-
International Journal ofRotating Machinery1994, Vol. 1, No. 1,
pp. 1-18Reprints available directly from the publisherPhotocopying
permitted by license only
(C) 1994 OPA (Overseas Publishers Association) Amsterdam
B.V.Published under license by Gordon and Breach Science Publishers
SA
Printed in the United States of America
Aspect Ratio Effect on Convective Heat Transfer ofRadially
Outward Flow in Rotating Rectangular Ducts
C. R. KUO and G. J. HWANGDepartment of Power Mechanical
Engineering, National Tsing Hua University, Hsinchu, Taiwan,
R.O.C.
Experiments were conducted to investigate the effects of
rotation and aspect ratio on the convective heat transfer of
radiallyoutward air flows in rotating rectangular ducts with a
uniform wall heat flux by using fiberglass duct walls lined
withseparated film heaters. The duct hydraulic diameter, heater
active length, and mean rotation radius were 4, 120, and 180
mm,respectively. Ranges of parameters were through-flow Reynolds
number, 1,000-15,000; rotation number, 0-0.32; rotationalbuoyancy
parameter, 0-1.2; and cross-sectional aspect ratio, 0.5, 1.0, and
2.0. The results showed that the higher the rotationnumber, the
greater the enhancement ofthe heat transfer rate especially at the
pressure side. The rotational buoyancy parameterdecreases the heat
transfer for low Re but enhances the heat transfer for high Re. The
largest heat transfer enhancement isseen for AR 1.0, and the
enhancement for AR 0.5 is greater than that for AR 2.0.
Key Words: Aspect ratio, convective heat transfer, rotating
duct, Coriolis force, centrifugal-buoyancy brce,
radiallyoutwardflow
HE study of internal convective heat transfer in rotat-ing ducts
is becoming of great significance for engi-
neers because of its potential applications in industry:
e.g.,cooling ofturbine blades and cooling ofelectrical machin-ery.
Increasing the turbine entry temperature is especiallyrequired to
improve the thermodynamic efficiency and toreduce the specific fuel
consumption for the compact de-sign ofadvanced gas turbine engines.
Increasing the poweroutput of electrical machinery is via the
increases in theelectrical and magnetic loadings in the stator and
rotorof machine. Moreover, high operating temperature mightcause
material degradation on rotating components andexcess ohm loss in
electrical conductors; thus efficient in-ternal convective cooling
technology introduced by flowsin radially rotating duct is
increasingly important. In aradially rotating heated rectangular
duct the flow struc-ture and the heat transfer mechanism are
simultaneouslyinfluenced by the rotation and the duct geometry.
Manyinvestigations on the effects of rotation and duct geometryon
flow and internal heat transfer have been reported overthese
years.By obtaining an approximate series solution from a per-
turbation equation in a rotating pipe flow, Barua [1955]
showed that two counter-rotating vortices induced byCoriolis
acceleration appear symmetrically in the duct.Mori et al. [1968]
studied the laminar convective heattransfer in radially rotating
circular ducts by assuming ve-locity and temperature boundary layer
profiles along thepipe wall. Subsequently, by using the same
techniques,Mori et al. [1971] analyzed the turbulent convective
heattransfer in a circular pipe. Table I lists recent
experimentalinvestigations on the internal convective heat transfer
inradially rotating ducts. Under uniform wall
temperatureconditions, Wagner et al. [1991a, 1991b] investigated
thelocal heat transfer .of radially outward and inward flowsin
rotating serpentine passages with smooth walls. Buoy-ant flow is
found to be favourable for heat transfer forboth pressure and
suction sides. However, the increasein heat transfer for the
inward-flowing passage was rela-tively less than that for outward
flow. Morris and Ghavami-Nasr 1991] observed that centrifugal
buoyancy is shownto influence the heat transfer response in a
rectangular-sectioned duct. Heat transfer is improved on pressure
andsuction sides as the wall-to-coolant temperature differenceis
increased for radially rotating outward flows. Han andZhang [1992]
reported the local heat transfer coefficient
-
C. R. KUO AND G. J. HWANG
Author Year
TABLEExperimental investigations on heat transfer in radially
rotating ducts
Duct Dh Max. L/Dh R/Dh Re. 10-3Type (mm) rpm
Max. B.C.Ro
Clifford et al. 1984
Harasgama and Morris 1988
Guidez 1989
Hwang and Soong 1989
Soong et al. 1991
Morris and Ghavami-Nasr 1991
Wagner et al. (a, b) 1991
Han and Zhang 1992
Hwang and Kuo 1993
Present study 1994
A 7.67 1,000 20 33 6.8-38.0 0.004 UHFA 7.67 1,850 20 33 7.0-25.0
0.103 UHFI-’1 7.5 1,000 20 33 7.0-21.0 0.042 UHFC) 5.0 2,000 20 70
7.0-25.0 0.049 UHF
I’-’-’! 10.66 5,000 11.5 27 17.0-41.0 0.2 UHF
I’-’l 4.0 3,000 30 30 0.7-20.0 0.428 UWT
I’--’] I’-’1 4.0 3,000 30 30 0.7-20.0 0.428 UWT
I-"-] 7.3 1,800 20 34 10.0-25.0 0.02 UHF
[’-I 12.7 1,100 14 33, 49 12.5-50.0 0.48 UWT
I-’-! 12.7 800 12 30 2.5-25.0 0.352 UWT, UHF
!-"1 4.0 3,000 30 45 1.0-15.5 0.32 UHF
i-"-I I"-I 4.0 3,000 30 45 1.0-15.5 0.32 UHF
in a square channel with smooth walls and radial outwardflow for
cases of uneven wall temperature with experi-ments. Hwang and Kuo
1993] conducted experiments onradially outward flows in a rotating
square duct with uni-form wall heat flux. Augmentation of heat
transfer on thepressure side is clearly observed.
To increase the effects ofrotation on internal heat trans-fer in
the rotating ducts, lightweight and high-strengthtest sections were
built for the requirement of high rota-tional speed up to 3,000
rpm. The interior wall surfacesof duct section were lined with
separated stainless-steelfilm heaters of 0.01-mm thickness for the
uniform wallheat flux. The purpose of his study was to investigate
theeffects of forced flow, rotation, and aspect ration on
theconvective heat transfer ofradially outward flows in
heatedrotating rectangular ducts. The aspect ratio used were
0.5,1.0, and 2.0. Regional average Nusselt numbers on thepressure
side, the suction side, and the side walls wereobtained.
GOVERNING PARAMETERS
The physical model and coordinates, as shown in Fig. 1,present
the rotation-induced inertial effects on forced con-vection of
radially outward air flow in a rotating rectan-gular duct.
Observing the force diagram, one discerns thatthe dominant forces
due to rotation are force vectors of2pU and pf22X because of U
>> V and X >> Y andZ. The Coriolis force term (2pflU)
induces cross streamswhich create additional mixing to the flow.
The centrifugal
force term (pf22X) generates centrifugal-buoyant radialsecondary
flow. The direction of this free convection flowis opposite to that
of the radially outward flow in a heatedrotating duct. An analysis
ofthe flow-governing equationsgives that the heat transfer
coefficient at a certain axial lo-cation in the heated region is
functionally influenced byother operating parameters. The results
are (Soong et al.[1991]):
Nuf f(Re, Refz, Rafz, Pr, L/Dh, R/Dh, a/b) (1)
The definitions of these dimensionless parameters arelisted in
the nomenclature.
In the present study, air with Pr 0.72 is used as thecoolant
fluid. The ratio of heated length and hydraulic di-ameter L/Dh 30.0
and the ratio of mean rotation radiusand hydraulic diameter R/Dh
45.0 are also fixed. Thus,Eq. 1 reduces to
Nu2/Nu0 f(Re, Ro, Ra*, AR) (2)
where the Reynolds number Re indicates the forced con-vection
effect; the rotation number Ro Re Re/Re2,a ratio of the relative
strength of Coriolis force to theinertial force, represents the
effects of Coriolis forceon forced convection; the rotational
buoyancy parame-ter Ra* Rafz/Re2 denotes the effect of
centrifugal-buoyancy; and the cross-sectional aspect ratio AR
revealsthe effect ofthe cross-sectional configuration ofthe
rectan-gular duct. All the physical properties needed in
calculat-ing these parameters is evaluated at the bulk
temperatureTb. The enhancement of heat transfer is presented by
theratio of the Nusselt number on rotating condition to the
-
ASPECT RATIO EFFECT
COOLING CHANNELS
DIRECTION OFROTATION
SUCTION SIDE
LEADING EDGE
TRAILING EDGE
RESSURE SIDE
TURBINE BLADE
GEOMETRY:
a=Height of ductb=Width of ductDh=4 mmL= 120 mmXo=120 mmR=180
mm
pg/y
2P U
Cross-stream Axial Velocity Suction SideFlow Profile (S.S.)
7- Pressure SideR 7 (P.S.)
FIGURE Physical model and coordinate.
corresponding nonrotating Nusselt number. The greaterthe Nusselt
number ratio, the larger the heat transfer en-hancement. On the
contrary, if the Nusselt number ratiois less than unity, the heat
transfer is depressed. To scalethe effects of rotation and to
deduce the heat transfer fromthe experimental data, nondimensional
parameter groupswere applied. Table II depicts the ranges of the
experi-mental variables and the corresponding
nondimensionalparameters used in the present study.
EXPERIMENTAL FACILITIES ANDTEST PROCEDURE
The experimental facilities, as illustrated in Fig. 2, con-sists
of four major parts: coolant air supply, test section,motor with
speed controller, and data acquisition system.Coolant air was
supplied from a compressor through flowmeters and rotary seal
assembly to the test section. Theflow meters ofdifferent flow
ranges, from 0.2 to 8.0m3/hr,
-
C. R. KUO AND G. J. HWANG
2.3.4.5.6.
Air inletFlow meterRotary jointSlip ringSafety glassTest
section
7. Support frame8. Motor9. Tachometer
10. Slip ring11. DC-power supply
1 2 2 3 4 5 6 7 8 910
ooo ooo
Air compressor 11 11FIGURE 2 Experimental setup.
Recorder
Computer
Variables
TABLE IIThe expedmental variables
Ranges
Flow Direction Radially Outward Flowrpm 0, 500, 1,000, 1,500,
2,000, 3,000Tb.o Tb.i (C) 15, 30, 45Tto (C) 40-130x/Dh 2.5, 10.0,
17.5, 25.0Pr 0.72Re 1,000, 2,000, 4,000, 8,200, 10,000, 15,500Re
53.4, 106.8, 162.2, 214.0, 320.4Ro 0-0.32Ra* 0-1.2AR 0.5, 1.0,
2.0
were used for indicating the coolant flow rate. A 0.3-mmtype-T
thermocouple was located at the duct inlet to mea-sure the inlet
coolant bulk temperature. A mixing chamber
with staggered rod bundles was attached to the exit planefor
providing a well-mixed condition for outlet bulk tem-perature
measurement by using another thermocouple setbehind the mixing
chamber.
Glass-fiber, reinforced plastic with a low thermal con-ductivity
(0.048 W/mC) for reducing heat loss was usedfor smooth duct walls.
Four pieces of 0.01-mm thicknessstainless-steel film heaters,
heated by electrical power sup-plier through slip rings, were
attached separately to eachinterior wall surface of the duct. At a
certain axial loca-tion, the wall surface temperatures were
measured via ther-mocouples which were firmly attached to copper
blocks.Morcos and Bergles [1975], Hwang and Chou [1987],and Chen
and Hwang 1989] proposed wall heat conduc-tion parameters for
analyzing duct wall thermal boundaryconditions: i.e.,
Kp (kwt)/(kairDh)
-
Material kw(W/mC)
ASPECT RATIO EFFECT
TABLE IIIEstimation for duct wall thermal boundary condition
kp kp(mm) (Based on air pure (Based on air mixed
conduction)* convection)**
Purpose
Fiber Glass 0.048
Stainless 16.3Film Heater
Copper block 111.0
3.0 1.199 0.199
0.01 1.357 0.1357
0.5 462.0 46.2
*Air thermal conductivity kair 0.03003 W/mC**Assumed the Nusselt
number for mixed convection is 10.
wall insulation
wall insulation
regional averagewall temperature
is the ratio of wall heat conduction and air pure
conductioninside a duct, and
2Kp (kwt)/(hDh)
is the ratio of wall heat conduction and air mixed convec-tion
inside a duct.
values for theTable III gives the estimated Kp and Kppresent
test facility. Because of Kp
-
C. R. KUO AND G. J. HWANG
A
CoolantOutletB
9
2.3.4.
Aluminum CoverThermocouplesFilled CottonDuet Wall
A
Section on AA
56789
10
7
Film HeaterMixing ScreenRotating ShaftCounter WeightRectangular
DuctCopper Bloek
6543 21
Section on BB
8CoolantInlet
FIGURE 3 Details of the test section.
showed in the present study that the estimated uncertain-ties in
calculating Nusselt number were mainly affectedby the local
wall-to-coolant temperature and the net heatflux added to coolant
from each wall. The measured vari-ables and their uncertainties in
the measurement could beexpressed as: Xi Xi (measured) 4-$Xi, where
the bestestimate of Xi is Xi (measured) and there was an
uncer-tainty in Xi that might be as large as ,Xi. For the case ofAR
1.0 and Ref 162.2, Fig. 4 showed the typicalvariations of local
wall-to-coolant bulk temperature along
the test duct for Re 1000, qnet 1,150W/m2, and Re10,000, qnet
10,500 W/m2, respectively. Uncertainty inthe Nusselt number
increased with the decrease in eitherthe wall-to-coolant
temperature difference or the net heatflux. It was found that the
largest uncertainty of 20 per-cent was observed for Re 1,000 at
x!Dh 25.0 on thepressure side because of the corresponding low
wall-to-coolant bulk temperature difference and low heat flux.
Theuncertainty in the Nusselt numbers was approximately 8percent
when Reynolds number was greater than 10,000.
-
ASPECT RATIO EFFECT
100
90
70
kRe = 1000 Re =10,000Rely= 162.6 Rely= 162.6, :SUCTION SIDEIq
:SIDE WALLC) :PRESSURE__
:3130.0 10.0 20.0 30.0 10.0 20.0 30.0
x//Dh X/DhFIGURE 4 Wall and coolant temperature variations along
the test section in the case of AR 1.0.
RESULTS AND DISCUSSION
The thermocouples were installed on two adjacent wallsof the
channel only. By rotating the radial channel, clock-wise or
counter-clockwise or switching the channel 90degrees, one was able
to obtain the data on pressure side,suction side, and two side
walls or for AR 0.5, and2.0. Experiments were first conducted to
determine the re-gional average Nusslet numbers for the nonrotating
casealong the four duct sides for a range of Reynolds num-bers (Re
1,000 15,000), positions (x/Dh 2.5, 10.0,17.5, and 25.0), and
outlet-to-inlet temperature differences(Tb,o Tb,i 15.0, 30.0, and
45.0C). For the case ofAR 1.0, Fig. 5 gives the results of
nonrotating condi-tion which are compared with the Dittus-Boelter
[1930]correlation in the turbulent flow regime and the Perkins
etal. 1973] correlation in the laminar flow regime, respec-tively.
The correlations are
Dittus-Boeiter [1930]
Nu0 NUTFD[1.0 + 2.0/(x/Dh)]
08forxDh >_ 10.0, where NUTFD 0.023Re Pr0’4 is forfully
developed flow in a circular duct with a uniform walltemperature,
and
Perkins et al. [1973]
Nu0 1/[0.277 0.152exp (-38.68)]
for square duct with a uniform wall heat flux, where8 x/(Dh Re
Pr) >_ 0.005. Fig. 5 shows that for thehigher Reynolds number
the Nusselt number is approxi-mately within 10 percent of that of
Dittus-Boelter 1930].However, the higher Nusselt number at x /Dh
25.0 wasaffected by the discontinuity of the uniform heat flux
ther-mal boundary condition at the duct exit region.
Figures 6(a), (b) and (c) indicate the effects of rota-tion on
heat transfer along the test section for selectedthrough-flow
Reynolds numbers with aspect ratios 0.5,1.0, and 2.0, respectively.
It is noted that the Nusselt num-ber ratios at the pressure side
were always greater thanthose at the suction side but this trend
was attenuated withthe higher Reynolds number. This was in
agreement withthe results of Mori et al. [1968, 1971] that
rotational heattransfer enhancement for laminar flow, in general,
wasmore prominent than that for turbulent flow. The
differentbehaviors on heat transfer over the pressure side and
thesuction side were due to the Coriolis-induced secondarycross
streams in the form of a vortex pair which impingedtoward the
pressure side, then caused a return flow whichcarried already
heated, relatively quiescent fluid from thepressure side and side
walls to near the suction side. Athigher rotation rate the strength
of the Coriolis-induced
-
C. R. KUO AND G. J. HWANG
25.0
20.0
10.0
’I ’IRe-1000Re-2000
" Perkin et al.( 973)o :P.S., :S.S.A ’Side wallI ill,
5 10-3 2 5 10
-2 2
x/(DhRe Pr)5 10
0.06
0.04
0.02
I
0.000.0 5.0
Present DataDittus and Boelter(1930)_Circular Duct
.I_ I I
10.0 15.0 20.0 25.0x/Dh
30.0
FIGURE 5 Comparison of Nu in laminar and turbulent flow regime
for AR 1.0.
cross streams was more intensified and this trend was
morenoticeable than that at lower rotation rate. It is also
seenthat the Nusselt number ratios at both pressure and
suctionsides dropped near duct outlet for most cases under
study.This was due to the increase of Nu0 near exit as shown inFig.
5. For the effect of duct aspect ratio, large aspect ratio(long
side aligned with the Coriolis force) gave larger shortside
direction Coriolis force gradient, but yielded a
greatercross-sectional flow resistance. Due to the combination
ofthese two effects, the largest heat transfer enhancementwas seen
for the case of AR 1.0, and the enhancementfor AR 0.5 is greater
than that for AR 2.0. For somecases, the heat transfer was
depressed on the suction side
because of the stabilizing effect of the vortex motion onthe
main flow disturbances.
Considering the effect of the Coriolis-induced crossstreams on
the main flows, Fig. 7 discloses the variationsof the Nusselt
number ratios with rotation number for thecase of aspect ratio of
1.0, along with a comparison to theexperimental results by Han and
Zhang 1992]. The resultsshow that both the pressure side and
suction side Nusseltnumber ratios of the present study at x/Dh 10.0
agreedfairly with those of Han and Zhang 1992] at x/Dh 9.0and 11.0.
Note that the results of Han and Zhang 1992]were based on the
following conditions: Ro calculated atrotational speed 400 and 800
rpm, Re between 2,500
-
ASPECT RATIO EFFECT
Re- 900
Re- 3600
P.So
30.0
Re- 9200
0.0 10.0 20.0
x/DhFIGURE 6 (a)
Re- 1800
Re- 7200P.S.S.S. Re0
53
162.42320.
Re-14,000
10.0 20.0
x/Dh30.0
-
10 C. R. KUO AND G. J. HWANG
2.0
Z 1.5
Re- 4000
Re-lO,O00
0.0 10.0 20.0 30.0
x/Dh
Re- 2000
Re- 8200
P.S.S.S.(C)
A53.4162.2320.4
Re-15,500
I,
10.0 20.0
x/Dh30.0
FIGURE 6 (b)
-
ASPECT RATIO EFFECT 11
Re- 900
Re= 3600
1800
7200
o 53.4162.2320.4
Re- 9200 Re-14,000
0.0 10.0 20.0 30.0 10.0 20.0 30.0
x/Dh x/’DhFIGURE 6 (c) Variations ofNua/Nuo withx/Dh based on
Tb,o Tb,i 30.0C for the cases ofaspect ratios of(a) 0.5, (b) 1.0,
and (c) 2.0.
-
12 C. R. KUO AND G. J. HWANG
X/Dh=25.0
AOp.s.
RoFIGURE 7 Variations of Nu/Nuo from Ro for AR 1.0 and Tb,o Tb,i
30.0C.
-
ASPECT RATIO EFFECT 13
and 25,000, R/Dh 30, and Dh 12.7 mm. The presentdata were based
on: Ro calculated at rotational speed500, 1,500, and 3,000 rpm, Re
between 1,000 and 15,500,R/Dh 45, and Dh 4 mm. This confirms that
Rois indeed an important heat transfer governing parameterin a
rotating channel. By either increasing the rotationalspeed or
decreasing the Reynolds number, a higher Ro canbe achieved. In the
entry region, x/Dh 2.5, it is alsoseen that the observed
enhancement in heat transfer forthe present developing flow was
less than that of higherx/Dh. This result was consistent with the
experiment ofMetzger and Stan 1977] for entry region heat transfer
ina rotating radial tube.To investigate the geometry effect of
cross-sectional
aspect ratio, Fig. 8 demonstrates the influence of aspectratio
0.5, 1.0, and 2.0 on the Nusselt number ratios. Theresults show
that the enhancement of the Nusselt numberratios for the case of AR
1.0 was always highest, andthe enhancement for the case ofAR 0.5
was higher thanthat for AR 2.0. This was due to the combination
effectof weak long side Coriolis force gradient for low AR andhigh
flow resistance for high AR. These phenomena canalso be observed in
heated horizontal rectangular ducts(Cheng and Hwang [1969] and
curved channels (Chengand Akiyama 1970]. The depression of heat
transfer onthe suction side was also seen for small Ro case.By
definition the rotational buoyancy parameter is af-
fected by the rotation number, wall-to-coolant tempera-ture,
eccentricity, and local positions. To highlight thesalient feautres
of the centrifugal-buoyant radial sec-ondary flows, three
outlet-to-inlet bulk temperature dif-ferences, Tb,o Tb,i 15.0,
30.0, and 45.0C, wereselected while other operating parameters were
held con-stant during each measurement. Figs. 9(a), (b), and
(c)illustrate the results of the variations of Nusselt numberratios
with rotational buoyancy parameters at axial lo-cation of x/Dh 17.5
and for aspect ratios 0.5, 1.0,and 2.0, respectively. As the
rotation number was fixed, itwas found that increasing the
rotational buoyancy param-eter decreased the Nusselt number ratios
at both the pres-sure side and suction side for low Reynolds number
flowsRe 1,000, but the trend was reversed for Re 4,000.Then, these
trends were diminished for higher Reynoldsnumber. These phenomena
can be found from analyzingthe mixed convection of the
buoyancy-induced opposingflows in a vertical heated tube for both
constant wall tem-perature and uniform wall heat flux: the buoyancy
forcestend to decrease the laminar heat transfer rate while
theyincrease the turbulent heat transfer rate (Abdelmeguid
andSpalding [1979]; Buhr et al. [1974]. With increasing
therotational buoyance parameters, the depressed effect onheat
transfer agree with those proposed by Morris and Ay-
han [1979], Clifford et al. [1984], Harasgama and Morris[1988],
and Soong et al. [1991]; on the other hand, theincreased tendencies
on heat transfer were found by Wag-ner et al. 1991a, b], Morris and
Ghvami-Nasr 1991], andHan and Zhang 1992]. It is seen that the
depression ofNusselt number for the centrifugal buoyancy force is
big-ger for the cases of AR 0.5 and 2.0 than that for AR
1.0. One may attribute this phenomenon to the largerperipheral
area of AR 0.5 and 2.0 for more heated fluidwith decelerated axial
velocity.
PRACTICAL IMPORTANCE
1. The study of internal convective cooling in rotatingducts is
of engineering importance for its applicationsto the cooling of
turbine blades and cooling of electri-cal machinery.
2. Rectangular ducts of aspect ratios other than 1.0 maybe
applied to the internal cooling passage near the trail-ing edge of
a turbine blade and in the cooling passagein a rotor of electrical
machinery.
CONCLUSION
The investigation has presented rotation effects,
Coriolis-induced cross streams, and centrifugal-buoyant radial
sec-ondary flows, on convective heat transfer of radially out-ward
flows in rotating rectangular ducts with AR 0.5,1.0, and 2.0.
According to an analysis with a wall heatconduction parameter (Kp),
four pieces of stainless-steelfilm heater of 0.01-mm thickness were
separately linedwith the interior wall surfaces of the fiberglass
duct to ob-tain the nearly uniform wall heat flux boundary
conditions.The results obtained and described in this experiment
arepresented as follows.
1. Due to rotation, the Coriolis-induced cross streamsimpinge
directly toward the pressure side, then cause areturn flow which
carriers already heated, relatively qui-escent fluid from the
pressure side and side walls to nearthe suction side. Therefore,
the Coriolis-induced crossstreams create additional mixing to the
main flows andenhance the heat transfer rate, especially at the
pressureside. Also, the enhancement of heat transfer rate is
grad-ually attenuated with increasing through-flow Reynoldsnumber
because of the effect of the turbulence becomingprogressively
larger than that induced by rotation.
2. The rotation number, effect of the Coriolis-inducedcross
streams on the forced main flows, performs an im-
-
14 C. R. KUO AND G. J. HWANG
X/Dh-17.5
2.0
1.5
X/Dh-lO.O
x/Dh-
P.S.
0
A
0 @V
AR
5z o#
;07
0.0 0.1 0.2 0.3 0.4 0.5
FIGURE 8 Variations of Nun/Nu0 from Ro for the cases of
different aspect ratio.
portant parameter to the internal convective heat trans-fer for
radially rotating duct flows and a higher valuecan be obtained by
either increasing the rotational speedor decreasing the
through-flow Reynolds number. Thehigher the rotation number, the
more intensified thestrength of the Coriolis-induced cross streams
and themore noticeable the enhancement of the heat
transferrate.
3. For high aspect ratio narrow duct, the short side Cori-olis
force gradient is large, but the Coriolis-induced crossstreams are
weakened by viscous force over the longerside walls. Therefore, due
to the combination of these twoeffects, the heat-transfer
enhancement on the pressure sidefor the largest for AR 1.0 and the
enhancement for AR
0.5 are larger than that for AR 2.0. For some cases,the heat
transfer is depressed on the suction side because
-
ASPECT RATIO EFFECT 15
4.0
z.01.5
.o4.03.0
2.01.5
.o4.03.0
2.01.5
.o
4.03.0
2.01.
.o4.03.0
2.0
.o0.7
II]
Re-14,000
P.S. o.oo.o11
Ro=0.004
IIIll
Re- 9200
P.S.S.S. Tb,o--Tb,
’ so-Iv so’odoJ-
Ro=0.006
IIII]
Re- 3600
Ro=0.015
0.033
o.o7 o0,0,
Re- 1800
IIIIIi1111
Re-
IIIII
IIII
0.084
o.o4 0
IIIll
0.085
Ro=0.031o,
900
Ro=0.059>
tttilIiiii
IIII!
0.166
IIIII
0.332
o %
Itltll Illlil tlllll
10-3
10-z
10-1
1 2
RaFIGURE 9 (a)
of the stabilizing effect of the vortex motion on the mainflow
disturbances.
4. Varying the difference of the outlet-to-inlet air
bulktemperature while other operating parameters were heldconstant
during the experiment, the increasing rotationalbuoyancy parameters
made the heat transfer rate de-crease for cases of Re 1,000 but
increase for cases
of Re 4,000, and these trends are then diminished forhigher
Reynolds numbers. These phenomena can be foundfor buoyancy-induced
opposing flows in a vertical heatedtube: the buoyancy effect
decreases the laminar heat trans-fer rate but increases the
turbulent heat transfer rate, andthis trend is more pronounced for
AR 0.5 and 2.0 thanthat for AR 1.0.
-
16 C. R. KUO AND G. J. HWANG
4.0
3.0-
2.0
1.5
1.0
4.0
3.0-
2.0
1.5
1.0
4.0
3.0
2.0
1.5
1.0
4.0
3.0-
2.0
1.5
1.0
4.0
3.0
2.0
1.5
1.0
0.7-4
10
Re-15 500P.S.S.S. Tb,o- T b,Ib *1 .o1IV 30.0
P.S. o.oo21 45.o0.0010
Ro=O.O04
_-l’"l’" ’’"":ll ’"’"’ 1’"’":’,Re=lO,O00
0.031O.OLO o,
Ro=0.005
Re= 40000.076
0.040
Ro=O.O14 O
IIII
Re- 2000 0o.ots
0.079
o,
Ittlll tlill[ ttlttllilliJ iliillo 0.312
Re= 1000 o.s7o
FIGURE 9 (b)
Nomenclature
aARbcpOhhx
height of rectangular ductcross-sectional aspect ratio a/bwidth
of rectangular ductspecific heat of air, J/(kgC)hydraulic diameter
2ab/(a + b)local heat transfer coefficient qnet,x/(Tw,x
Tb,x),W/(m2C)
kairKpkwLMNuNuoNuP.S.
thermal conductivity of air, W/(m C)wall heat conduction
parameter kwt/(kairDh)thermal conductivity of wall, W/(mC)actively
heated length of duct, mmmach numberNusselt number hx Oh /
kairNusselt number for nonrotating conditionNusselt number for
rotating conditionpressure side
-
ASPECT RATIO EFFECT 17
FIGURE 9
Re- 9200
Ro=0.006
Re= 3600
Re- 1800
,JillIII
3.0
2.01.5
.o
Ro=O.031
oililll
9OO
0Ro=0.059
3.0
z.o1.{5
.o
Re=
illl
10-Ra.7 Illl10-4 10-a
0.166
0.085
iill, ll,,,IIII
0.3320-
0 V(
v o_v,i v
1, lilil *1 I-10
-11 2
(c) Effects of Ra* on Nu/Nuo for the cases of aspect ratios of
(a) 0.5, (b) 1.0, and (c) 2.0.
PrqnetRRa
Ra*
ReRo
Prandtl number Cp/L/kairnet wall heat flux, W/mmean rotation
radius Xo + L/2, mmrotational Rayleigh numberRo2Re2(e -t-
x/Dh)[(Tw,x Tb,x)/Tb,x] Prrotational buoyancy parameter
Raf/Rethrough-flow Reynolds number pUoDhllZrotational Reynolds
number pflD/lRotation number Reu/Re 2Dhl Uo
S.S. suction sidewall thickness, mmbulk temperature of air,
Cinlet air bulk temperature, Coutlet air bulk temperature, Cair
bulk temperature at local position x, Crecovery temperature of air,
Cduct wall temperature at x, Cmean air velocity, m/s
-
18 C. R. KUO AND G. J. HWANG
U,V,W
Xo
X,Y,Z
velocity component, m/sstreamwise distance from heated channel
inlet, mmdistance between rotating axis and the heated channel
inlet,mmsystem coordinates, mm
Greek letters
eccentricity Xo/Dhtemperature recovery factorair dynamic
viscosity, kg/msair density, Kg/mrotational speed, rad/s
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