Fluid Flow and Heat Transfer In an Axially Rotating Pipe: The

Fluid Flow and Heat Transfer In an Axially Rotating
Pipe: The Rotational Entrance
B. WEIGAND one! H. BEER
InstiM fUr Technische Thennodynamik
Technische Hochschule Dermstadt
Petersenstrasse 30
6100 Darmstadt. Germany
ABSTRACT
The complex tnteractlons between turbulence and rotation in the rotatlonal
entrance reclon ot • pipe. rotaUng about Ita axis. are examJned .
By ••• umlne: a unlvenal tangentla.l veloctty pronle and with the use of a
modified mlxlnc leneth theory. the development ot the axial velocIty protUe
and the heat tranlter coemelent alon& the rotational entrance length are
calculated. The theoretical results a r e compared with experimental flndings
Reich [8].
0'
I. INTRODUCTION
Fluid now and heat transrer tn rotaUn, pIpes are not only of considerable
theoretical Interelt. but also or creat practical Importance. An obvious
technIcal appllcatlon Is a rotatlne power transmission shatt that is
10ng1tudinally bored, and through which a fiuld Is pumped tor cooling of
turbine biades or tor other purposes.
When a fiuld enters a pipe rotatlne: about Its axis, tangential torces acting
between the rota tine pIpe and the nuld cause the nutd to rotate with the
pipe, resulttne tn a now pattern which Is rather dIfferent from that
observed in a nonrotaUni pipe. Rotation was found to have a very marked
innuence on the suppression ot the turbulent motion because ot radially
growlnl centrifugal torces.
In 1Sf29 Levy (1) studied experimentally the now In rotaUng pipes.
Murakami tnd Klkuyama (2) measured the time-mean velocity components
and hydrauUc losslS in an axially rotaUng pipe when a rully developed
turbulent now was introduced into the pipe. The pipe rotation was found
to suppress the turbulence In the now, and also to reduce the hydraulic
loss. With lnereasin&: rotational speed the axtal velocity distribution nnally
approaches the Hagen-Potseullle now . Klkuyama et al (3) calculated the
veloelty distribution In the tully developed region or a rotating pipe with
the help of a modified mlxlne: lenlth theory proposed by Bradshaw (4).
They assumed the tangenUal velocity to be a paraboUc distribution
v,/ v,. = (rIR)- In the tully developed region, which was well confirmed by
experIments (2). [3],
ReIch and Beer [5) examined experimentally and by analysIs the effect or
tube rotation on the veloelt)' dIstrIbution and on the heat transter to a
nutd nowinE Inside a tube. The now was thermaly and hydrodynamically
rully developed . For their calculations they used a sUghtly different mixlnE
length formula as proposed In (3] . With an Increase in the rotation rate
N :z v,wIVzo they observed a remarkable decrease In heat transrer. The
experimental observations were in close agreement w1th their theory. The
remarkable decrease In heat transter with IncreasIng N results from the
325
atrong relam.1narlzatlon c:auled by the c:entrttugal torcel. For growing values
of N, the errect of rotation becomes more pronounced and the Nusselt
number approaches gradually the value tor laminar pipe now .
For a hydrodynamic tully developed now the thermal entrance region was
examined In [&J. The analytlc:al investigation ahowed that the problem
could be reduced to a Graetz-Problem. With increasing rotatlonal Reynolds
was observed. The
number Re, a remarkable decrease In heat transrer
thermal entrance length lncreasea remarkably with growing Re.,. (Par
Reo ... 60000 and N := 3, the now needs nearly 200 pipe diameters to get
thermally tully developed.
The rollowlnc study wUl tocus attention on the nuld now and heat
transrer Inside an axially rotaUng pipe tn the combined hydrodynamic and
thermal entrance recton . Heretofore the interactlon at centrifugal torces
and turbulence was Investigated only In the hydrodynamic rotational
entrance by experUnents [8], [18] .
2 . ANALYSIS
2. 1 The Conservation Equations In CyUndrlcal CoordInates:
The problem under consIderation Is ot great practical Importance, because
In most rotating machinery the geometric: connguratlons are not large
enough to guarantee tully developed now conditions.
Because at steady no .... conditions and cyllndrlcal symmetry, there are no
variations In time and In tangential direction. By assuming an
incompressible, Newtonian nuld .... ith constant nuld properties, the
conservation equations tor the time smoothed variables may be written as
tollows:
av
"r
h"z
--L + - + - - 0
3r
r
3z
(1)
,
•...L
r
0
!~
p
3r
av
••
+ -L...l + 'z ~]
p[Yr ----t
3r
r
az
av
p['r
P
--L +
ar
'z
:z]
o
Cp['r -ailTr + 'z az
ilT]
_
1 a
r a ar [r'
0--
~ +!!...
az
r
1a
--rar
ar
[r
T
Tn
[ rqr
• J
r ,)
(3)
J
(S)
By deriving the eqs. (2) - (5) the usual boundary-layer BIIsumptions were
made , Which are a common treatment at the conservation equations tor pipe
now
[7]. [14] .
The components ot the stress tensor, which appear in eqs. (3) and (4) can
be written tor a Newtonian nuld as tollows
_ I'
•
!... [ :t
ar r
av z
r
I' -
ar
-
p V
r
I _ =•.....-:.•
P
'v
z
r
:-7.
(6)
,
(7)
t
328
r
Cyliodrical coordinate IYltea
rIGUIl 1.
The coordinate system and the components or the stress tensor are
mUltrated in nCo 1. The radial component of the heat nux vector Is:
•
q
M
ar +
• - t -
r
P
---ep "'T'
r
(8)
The parabollc nature ot the boundary-layer equations requires that
boundary conditions be provided on three sides or the lolution domatn. In
addition to Initial condItions tor the momentum and energy equations,
symmetry 11 assumed. At the pipe wall the heat nux 1. prescribed and the
zero sUp condition holds .
M
•
r • I
•
or • 0 ;
• 0, • 0 ; - t - . q
.
r • 0 :
I
•
0 :
•, °,. ·
or • 0 ·• °, • 0
•·
p • p
T• T ;
• ·•
•
ar
iIY,
-·0 ;
ar
rw
M
-·0
ar
.z - her)
(9)
Accordln, to eq. (g) It 11 assumed that the nuld enters the rotaUng pipe
section with a tully developed. turbulent velocity prante. This iniUal
condition wa, chosen to compar. results with measurements ot Reich (8) .
2.2 The UnIversal Tangential Velocity DIstribution
FIg. 2 ahows experimental tangenUal velocity profiles tor various z/D and
difterent values of the rotation rate N. It is obvIous, that the profiles are
only sUghtly or not at all Innuenced by the now-rate Reynolds number
and by N.
This tact was also recognized In (8) and un. Inereas1ng rotational
Reynolds numbers resUlts In a growing shear stress in the tangenU ..1
directlon which leads to an only IUght dependence at the tangential
velocity pront. on N. This _111 Inttuence the entrance length which Is
needed tor the development ot the tangential velocity protne. Theretore. It
Is reasonable to Introduce a quite universal prattle to descrIbe the
tangenUal velocity distributIon:
.
0,.
...L. i(2 +
-
_
f(I))
,
f(i) • ; + 9.5 .-0.019
with;' :; r/R and
z :;
(10)
(11)
I
z/R.
327
I..,.... :
AVD_ It
AlIO_ ...
• lID. III
A:M)· ...
D ZIt).12.
00
-f
1.0
\0
V
1,0
........,
(
~ 1··'1
I
l~ :
0.5 ovo_ •
.vo. It
o lID • •
AVD. It
6W ....
A ZID....
.110• •
D VDet:,.
-1
1.0
0
0,5
-f
\0
FIGUII 2. TaDgential .elocity 4iltributioB a•• fUBctioD of the axial
coor4iDate lID.
The universal tangential velocity pronle II assumed to be only a tllnction
of the rotatlonal length. Fie. 2 showl that this asaumptlon Is well
conftrmed by txperlmentt [8) . which were performed wIth air. Because the
tangenUal velocity dlstrlbutlon I, now given by eq. (10) and the prenure
Is not a function ot the tangenUal coordinate, eq. (3) can be omlted.
However, It has to be pointed out that an extrapolation ot eq. (to) to
N ) 3 Is admissible only It experlment.1 nndln&s which connrm the
unlver.anty of eq. (to) for lareer values ot N were avallable.
2.3 The Turbulence Model
In order to nnd a lolution ot the dIfferential equations (t) -(5) with the
boundary conditione un, a relation between the Reynolds ,hear ItrelSes
and the time mean velodty components mUlt be eltabUshed. This w111 be
done by the use ot • mixing lencth hypothesis proposed by KooslnUn (10].
This turbulence model I, well connrmed tor nuld now and heat transfer In
rotating pIpes (3), (6), [6). It has to be pointed out, that luch a elmple
turbulence model can be used lucesatully only, it the tangenUal velocity
pronle Is known . Without know led,. at the tangential velocity distribution
the laminarizatlon phenomena can only be calculated applyinc the etress
equations turbulence model, whleh ••• shown by Htrai et al. (til tor tully
developed fiow condItions.
Usinc a modlned Prandtl'. mixIng length theory,
proposed by Kooslnlln
.a
328
[10], the axIal ahear .te ••
[~+
Tn •
P I' [ [ : .
r ]:.
~r
~ I rr ]:;
Tr:r
In eq.
f +[
(7)
r :r [
can be .rltten ••
~ If
(12)
where ) I. the hydrodynam.1~ mlx!nl lencth.
The radial eomponent or the heat nux vector eq. (8) I, tound to be
qr • [ - k -
P
cp 1 lq [ [
+ [ r :; [
(ll)
where lq denote. the thermal mlxlne 1enlth.
In a rota tine system the turbulence, I.e. the mixing length 1 and lq are
markedly eNeeted by the eentrltugal torees [8), [9) . To describe the
suppressIon ot turbulence wIth radially growlnl centrifugal torces , the
mIxln, length I. or a nonrotatlnl pipe must be modified tor the now in a
rotating tube. Bradshaw [4] proposed the tollowtng equation:
L.
1.
(1 - _ Ii)«
(14)
where. 18 • constant (01 = 2 used tn [S]).
The Richardson number Rt appearing in eq. (14) describes the ettect or
tube rotation on turbulence. Rt Is denned by
y
~
2 J
-
(y
r)
(15)
Without rotation. Rt. 0, there exists a tully turbulent pIpe now. It
Rt ) O. I.e. tor a rotatlna tube with a radially ,rowIng tangential veloe1ty.
the eentrttugaJ torees luppres8 the turbulent nuctuatlons and the mixing
lencth decreases. It Rt (0. e., . tor nows over spinning surtaces. the
turbulence will be enhanced by rotation:
The function _ In eq. (4) Is a con.tant tor tully developed now. as it was
which means
shown In [5) by checking the boundary value ot lU tor N ..
the extstence ot quasi laminar fiow tor large N. This results In ~ = 1/6. In
the case ot hydrodynamIcally developinc now. the same procedure yields an
expression tor _. that 1s elven by
-t
_(i) •
(1
-
+ f(z~)
2
• .. •
6 + 21(z)
(16)
ot eourse. the value ot I(Z) according to eq.
4J
II
I
lI.fIlll l
'.2f1ti
•
!.t.96e-o.0225 !
t
_l.
FIGUlE l.
Function'
o
110
329
(6)
can only present an
upper 11mlt. pretending a tully laminar now In a near wall reclon for laree
values of N. This will overpredlct the lamlnarlzatlon phenomena In the
rotational entrance region, because It lenores any Interactlon between the
laminar wall region and the fUlly turbulent pipe fiow In the central core,
which Is not yet etfected by the radIally cro.lnc centrlfUcal foreea
(ng. 2). Only for large valuea of zlD. If the centrifugal torces Infiuence
the whole cross aectlonal area ot the tube. there must be a transition to a
laminar Uke proftle tor N ~ - . Thl. means that "(I.) muat take values
between 1/ 6 and that or eq. oe) ror nnlte valuea ot z/D. In order to
ov_ercome the above mentioned dlmcultlel. a aim pie empirical expresalon tor
"Cx) wal developed, which Is
,(i) •
t.
1. 96 .-0 . 0225
i
Fig. 3 ahow. a comparlaon at
respectively.
(17)
'C%.)
accordln& to eq. (16) and to eq. (l7),
A common conelatlon between the mIllng lenlth In a nonrotatine pipe I.
and the radial coordinate r 1. Nlkuradae'. mIxlne length expression [12].
which 1. multiplled by the van Drtest dampln, tactor. In order to describe
the disappearance of the mixIng leneth near the tube .all, In the viscous
Bublayer.
~o
•
[
1 - .-Y·'l6
I[
0.1. - 0.08 [
~
}' - 0.06 [
~ }'
I
(11)
with the dlmenslonle" dIstance from the wall
y+ ••• (I - r)
(19)
y
and the friction velocity
...
/
ITr (r'
.)1 .
.!....!C!Z"p,-_--!.
(l0)
The Infiuence ot the eentrttucal torces on the thermal mIxinC length Is
expreaaed by the lame e~uatIon a. the Innuence on the hydrodynamic
mixing lencth.
~.
lqo
(1 - , 11)'
Ill)
where lq. Is the thenaal mixlnc lencth for a nonrotatIne pipe whIch can be
expressed by the turbulent Prandtl-number [12] as
1
;r- •
t
1
-i!
1 • 1.53 - 2. 82 r-a + 3.15 r-, - I . "
0
-.
r
according to experlmenU performed by Ludwiec (13) tor air
2.4 The DlmenllonIe.s
Ill)
(Pr
= 0.11) .
E~uationl
In eq. (2) the radial prelJure cradlent 11 balanced by the radJaUy Irowlnc
tancential veloetty. which relults In a radlaUy increastng pressure
dltferenee. The maximum ot the radial pressure dltference between pipe
•• U and ~en~er wU1 be attaIned tor a tully developed tancentlal velocity
pronle. Reich Ihowed (8] that this pressure difference Is negligible for
330
alrfiow . Theretore. the pressure was assumed to be constant over the pipe
cross .ectlon.
Introducing the tollo.ln, dimen.lonless quantltlea
- •• / V
'. • ••
- • Yr fo;.'
'r
"', • Y,/Yzo
-
V••
r · r/.
8•
.e.
T - T
•
fir.
.
•
I
~
A
'J
•
fo;.'
• ...I
.zo
I
I-r
J- • --.-
Aae a
-
p.
1
P - p.
p
v
•
Ie
Pr • -
•
••
,
•
•
• ..!.L
v
./.
"zo •
....
... "',1iI
(2l)
Into the eqs. 0). (2) - (5). re.ults In the follo.lng dlrterential equations
(2()
-
-
a. Z ... a. z
- 1 & '["
• ray
-::- +.,.Z&z
-:-. - ~
+ -=ray
'""':' r[l
i)z
...
••a+I
•Jay
!!. +;.~.
Cl!. LPrray
~ !:. [<[1 •
(25)
(26)
with ••+ denned by
(27)
lnsertinc the dlmen.ionless quantities into the boundary conditions eq. (9)
yields:
y • 0
• 0 ; ~8
:. .0;
-1
y
-
•-•
a..
-=-·0
ay
: p-
• • 0
0 :
e•
ay
~.
.
• "!
ay
1 ;
•0
;z • h(r)
(28)
In order to make eqs. (24) - (27) aceeslble to a numerIcal analysIs and to
ensure that the equations have a form that i. common to plane and
aXisymmetric Oow. the Mangler transformation (14] Is applled:
dr • di
(29)
Thl, results In
~V
n'
-.!. + --I. • 0
~z
v. ~z
a.
(lO)
~
+" ~z. _ !g + "_
Yay
~.
ay
(ll)
331
a!.
'f
It.
'f
a!.
J3y
L
[b a!)
'31
t_
Pr3y
(ll)
with the tollo,,1n& quantities
b
•
.
-.
~.
-
p
•
• r (1" )
Introducing the Itream tunction. denned. all
eq • .
" . - -a,
a,
,
'f •
aJ
(30) -
(32)
Ill)
J
ilr
may be .rltten
&•
• .!f + itr a·, _ aa, .!r
h
lIf liar lIf' ai
• Pr
Ilt)
[a! a! _ 3! a!J
3yh
(lS)
3yh
The presence or ~/az In eq. (34) Introduces an additional unknown to the
system elven by eq. (34) and (35). Thus another equation 11 needed and 11
provided by the conlervation of ma.. In Inteeral torm (141 .
'fl' I' • l
witb 'L •
•
I w, r 4r
~
•
! .fiO;' .
l.fiO;'·
r('L)
(l6)
At the centerllne. symmetry require. Iv,:lar and aTIIr to be zero In
untranafotmed coordinates. In terma or dimension lese variables the
parameters aV,fay and
at the unterllne are indeterminate. Theretore.
a8/~
the boundary cORditon. at the centerllne were calculated from eqs. (34)
and (35). aeeordlnc to (15).
The boundary condltlona are:
, • 0
:
r.
0
.•
. !!.-1
'aJ
~. 0
ar
a',
/.e•. [~ • !
-.a.
aJ'
38 - .fiO;'
-.
3
lIf
-
1'-0
:
~.
0
;
a_ [a!J ']
l3z3y
l
3' a8
31 a.
Pr-=-=
• • 1
;
r •
gheD.
(31)
2.6 The Numerlcal Method
In order to obtaln solutions 01 eq. (34) and (36) wIth the boundary
condltlons eq. (37), an ImpUdt nnite-d1fterence method Is applled, whIch Is
known in Uteratur. a. the Keller-box method. A detaUed delterlption can
be found In (4) and (15]. BecaUI. the box leheme Is a common method to
lolve parabol1c dlrterential equations, only a brlet outline I, provIded here.
332
In consequence of the ..sumptlon ot an incompressible nuld .Ith constant
propertIes. the equatton or motlon and the energy equation are uncoupled
and may be lolved separately.
The velocity distribution. The momentum eq. (34) will be nut reduced to a
nrat-order Iystem of dIfferential equations
"
(l8)
• U
U· • V
[bV• I, ...Iz
(l9)
~
aT
"0)
.~+O"",,:-V~
az
az
where primes denote dIfferentiation with respect to }'.
boundary condItions belong to eqs. (38) - (40)
, •°
v • -!Z ,fIiO'
r~
--I
az
+!
3_
Z 3.
The
(u')l
rollowlng
(U)
For 0 , % , %N' 0 , 7 , '1L' a posilbly nonunltonn net is placed:
•
0
i0
• ° •• • •.-, + t
·0
-
l'J
• "J-'
+h
D
j
•
Il.
1.2 •••••• I
J •
1,2, •••.• J, "1· Yl
-
(U)
with kIt and h J denoting variable dlltances between nodes In the i and y
directlon.
[qs. (38) and (30) were approximated by central-dlfference quotlents and
averages about the midpoint an. 'l' J-')' whUe eq. (40) w.. centered about
the mldpolnt
l'J-')' After approximating the boundary condItions,
eq. (41) . •Ith central ditterence quotients, a system or 3J+3 nonlinear
equations tor the 3J+3 unknowns (F~. Ul, Vl) il obtained. The system can
eaaUy be lolved by the block eUmfnatioD atter linearIzation by applying
Newton'. method. The pres.ure gradient appearing in eq. (40) w.s treated
as a nonUnear etgenvalue. Detalls or the numerical method are found in
(J.-".
1i51.
The temperature distribution. The method used to obtain solutions of
eq. (36) II ,lmUar to that described In the previous sectlon.
Reductnc eq. (35) to a nrst-order system of differentIal equations gtves
P • 8'
(tl)
r . Pr [u 3! _ P a!1
[b •P
Iz
(CC)
3z
with the boundary conditIons
,. ° .
P• - 1
Fe; Pr
39
(tS)
0-=
2
h
Eq. (43) was approximated by centrai-ditterenee quotIentl and averages
about the midpoint (in' YJ-")' whUe eq. (44) waa centered about the
midpOint (zn-'" V _~) usIng the same net defined by eq. (42.). The solution
of the resultlng linear system ot equations w.s obtained by the aame block
eUmlnation method used with the momentum equation.
p. - -
333
The Nusselt number. The Nussell number, bued on the local dIfference
between wall temperature and the nuld bulk temperature I, defined by
lu •
2 I n/~l.
T - T
•
(U)
•
with the dennltlon of the nuld bulk temperature for an IncompressIble
nuld aceordln, to
!
"I T 41
· ! ",
T •
(t7)
41
inserting eq. (47) into eq. (44) results In the followln, expression for the
Nusselt number
lu •
- I.
I
(U)
'FI ·u a df - Fa·..
•
2
3. RESULTS AND DISCUSSION
3.1 Velocity DIstribution
The .ff.eu of tube rotation on the axial veloelty distribution In the
rotational entrance reclon are shown tn nC. " for varlous rotation rates N
and dIfferent now-rat. Reynolds numben. Ezperlmental relult' of Reich
[8] are plotted for comparison. Generally the calculated profiles of the
axIal veloelty are In good acreement with experiments. However, the
entrance veloelty pronle deviates trom the experimental relulU. The
experimental result •• how a more turbulent profile. W• • uppos., that either
the entrance region of 40 pipe diameten ••• to short to produce a tully
developed turbulent entrance velocity pronle, or the mtx!n, length
hypothesi, Is In error for such loW Reynolds numbers.
For Inere ..1n, now-rate Reynolds numbers (Reo - 20000), where the
entranee pronle Is In doser a,reement with the data, the development of
the axial veloe1ty pronle In the rotational entrance reclon 1. In excellent
agreement with experimental nndlngs.
The pronles Ihows, that there exists a potenttal cor. In the pipe. which 1.
not affected by the tube rotaUon for low '1./0. The veloelty pronl .. adopt a
nat .hape near the center recloR for zlD , 80. The fiuld In this reclon 1.
only accelerated by tube rotation . Por zlO > 80 the centrltucal forces
lnnuence the eenter region (nC. 2) and the shape of the pronle. In the
core region Is altered by rotation. Thl, typical development of the axial
veloelty pronle was al.o mentioned by Nl.hlborl et al. [IS].
Por the calculation. of the veloe1ty distribution. and temperature pronles
a .trongly nonuniform crld with approximately 60 Crld points in radJal
direction .a. used. It enable. to predict the atrone variations of velocity
and temperature In the near .aU reclon with load accuracy.
334
2,t
.
T,,",, :
ZIO : 11O
?.
V
Thwy:
·no
/ lID. 10
f
I
...
,
[1pft'iMnt :
[~t :
elm.
o liD-
• lID - •
10
.. liD. 10
A lID. U
OliO", 10
.. lID. 20
A ZIO I' ...
• lID- 10
D
'.
vo- 120
IR••• soool
•••
• I/O.
,...,
-r
u
u
lID _no
V,.
-r
l.'
ThIot,.
~
eo
D VO l' lID
n-y.
llOcaf20
~
V,.
I
'.'
f
...
'.'
,
-;
ezm·
o VOl'
0
• zm.
10
.. ZlO. 20
6 lID. 40
IRe.·1OOOO1
FIGURE ,.
ZlD" 10
A
llll . 4.0
• lID. 110
o lID-12D
D ZlO.ll.
'.'
0
•
o
.. liD . 20
• I/O. 10
,
,_.
,.,'1
-.
to
'.
IAII " 20000 1
•••
,···1
-.
tI
Axial geloeity distribution II I function of z/D.
FIg. 5 shows tully developed velocity protlles tor two flow-rate Reynolds
numbers and dUterent rotation rates N. compared with experimental results
of [8]. ThIs may be considered as an other proof tor the valtdlty or the
applied method .
336
11..-----------,
...
~
..
.!L.
v..
I
E.,.,....:
• • ·0
0"...'
..
..',
I"
I s:
0 •• "
••• 1
••• I
••• J
. .
IRo,"20000 I
..
.L.:==-_--1
-r
FIGURE 6. Axial veloC:1ty distribution tor tully developed no. (&/0 .. -) al
& tunctlon of the rotaUon rate N.
3.2 Richardlon Number and Mixing Length
To lllu.trate th.' ertect of turbulence luppreeslon due to tbe pipe rotatlon
in the rotational entrance. dl.trlbutlona of the Richardson number and the
mlxlnc Ieneth are depleted In fiC. G and nC. 1 re.pectlvely. tor varioul
values or z/D.
WIth Irow1ng zlD
the Innuence or the centrlfu,a. forcee expands
increasingly .croes the crol. .actlon of the pipe. For z/D > 80 the whole
cross aectlon 11 lelzed by the radially crowIng centrltu,.l tore ••.
The development of the Richardson number explains the appearance ot •
Ri
21J
IRe g=2oo001
IN=ll
4,0
Z/D=l20
Ri
•2,0
t
1,0
0,5
-
-r
0,5
1,0
rIGUlI ,. The Itch.rdlOD uuaber Ii for
~.riou.
lID
-F
1,0
-RI
0,14
IRe a=50001
f
0,07
ZlD=120
ZID = 80
Z/D: 40
Z/D= 20
Z/D= 10
ZlD= 0
°o~------~----~
0,5
_ 1,0
-r
lIGUlI 1. The aiziDg leDgth 111 a •• functioD of z/D .
potential core In the axial velocity, because In the region 0 , z!D , 80 no
suppres.ion or turbulent nuctuat10ns takes place near the pipe center and.
theretore. a tully turbulent fiow II eltabUshed In this region. Because ot
lam.1narlzaUon or the now dose to the wall. the velocity gradient decreases
at the •• U and the central core Is accelerated tor reason or conservation
or mus, eq. (3d). Fig. 7 ahow. the tnnuence or the tube rotation on the
miXing lencth 1 tor Reo = 6000 and N:I: 3. For turbulent pipe fio"
(zID ::II 0). the mixing length distribution Is that one given by Nlkuradse
with the van DrIest damping tactor. eq. (8), The mixing length
distribution. obtained trom eq. (14). elucidates the turbulence suppression
.lth Increasing Z/D.
3.3 The Nusselt Number
In nc. 8 the Nusselt number Nu. tor tully developed now Is plotted as a
functton ot the now-rate Reynolds number tor various valuea ot the
rotatton rate N and tor Pr = 0.71. With Increas1ng N a remarkable decrease
In the Nusselt nwaber can be observed. Por N . . the Nusselt number
approach •• Iradually the value tor laminar pipe now. which Is Nu. = 4.38
tor conltant heat nux at the .alL
Pte. V Ihowl the local NUlaelt number Nu tor air (Pr = 0.71) devlded by
Nu. tor different tlo.-rate Reynolds numbera and various N. Because
lamlnarlzation tak.es place. the now Is not tully developed tor zlD = 120
and N = 3. This I, a very Interestlng consequence ot the lamlnarlzatlon
phenomena. With Increaslnl rotatlon rate N. the thenna} entrance region Is
markedly enlarled. which w.. first recognized In (6] . For N". the
thermal entrance length LID will approach 0.05 RenPr. which Is the
entrance length tor laminar pipe now .
337
-?:
,-
...
~-
I"'~. ~-
~
::~
'0: ~ ,......
~: ~
'0' " ~ t-~
V
•
=
rIGUlI 8.
flow IS •
wall) .
lua,ett nuaber .u. for ther•• l aDd by4rodynaatc fully 4••elope4
of 1'0 witb • at par ..eter (conatant beat flux at tbe
fUD~tioD
•'r---~~------r=~
1..,,_ I
'0 ,
'0 ,
'0 ,
II· •
",'
"'.
I
•
•
...
...
"'I
•
2~~
2~~~
.1-===
1),
.!---=
z
-0o
.
50
-Z
W'r---~~,-----~~
I~--I
It • •
'"
"'.
I •
•
..
'0 ,
'0 ,,
...
...
"'I
SO
110
.r---~~'-----r=~
I..., tIICID I
... •
o
,
'0 ,
•
0
,
•
2:~~~ 2:~~
'0
50
--
Z 100
'.
o
SO
-l
o
100
FIGURE 9.
NUlselt number al a tunction of z/D with N •• parameter
(constant heat nux at the .aU).
338
4 . CONCLUSIONS
A theoretical model •• s presented to predict the eomplex Interactions
between turbulence luppre•• lon and tube rotation in the rotatIonal
entrance reelon or • pipe rotating about ItI axis. With the assumption or a
quite universal taneentlal profile and the us. of • modlned mixing length
hypotheals, which takes Into account the turbulenee ,uppression due to
centrltueal rore.a. the axial velocity distribution and the Nusselt number
were calculated. The theoretical results were verlned by experimental
findings [8] and a generally good agreement .as round. It could be stated
that tube rotation w111 enhance the thermal entrance length, becau.e or
no. laminarizetion.
An eItr.polation to N > 3 1. adm1 •• 1ble only It tuture experimental results
will confirm the universality at the tangential velocity pronles according
to eq. (to.)
JOIIDCLlTUlI
a
0.
D
r
k
L
I, 1 ..
•to
Ro,
Ii
r
theraa1 4iffu.ivity
.pecific heat at
conatant pre •• ure
pip. 4i..eter
Itre..-function
theraal conductivity
pip. lengtb
by4rodynuic an4 theraal
alxiDg length in the
rotating pipe
hy4rodyaaalc and theraal
.ising l.ngth in the
nonfotatinG pipe
rotation rate
lu .. e1 t nuaber
PrandU nUber
turbulent Pran4t1 nuaber
pre. lure
heat flux deD.lty
heat flux 4en.ity
at tho pipe wall
pip. radiu •
flow-rate leYDold. nuaber
rotational leynold. Duaber
lichard,oD nuaber
rad1al coordinate
.,.
..
..
'
., .
.Y.r
y+ -
!
'. !'
• +
•
,.8
,T,
y
p
J
teaperature
t.aperature fluctuation
bult teaperature
tia. aean .elocity in
r, , . I -4irection
tangential velocity
of the pipe wall
dia.uionless
velocity coaponent,
aean &xial velocity over
tbe pipe cross .ectioD
velocity fluctuation
friction velocity
diaeDsioDles. ra4ial
distance froa the wall
axial coordinate
di.enaioDless
axial coordinate
ed4y diffu.ivity for
aoa.ot08
diaenlionle,s t.aperature
dynaaic vi,colit,
tineaatic viscosity
shear stresa
function according to
eqs. (16) ••d (17)
density
nruoclS
1. Levy r .• Str6aungser.cbeinungen in rotierenden lobren, ¥.Dr For.cbu~g.­
Irb.lten lui d~. Geblet de. IDg~Dj~urW~$~D', Vol. 322. pp. 18-45, 1929
2. Kurat..i, K., I. lituy... , Turbulent rlow in' Axially Rotating Pipe • •
J, 01 Fluid ZngDg •• vol . 102, pp . 91 - 103, 1980.
3. likuyaaa I . , B. Burakaai, I. li.hibori , I. Ba.4a, rlo. in an laially
lotatin; Pip• • Bull 01 tb. JSal. yol . 16. pp. 506 - 513 . 19'3.
t. Bra4,ha•• P., Th. ADalogy B.t.e.n Stre.. line Cur,ature In4 Buoyaney
in Turbulent Sbear rIo., J . 01 ,luid x.ebaDie• • '01 36, pp. 111 - 191
1969.
rIo. an4 I.at TraD.tar in an axially
rotating pip., fb. S.coDd Iat . S,.p. OD Trl••port Pb'DO••D., Dzu..ic.
aDd ~.i'D' 01 'ot.tiD, •• cbiD.rr. '01 . 1. pp. 298-313 , 1988 .
5. leich, G., B. B.er , rlui4
6. Veigand . I • • B. B•• r, Vlr•• ObertraGung in .in•• axial rotler.nden,
dureh.tr 6.t.D lobr ia Bereicb 4e. tb.ralleh.o linlautl, Vlr•• - UDd
Stoll~rtr.vu.g. Yol. 1(. pp . 191-101. 19'9.
1. B.DD.et •• D• • leat Tranlter by .ageo-Poi.euill. rlow in tb. ther.al
4fTelop.ent regioD with axial conduction, Vlr.,- UDd StoltOb.rtra,uDf.
Yol . 1. pp . 177 - 18(. 196••
•• leich. G• • Str6aunG und "r.eGb.rtragung in .in•• axial roti.r .nden
lobr, Doctoral The.i. , TB naraltaat, 1988.
, . likur .... I •• 8. 8urat..i. I. 11.hibori, DeT.lopa.nt of rbr •• Di •• n.ional Turbul.nt Boundary Llyer in an lzially lotating Pip. , J.
01 Flu1d. Ing.g •• yol . 105. pp. 15. - 160. 19'3.
10. loo.inlin, • • L., • •1. Launder. B.I . Shar.I , Pr.diction of loa.otua,
leat and Xa •• Tran.f.r in Swirling Turbul.nt Bouodary Lay.rl.
J. 01 Beat fr&D.I.r, Tol . 96, pp. 20' - 209 , 1974.
11 . Birai. S., T. Tatagi, B. Bat.uaoto, Prediction. of tb. Laainlrisation
Ph.noa.nl in In 1sillly lotating .ip. Flow. J. 0/ Fluid. JDgn,.,
yol. 110. pp . (1( - (30. 19'1.
12. SchlichtinG, B• • Gr.Dz.cb1cbttbeori •• Id .4 • • Braun larl.rub •• 1912.
13 . Ludwieg, I . , •• ,ti..ung 4e. Verhlltnil ••, 4.r lu.tlu.ehtoeffizienteD
fOr Vlr.e un4 Iapull b.i turbulenteD Gr.Dz.ebiehten. z. t~r Flu,.1 •••• yol ••• pp. 73 -11. 1956 .
1'. C.b.ei. T•• I.C. Chaag , 1 GeDeral lethad ror CalculatiDg Koa.ntua In4
leat Tran.f.r iD L.. tnar and Tur hulent Duct rIo••• • u..ri c.l ••at
rr'D.rer. '01 1. pp. 39 - 68. 191• •
15. C.beei. T., P. Brad.haw, .br.ical ID4 CoapQtatioDll ',peet. of
COD•• ctiT. Be.t TraD.fer, SpriDger, New ror., 198'.
16. li.hihari . I •• I. liku' .... I. Kurlt ..i , Laaialri!atioD of Turhulent
rlow in the Inlet aeOiOD of In lxially lotatiDG .ip.,
Bull. 01 tbe JS~. yolo 30 . pp . 155-160. 1987.