p r o b l e m
two
cer, t e r s ( p o l e s ) a n d
r
a
*
1
=
er
2
F ( r. 2 2
L
rZ
+
t
-- e z )
2
2
-. . . . . . . . .
--
t
i
2
J
,
2
calculated
4rZr z ]
[(I+~2--C~2' --4~2] /2 =
is
I/2
2
~ - = E~-
e as
a
~
sL~
the distance be expressed
(2)
r-
where
6 =-El
,
:r"
e
c~
= - -
(3)
~"
Assuming that the flow follows that the velocity
has neitheru has only
spin one
nor radial motion non-zero component
--
while
the
other two vanish, = u = 0
u~.
z a
7-
'4)
'
, the
=
(-)
=
0
continuity
and
Navier-Stol:es
equations (c~
z
a,-,
ap ,
,,,,
[ (ico s }', <-cos
~)]
=
2
0
~
(6)
2
2
@ U= (
+
0
El
z
a
of
z ) = ! OF._._~ 2
W
7..'
~z
ac
where p is the The bc)un~ar-)~
modified pressure and u the dynamic viscosity. conditions i:or the a>'ial velocity u are:
(i>
the
No
sli0
on
it ,
i.e.
C
In t h i s c a s e become ~u
u
inner
and
outer
cylinders, i.e.
[i]]
LAMINAR
FLOW BETWEEN
CYLINDERS
) = O,
(8)
= O,
(9)
Uz(~'~2)
~or all 0 _< ~ _< 2n (//)The symmetry of the solution in t h e both sides of the straight line passing the cylinders gives u (~,~) = u ( - ~ , ~ ) , z
for- a l l which (iii)u
normal through
cross section the two axes
implies
and
that
(I0)
(z <
("
~i
<
u must be an even f u n c t i o n
in
z
must be p e r i o d i c
~.
in ~.
SOLUTIONS t h r e e e q u a t i o n s (5) and ( 6 ) , i t is obvious that u must be a ~unction of ~ and E , whereas
From t h e f i r s t axial velocity
z
m o d i G i e d p r e s s u r e p d e p e n d s o n l y on z . T h i s means t h a t e q u a t i o n (7) s h o u l d be c o n s t a n t , i . e . d~ = -C dz Substituting in e u u a t i o n 2
C
each s i d e
"-0 .
(7),
the the of
(II)
we
get
2
a u
a u
M a *z
"- +
= = -
2
2
a~ where a l i s speed.
on of
z
0 -< ~ < 2~
2
47
,
(12)
a material
characteristic
2
a(
given
by
(2)
and M=CrZ/~H 2
is
E q u a t i o n (12) i s a n o n - h o m o g e n e o u s p a r t i a l differential equation of the second o r d e r . The general periodic solution o~ this equation, which satisfies the boundary condition (1O),has the form U
M
=
a ~z
- 4
(coBb ~ - c o ~ ~)
+
(s o C + f ~ o )
O0
+~. (e The O o , ~ o , e and f conditions analysis.
(S)
Introducing
r i
m(
•
~c . ~ h
mC)cos m~
(13)
(9);then
m a k i n g use o f
t h e methods
of
Fourier
~imensionless parameters =~nh ~2 (14)
K s~n6
2
e r
boundary
the
c a n be d e t e r m i n e d by a p p l y i n g
and the
cosh
z
--
r
El coth
~2
--
coth
csch
~"
--
c~ch
El O<
i
~
<
i
(15)
48
as
Q. EL-FAROUK,
M. FARAH a n d A. B A D R A N
-For- t h e c o n s t a n t a n d 6 [9, a ~ ] c a n
The expressions .Functions in m
=
x
-
o
(X z
/90=
[i
Xz
/ -
X2
x
in
I)
(a
iXZ+l 2 X2 _
+
coe.F~icients in the be deduced, namely
1
2a
i
6
+
o
(X zm
X (X 2
-
i)
+ I -+ Xz 1
m
2a
'
}
-
I)Z
+
I)Z
ore]
(16)
Z (×am
X2
=
(i3)
] 6X)
ox] ,
a 1
series
X
6
(X z
-
I)
Z m
where a=~(1-6) Z
=
,
-(a
A=
+ 6_
I-a z -6 z 2~6
)m
, X=(A+~A--i)
Z =(a
+
6X)m
_
XZm(~
+ 6_
X The series solution (13),together with ~ormulae enables o n e t o ~:ind t h e velocity profiles .Forvalues o~: ~ a n d 6 t o a r b i t r a r y accuracy, so that more transparent .For s u c h e c c e n t r i c configuration. The average ~lux velocity u can be calculated -
u
1
=
-$$
u
S
where
s
=
(17)
ds
(16) and (17), all permissiblp results become ~rom
,
the
2
(i8)
the
cross
section
area
between
the
cylinders.
I
.For u
an0
evaluating
the
surface
integral,we
obtain
z
Q~
=
u_ =
H~
M
(J z -
~ J
t
)
(~.)
.j
_
j
L
-4a
[z a= X ° X2 _
+8(~) [
)+
_a(jz_
1
2
+
Jzln
-
jz)
1
2
(J3)-J1 In
!
i)
(xz_l) z
3
-"~=z X~ where =
(a
+6XI~,
3
i]'
j2
HX=s ~ n)"2~ z
2
23 3
Z
J
=e~ i
(
= I
(e 3
(X)]
I]
I
(j2
J
~:ormula
z
n(rZ-rZ .)is.
Substituting
)m
X
+
~
rrh
)
]
(19)
LAMINAR FLOW BETWEEN CYLINDERS
2
Jz
= ¢sch
i
~'2
Hi
.,,
(~)~=[2 ~. (I+~) (em _ ~c~ ) _ and
49
( 1+4/3° )I
O0
(~) ]
[ , E'~'(~: ~ T.- =e rn~ +
2f: i
r~=2
PJSFERENCES
[1] ARFKEN, G. ( 1 9 6 8 ) , M a t h e m a t i c a l M e t h o d s Got Physicists,Academic Press International Edition. [2] B A T C H E L O R , G.K. (1979),An Introduction to Fluid Dynamics, Cambridge University Press,Cambridge. [3] B E C K , F . ( 1 9 6 0 ) , W a r m e u b e r g a n g und D r u c k v e r l u s t in K o n z e n t r i s c h e n E x z e n t r i s c h e n R i n g s p a l t e n bei E r z w u n g e n e r S t r o m u n g und Freier Konvektion. [4] C H E N G , K . C . , a n d H A W A N G , G . J . ( 1 9 6 8 ) , A I C h E J . , V o l . 1 4 , N o . 3 , p p . 5 1 0 512. [ 5 ] FARAH,M.N.and BABRAN,A . E I - M ( 1 9 9 0 ) , The Flow o f t h e Laminar Motion o f a Viscous I n c o m p r e s s i b l e F l u i d between Two P a r a l l e l E c c e n r t r i c . c y l i n d e r s (by Using C y l i n d r i c a l C o o r d i n a t e s ) , M.Sc Thesis. [ 6 ] HAPPEL, J . , and BRENNER, H., (1965). Low Reynolds Number Hydrodynamics,Prentice-Hall,Englewood C l i f f s , N.J.,pp.35-37. [ 7 ] HEYDA,J.F, (1958),Heat T r a n s f e r i n T u r b u l e n t Flow Through non Concentric Annuli Having Unequal Heat Release ~rom the W a l l s - G e n e r a l E l e c t r i c C o . , R e p o r t APEX-391. of a Hollow [8] MAC D O N A L D , H . M . (1894),0n t h e T o r s i o n a l S t r e n g t h ShaEt. Proc. Camb. P h i l . S o c . 8 , p p . 6 2 - 6 8 . [9] P I E R C Y , N . A . V . , H O O P E R , M . S . , a n d WINNY,H.F. (1933), V i s c o u s Flow through Pipes with Cores,Phil.Mag.,Vol.15.No.7,pp.647-676. [ 1 0 ] R E D B E R G E R , P . J . , a n d C h a r l e s , M . W . (1962),Can. J.Chem. Eng.,Vol. 40, pp. 148-151. [ I I ] S N Y D E R , W . T . , and GOLDSTEIN,G.A., (1965),AIChE J.Vol.ll,pp.462467. r4.~ L~]VOLLBRECHT,H.und OBERSTEDT,H.W. (1959),Rohrenwarmetaucher mit V e r d r a n g e r k o r p e r n , D e c h e n a - M o n o q r_a p h i e n , B d . 3 3 , ~ 2~5 • . [13]WHITE,F.M., (1974),Viscous Fluid Flow, Mc G r a w - H i i l Book Co., New Y o r k , p a g e s 125-128.
0093-6413/92
A
STUDY
$8.00 + .00
ON
Vol. 1 9
(1) ,48-49,
LAMINAR MOTION OF A VISCOUS BETWEEN TWO PARALLEL ECCENTRIC O.EI-F.Badr,M.N.Farah
D e p t . o F Math.., F a c u i t y o F
1992
CopyPlght (c) 1991
and
Science
P P i n t e d In the U S A
Pe~.mon
INCOMPRESSIBLE CYLINDERS
Press plc
FLUID
A.Ei-M.Badran
,M a n s o u r e
Uni v e r s i t ~.,E g y p t
(Received 6 June 1991; accepted for print 26 AuEust 1991) SUMMARY The flow o~ incompressible viscous fluid between two eccentric cylinders under the action of a difference between the pressures imposed at the ends of the cylinders is analyzed. Using bipolar coordinates,the resulting boundary value problem is s o l v e d analvtically,and the average ~lux velocity is calculated for various values of the geometric parameters of the problem. INTRODUCTION In m a n y branches o~ nuclear research [12],the exchange oheat inside the reactor o~ the heavy water demands, beside a specific heat eneroy ,an exact chec"~:, o n t h e limited values o~ the decrease ir~ 0 r e s s u r e inside t h e r e a c t o r . It a l s o demands exact information about the volume o,~ t h e w a n t e d heavy water which must be a= little as oossible .With regard to the last demand,speciai technical constructions o E t h e h e a t e~-:changer a p p a r a t u s shou.ld be needed. Such apparatus may be composeO of arranged cor, c e n t r i c or eccen+~,-ic c y l i n d e r s ~ s o that the inner space between them is very smmil to allow flow of little heavy' w a t e r . The simple idea upon which this apparatus is d e s i g n e d is t o i n s e r t a cylindrical body inside .÷ p i p e t o 6 o r m a c o n c e n t r i c space between the body and the oip_~ However,in prar+ice ~t m a y n o t b e D o s s i b i e to a-,,o~d the ~ormation of an eccentric space in-tea.u oE the idea! ,-.'-,nren+r~.~ one.
i~_ i s w e ! l - k n o w n rM, ~ . ±.:,J _ that such ec,..-.entrici~v_, r a u s e s _ a d e c r e a s e in pressure o.~ the fluid,so that it should be taker into _ a ~~m e n t . consideration ~or an exact ~u~-~ The probiem of the lamin_~r ~!ow oE an incompressiDie ~;!t~id b e t w e e r ~ t,~o e...-_centric c y l i n d e r s has been studied in many earlier work,.Mac Donai0 [8] developed the method of the torsional strength to -a.icuiate the volume flux and the frictional forces of the inner and outer cylinders. Some authors [6~9,!3] have used con,~ormal mappino t.o t r a n s f o r m the problem to a oeom.e~rv o~ c~ncent~ic annu!us,Gor which solution was a!read,,, known. Others oer#ormed calculation usinq_ cylindrical ~~-..,~5 ] o;5iu--'_ _ _~rr,n r ~,.~_n a r e s [ 4 , 7 . I0. i ! ] . However,computations based or, these ~.~,r~:'~ m a v produce as much as 20 % error [6,~3]. in t h e o r e s e n t wo.~'i,:,an._=.ivtica.', s c , ! u t i o q ~ n r t h e ~,robie~T. i~ four, d by. usino., b ~ p o l _ ~ r r o n.r d i.n a.t . e.~ _ . a .n d . ~ o n ~ v S n q ~ . t h e ,,~-~~--'~,_,,_,s c.~ -flDu~--er a n a i , / s i s. The advantaoe o-F t h i s solution is these c a]culat!ons can be carried out to a higher decree oE accuracy .Fo~- a l l value.=_ c,E the ._Teometrir_ param~+~_.__.-= . . .~ . . ~ . ' ~ , - p n * r i c i t v ).~ aod 5 iratio o E radii~,. 45
46
M. FARAH and A. B A D R A N
Q. EL-FAROUK,
MATHEMATICAL
FOP.MULATIONS
We considerthe problem of steady laminar ~low of a n incompressible viscous Eluid passing through the interspace between two eccentric parallel cylinders with radii r and r
I
(r < r
i
2
Z
).
Using bipolar coordinates between the axes of the two fol lows: a r
=
arid
i
t-
(~.C,z) the radii and cylinders-fig. (1)-can
[I
2
e=a(coth
[z
- ¢oth
J
,
sLr, h
~z
(1)
~rI )
('2
}'
)
Fig. (I) G e o m e t r y ~ is ~ h e d i s t a n c e ~rom the equation
between
the
x
o F the p r o b l e m
two
cer, t e r s ( p o l e s ) a n d
r
a
*
1
=
er
2
F ( r. 2 2
L
rZ
+
t
-- e z )
2
2
-. . . . . . . . .
--
t
i
2
J
,
2
calculated
4rZr z ]
[(I+~2--C~2' --4~2] /2 =
is
I/2
2
~ - = E~-
e as
a
~
sL~
the distance be expressed
(2)
r-
where
6 =-El
,
:r"
e
c~
= - -
(3)
~"
Assuming that the flow follows that the velocity
has neitheru has only
spin one
nor radial motion non-zero component
--
while
the
other two vanish, = u = 0
u~.
z a
7-
'4)
'
, the
=
(-)
=
0
continuity
and
Navier-Stol:es
equations (c~
z
a,-,
ap ,
,,,,
[ (ico s }', <-cos
~)]
=
2
0
~
(6)
2
2
@ U= (
+
0
El
z
a
of
z ) = ! OF._._~ 2
W
7..'
~z
ac
where p is the The bc)un~ar-)~
modified pressure and u the dynamic viscosity. conditions i:or the a>'ial velocity u are:
(i>
the
No
sli0
on
it ,
i.e.
C
In t h i s c a s e become ~u
u
inner
and
outer
cylinders, i.e.
[i]]
LAMINAR
FLOW BETWEEN
CYLINDERS
) = O,
(8)
= O,
(9)
Uz(~'~2)
~or all 0 _< ~ _< 2n (//)The symmetry of the solution in t h e both sides of the straight line passing the cylinders gives u (~,~) = u ( - ~ , ~ ) , z
for- a l l which (iii)u
normal through
cross section the two axes
implies
and
that
(I0)
(z <
("
~i
<
u must be an even f u n c t i o n
in
z
must be p e r i o d i c
~.
in ~.
SOLUTIONS t h r e e e q u a t i o n s (5) and ( 6 ) , i t is obvious that u must be a ~unction of ~ and E , whereas
From t h e f i r s t axial velocity
z
m o d i G i e d p r e s s u r e p d e p e n d s o n l y on z . T h i s means t h a t e q u a t i o n (7) s h o u l d be c o n s t a n t , i . e . d~ = -C dz Substituting in e u u a t i o n 2
C
each s i d e
"-0 .
(7),
the the of
(II)
we
get
2
a u
a u
M a *z
"- +
= = -
2
2
a~ where a l i s speed.
on of
z
0 -< ~ < 2~
2
47
,
(12)
a material
characteristic
2
a(
given
by
(2)
and M=CrZ/~H 2
is
E q u a t i o n (12) i s a n o n - h o m o g e n e o u s p a r t i a l differential equation of the second o r d e r . The general periodic solution o~ this equation, which satisfies the boundary condition (1O),has the form U
M
=
a ~z
- 4
(coBb ~ - c o ~ ~)
+
(s o C + f ~ o )
O0
+~. (e The O o , ~ o , e and f conditions analysis.
(S)
Introducing
r i
m(
•
~c . ~ h
mC)cos m~
(13)
(9);then
m a k i n g use o f
t h e methods
of
Fourier
~imensionless parameters =~nh ~2 (14)
K s~n6
2
e r
boundary
the
c a n be d e t e r m i n e d by a p p l y i n g
and the
cosh
z
--
r
El coth
~2
--
coth
csch
~"
--
c~ch
El O<
i
~
<
i
(15)
48
as
Q. EL-FAROUK,
M. FARAH a n d A. B A D R A N
-For- t h e c o n s t a n t a n d 6 [9, a ~ ] c a n
The expressions .Functions in m
=
x
-
o
(X z
/90=
[i
Xz
/ -
X2
x
in
I)
(a
iXZ+l 2 X2 _
+
coe.F~icients in the be deduced, namely
1
2a
i
6
+
o
(X zm
X (X 2
-
i)
+ I -+ Xz 1
m
2a
'
}
-
I)Z
+
I)Z
ore]
(16)
Z (×am
X2
=
(i3)
] 6X)
ox] ,
a 1
series
X
6
(X z
-
I)
Z m
where a=~(1-6) Z
=
,
-(a
A=
+ 6_
I-a z -6 z 2~6
)m
, X=(A+~A--i)
Z =(a
+
6X)m
_
XZm(~
+ 6_
X The series solution (13),together with ~ormulae enables o n e t o ~:ind t h e velocity profiles .Forvalues o~: ~ a n d 6 t o a r b i t r a r y accuracy, so that more transparent .For s u c h e c c e n t r i c configuration. The average ~lux velocity u can be calculated -
u
1
=
-$$
u
S
where
s
=
(17)
ds
(16) and (17), all permissiblp results become ~rom
,
the
2
(i8)
the
cross
section
area
between
the
cylinders.
I
.For u
an0
evaluating
the
surface
integral,we
obtain
z
Q~
=
u_ =
H~
M
(J z -
~ J
t
)
(~.)
.j
_
j
L
-4a
[z a= X ° X2 _
+8(~) [
)+
_a(jz_
1
2
+
Jzln
-
jz)
1
2
(J3)-J1 In
!
i)
(xz_l) z
3
-"~=z X~ where =
(a
+6XI~,
3
i]'
j2
HX=s ~ n)"2~ z
2
23 3
Z
J
=e~ i
(
= I
(e 3
(X)]
I]
I
(j2
J
~:ormula
z
n(rZ-rZ .)is.
Substituting
)m
X
+
~
rrh
)
]
(19)
LAMINAR FLOW BETWEEN CYLINDERS
2
Jz
= ¢sch
i
~'2
Hi
.,,
(~)~=[2 ~. (I+~) (em _ ~c~ ) _ and
49
( 1+4/3° )I
O0
(~) ]
[ , E'~'(~: ~ T.- =e rn~ +
2f: i
r~=2
PJSFERENCES
[1] ARFKEN, G. ( 1 9 6 8 ) , M a t h e m a t i c a l M e t h o d s Got Physicists,Academic Press International Edition. [2] B A T C H E L O R , G.K. (1979),An Introduction to Fluid Dynamics, Cambridge University Press,Cambridge. [3] B E C K , F . ( 1 9 6 0 ) , W a r m e u b e r g a n g und D r u c k v e r l u s t in K o n z e n t r i s c h e n E x z e n t r i s c h e n R i n g s p a l t e n bei E r z w u n g e n e r S t r o m u n g und Freier Konvektion. [4] C H E N G , K . C . , a n d H A W A N G , G . J . ( 1 9 6 8 ) , A I C h E J . , V o l . 1 4 , N o . 3 , p p . 5 1 0 512. [ 5 ] FARAH,M.N.and BABRAN,A . E I - M ( 1 9 9 0 ) , The Flow o f t h e Laminar Motion o f a Viscous I n c o m p r e s s i b l e F l u i d between Two P a r a l l e l E c c e n r t r i c . c y l i n d e r s (by Using C y l i n d r i c a l C o o r d i n a t e s ) , M.Sc Thesis. [ 6 ] HAPPEL, J . , and BRENNER, H., (1965). Low Reynolds Number Hydrodynamics,Prentice-Hall,Englewood C l i f f s , N.J.,pp.35-37. [ 7 ] HEYDA,J.F, (1958),Heat T r a n s f e r i n T u r b u l e n t Flow Through non Concentric Annuli Having Unequal Heat Release ~rom the W a l l s - G e n e r a l E l e c t r i c C o . , R e p o r t APEX-391. of a Hollow [8] MAC D O N A L D , H . M . (1894),0n t h e T o r s i o n a l S t r e n g t h ShaEt. Proc. Camb. P h i l . S o c . 8 , p p . 6 2 - 6 8 . [9] P I E R C Y , N . A . V . , H O O P E R , M . S . , a n d WINNY,H.F. (1933), V i s c o u s Flow through Pipes with Cores,Phil.Mag.,Vol.15.No.7,pp.647-676. [ 1 0 ] R E D B E R G E R , P . J . , a n d C h a r l e s , M . W . (1962),Can. J.Chem. Eng.,Vol. 40, pp. 148-151. [ I I ] S N Y D E R , W . T . , and GOLDSTEIN,G.A., (1965),AIChE J.Vol.ll,pp.462467. r4.~ L~]VOLLBRECHT,H.und OBERSTEDT,H.W. (1959),Rohrenwarmetaucher mit V e r d r a n g e r k o r p e r n , D e c h e n a - M o n o q r_a p h i e n , B d . 3 3 , ~ 2~5 • . [13]WHITE,F.M., (1974),Viscous Fluid Flow, Mc G r a w - H i i l Book Co., New Y o r k , p a g e s 125-128.