Aspect Ratio Effect on Convective Heat Transfer of Radially Outward Flow in Rotating Rectangular Ducts

Experiments were conducted to investigate the effects of rotation and aspect ratio on the convective heat transfer of radially outward air flows in rotating rectangular ducts with a uniform wall heat flux by using fiberglass duct walls lined with separated 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; rotational buoyancy parameter, 0-1.2; and cross-sectional aspect ratio, 0.5, 1.0, and 2.0. The results showed that the higher the rotation number, the greater the enhancement ofthe heat transfer rate especially at the pressure side. The rotational buoyancy parameter decreases the heat transfer for low Re but enhances the heat transfer for high Re. The largest heat transfer enhancement is seen for AR 1.0, and the enhancement for AR 0.5 is greater than that for AR 2.0.

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 of turbine blades and cooling of electrical machin- ery.Increasing the turbine entry temperature is especially required to improve the thermodynamic efficiency and to reduce the specific fuel consumption for the compact de- sign ofadvanced gas turbine engines.Increasing the power output of electrical machinery is via the increases in the electrical and magnetic loadings in the stator and rotor of machine.Moreover, high operating temperature might cause material degradation on rotating components and excess ohm loss in electrical conductors; thus efficient in- ternal convective cooling technology introduced by flows in radially rotating duct is increasingly important.In a radially rotating heated rectangular duct the flow struc- ture and the heat transfer mechanism are simultaneously influenced by the rotation and the duct geometry.Many investigations on the effects of rotation and duct geometry on flow and internal heat transfer have been reported over these 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 by Coriolis acceleration appear symmetrically in the duct.Mori et al. [1968] studied the laminar convective heat transfer in radially rotating circular ducts by assuming ve- locity and temperature boundary layer profiles along the pipe wall.Subsequently, by using the same techniques, Mori et al. [1971] analyzed the turbulent convective heat transfer in a circular pipe.Table I lists recent experimental investigations on the internal convective heat transfer in radially rotating ducts.Under uniform wall temperature conditions, Wagner et al. [1991a, 1991b] investigated the local heat transfer .ofradially outward and inward flows in rotating serpentine passages with smooth walls.Buoyant flow is found to be favourable for heat transfer for both pressure and suction sides.However, the increase in 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 shown to influence the heat transfer response in a rectangularsectioned duct.Heat transfer is improved on pressure and suction sides as the wall-to-coolant temperature difference is increased for radially rotating outward flows.Han and Zhang [1992]

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in a square channel with smooth walls and radial outward flow for cases of uneven wall temperature with experiments.Hwang and Kuo 1993] conducted experiments on radially outward flows in a rotating square duct with uni- form wall heat flux.Augmentation of heat transfer on the pressure side is clearly observed.
To increase the effects ofrotation on internal heat trans- fer in the rotating ducts, lightweight and high-strength test sections were built for the requirement of high rota- tional speed up to 3,000 rpm.The interior wall surfaces of duct section were lined with separated stainless-steel film heaters of 0.01-mm thickness for the uniform wall heat flux.The purpose of his study was to investigate the effects of forced flow, rotation, and aspect ration on the convective heat transfer ofradially outward flows in heated rotating rectangular ducts.The aspect ratio used were 0.5, 1.0, and 2.0.Regional average Nusselt numbers on the pressure side, the suction side, and the side walls were obtained.

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 that the dominant forces due to rotation are force vectors of 2pU and pf22X because of U >> V and X >> Y and Z.The Coriolis force term (2pflU) induces cross streams which create additional mixing to the flow.The centrifugal force term (pf22X) generates centrifugal-buoyant radial secondary flow.The direction of this free convection flow is opposite to that of the radially outward flow in a heated rotating duct.An analysis of the flow-governing equations gives that the heat transfer coefficient at a certain axial lo- cation in the heated region is functionally influenced by other 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 are listed in the nomenclature.
In the present study, air with Pr 0.72 is used as the coolant fluid.The ratio of heated length and hydraulic di- ameter L/Dh 30.0 and the ratio of mean rotation radius and hydraulic diameter R / D h 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/Re 2, a ratio of the relative strength of Coriolis force to the inertial force, represents the effects of Coriolis force on forced convection; the rotational buoyancy parameter Ra* Rafz/Re 2 denotes the effect of centrifugal- buoyancy; and the cross-sectional aspect ratio AR reveals the effect ofthe cross-sectional configuration ofthe rectan- gular duct.All the physical properties needed in calculat- ing these parameters is evaluated at the bulk temperature Tb. corresponding nonrotating Nusselt number.The greater the Nusselt number ratio, the larger the heat transfer en- hancement.On the contrary, if the Nusselt number ratio is less than unity, the heat transfer is depressed.To scale the effects of rotation and to deduce the heat transfer from the experimental data, nondimensional parameter groups were applied.Table II depicts the ranges of the experi- mental variables and the corresponding nondimensional parameters used in the present study.

EXPERIMENTAL FACILITIES AND TEST 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 flow meters and rotary seal assembly to the test section.The flow meters of different flow ranges, from 0.2 to 8.0 m3/hr, 2. 3.  were used for indicating the coolant flow rate.A 0.3-mm type-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 plane for providing a well-mixed condition for outlet bulk tem- perature measurement by using another thermocouple set behind the mixing chamber.Glass-fiber, reinforced plastic with a low thermal con- ductivity (0.048 W/mC) for reducing heat loss was used for smooth duct walls.Four pieces of 0.01-mm thickness stainless-steel film heaters, heated by electrical power sup- plier through slip rings, were attached separately to each interior 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 boundary conditions: i.e., Kp (kwt)  is the ratio of wall heat conduction and air pure conduction inside a duct, and 2 Kp (kwt)/(hDh) is the ratio of wall heat conduction and air mixed convec- tion inside a duct.values for the Table III gives the estimated Kp and Kp present test facility.Because of K p << 1.0 for fiberglass and film heater, both of them can be considered as insu- lators.When the heater is supplied with electrical power, a boundary condition of nearly uniform heat flux can be of the achieved.On the other hand, the high K p and cooper block make the measurement of temperature in the block to be the regional average value.The duct hydraulic diameter, heater active length, and the mean rotation radius were 4, 120, and 180 mm, re- spectively.This gave a ratio of heater active length to hydraulic diamter of 30, which covered most of laminar flow entrance region and both the turbulence flow entrance and fully developed regions, and a mean rotation radius to hy- draulic diameter ratio of 45, which was reasonably large as compared with the ratio in a gas turbine or an electrical machine.Further construction details of the test section are shown in Fig. 3.The test assembly was encased in an elliptical aluminum tube with the internal void space filled with insulating ce- ramic cotton and was subsequently bolted on the rotating shafts so that the duct axis was perpendicular to the shaft axis.The whole model was installed in a test cell enclosed by a support frame with safety-glass plates and ventilation openings, and it was driven by a controlled electric motor.
An inverter which adjusted the electric current frequency was used for controlling the rotational speed detected by a photoelectric tachometer.A slip ring located at the other end of the shaft was to transmit the detected data from the thermocouples to a recorder, as shown in Fig. 2.

DATA REDUCTION
In an experiment with either a large flow rate or a large ro- tational speed, the compressibility correction for the mea- sured temperature of coolant flow must be performed.By denoting Tr as the recovery temperature, the fluid temper- ature may be corrected by /( T Tr l q- 2 ?' 2Cp where k is the specific heat ratio and ?' is the temperature recovery factor (Schlichting [1979]): ?, pr 1/2 for laminar flow ?, pr 1/3 for turbulent flow Note that in the present study a temperature correction up to 2.5C was found for the case of Re 15, 000 (M , 0.15) and 1.5C for f2 315 rad/s (3,000 rpm).At a certain axial location the heat transfer coefficient hx was evaluated as the ratio of the net heat flux qnet,x to the temperature difference between the heated wall tem- perature Tw,x and the coolant bulk temperature Tb,x: i.e., hx qnet,x/(Tw,x Tb,x).The net heat flux imposed on the coolant by convection was obtained by subtract- ing the external heat loss from the electric power supplied to the heater.The external heat losses were attributed to both conduction to the structure support and convection to the ambient air, and were estimated under no-flow con- dition by measuring heater power setting over ranges of wall temperature.The no-flow condition was achieved by filling insulation material in the channel.Based on a ther- modynamic energy balance, the rise of local coolant bulk temperature was determined step by step from the inlet coolant temperature by adding the net heat flux to the coolant along the duct.
Invoking the root-sum-square method introduced by showed in the present study that the estimated uncertain- ties in calculating Nusselt number were mainly affected by the local wall-to-coolant temperature and the net heat flux added to coolant from each wall.The measured vari- ables and their uncertainties in the measurement could be expressed as: Xi Xi (measured) 4-$Xi, where the best estimate of Xi is Xi (measured) and there was an uncer- tainty in Xi that might be as large as ,Xi.For the case of AR 1.0 and Ref 162.2, Fig. 4 showed the typical variations of local wall-to-coolant bulk temperature along the test duct for Re 1000, qnet 1,150 W/m2, and Re 10,000, qnet 10,500 W/m2, respectively.Uncertainty in the Nusselt number increased with the decrease in either the wall-to-coolant temperature difference or the net heat flux.It was found that the largest uncertainty of 20 per- cent was observed for Re 1,000 at x!Dh 25.0 on the pressure side because of the corresponding low wall-to- coolant bulk temperature difference and low heat flux.The uncertainty in the Nusselt numbers was approximately 8 percent when Reynolds number was greater than 10,000.

RESULTS AND DISCUSSION
The thermocouples were installed on two adjacent walls of the channel only.By rotating the radial channel, clock- wise or counter-clockwise or switching the channel 90 degrees, one was able to obtain the data on pressure side, suction side, and two side walls or for AR 0.5, and 2.0.Experiments were first conducted to determine the re- gional average Nusslet numbers for the nonrotating case along the four duct sides for a range of Reynolds num- bers (Re 1,000 15,000), positions (x/D h 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 of AR 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 et al. 1973] correlation in the laminar flow regime, respectively.The correlations are Dittus-Boeiter [1930]   Nu0 NUTFD[1.0 + 2.0/(x/Dh)] 08 for xDh >_ 10.0, where NUTFD 0.023Re Pr 0'4 is for fully developed flow in a circular duct with a uniform wall temperature, and Perkins et al. [1973] Nu0 1/[0.277 0.152exp (-38.68)] for square duct with a uniform wall heat flux, where 8 x/(Dh Re Pr) >_ 0.005.Fig. 5 shows that for the higher 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 was affected 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 selected through-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 than those at the suction side but this trend was attenuated with the higher Reynolds number.This was in agreement with the results of Mori et al. [1968,1971] that rotational heat transfer enhancement for laminar flow, in general, was more prominent than that for turbulent flow.The different behaviors on heat transfer over the pressure side and the suction side were due to the Coriolis-induced secondary cross streams in the form of a vortex pair which impinged toward the pressure side, then caused a return flow which carried already heated, relatively quiescent fluid from the pressure side and side walls to near the suction side.At higher rotation rate the strength of the Coriolis-induced cross streams was more intensified and this trend was more noticeable than that at lower rotation rate.It is also seen that the Nusselt number ratios at both pressure and suction sides dropped near duct outlet for most cases under study.
This was due to the increase of Nu0 near exit as shown in Fig. 5.For the effect of duct aspect ratio, large aspect ratio (long side aligned with the Coriolis force) gave larger short side direction Coriolis force gradient, but yielded a greater cross-sectional flow resistance.Due to the combination of these two effects, the largest heat transfer enhancement was seen for the case of AR 1.0, and the enhancement for AR 0.5 is greater than that for AR 2.0.For some cases, the heat transfer was depressed on the suction side because of the stabilizing effect of the vortex motion on the main flow disturbances.
Considering the effect of the Coriolis-induced cross streams on the main flows, Fig. 7 discloses the variations of the Nusselt number ratios with rotation number for the case of aspect ratio of 1.0, along with a comparison to the experimental results by Han and Zhang 1992].The results show that both the pressure side and suction side Nusselt number ratios of the present study at x /Dh 10.0 agreed fairly with those of Han and Zhang 1992] at x/Dh 9.0 and 11.0.Note that the results of Han and Zhang 1992] were based on the following conditions: Ro calculated at rotational speed 400 and 800 rpm, Re between 2,500 Re-900 Re-3600 P.So and 25,000, R/Dh 30, and D h 12.7 mm.The present data were based on: Ro calculated at rotational speed 500, 1,500, and 3,000 rpm, Re between 1,000 and 15,500, R/Dh 45, and Dh 4 mm.This confirms that Ro is indeed an important heat transfer governing parameter in a rotating channel.By either increasing the rotational speed or decreasing the Reynolds number, a higher Ro can be achieved.In the entry region, x/Dh 2.5, it is also seen that the observed enhancement in heat transfer for the present developing flow was less than that of higher x/Dh.This result was consistent with the experiment of Metzger and Stan 1977] for entry region heat transfer in a rotating radial tube.
To investigate the geometry effect of cross-sectional aspect ratio, Fig. 8 demonstrates the influence of aspect ratio 0.5, 1.0, and 2.0 on the Nusselt number ratios.The results show that the enhancement of the Nusselt number ratios for the case of AR 1.0 was always highest, and the enhancement for the case of AR 0.5 was higher than that for AR 2.0.This was due to the combination effect of weak long side Coriolis force gradient for low AR and high flow resistance for high AR.These phenomena can also be observed in heated horizontal rectangular ducts (Cheng and Hwang [1969] and curved channels (Cheng and Akiyama 1970].The depression of heat transfer on the 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 the salient 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, were selected 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 number ratios 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, it was found that increasing the rotational buoyancy param- eter decreased the Nusselt number ratios at both the pressure side and suction side for low Reynolds number flows Re 1,000, but the trend was reversed for Re 4,000.Then, these trends were diminished for higher Reynolds number.These phenomena can be found from analyzing the mixed convection of the buoyancy-induced opposing flows in a vertical heated tube for both constant wall tem- perature and uniform wall heat flux: the buoyancy forces tend to decrease the laminar heat transfer rate while they increase the turbulent heat transfer rate (Abdelmeguid and Spalding [1979]; Buhr et al. [1974].With increasing the rotational buoyance parameters, the depressed effect on heat 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, the increased tendencies on heat transfer were found by Wagner et al. 1991a, b], Morris and Ghvami-Nasr 1991], and Han and Zhang 1992].It is seen that the depression of Nusselt 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 larger peripheral area of AR 0.5 and 2.0 for more heated fluid with decelerated axial velocity.

PRACTICAL IMPORTANCE
1.The study of internal convective cooling in rotating ducts is of engineering importance for its applications to the cooling of turbine blades and cooling of electri- cal machinery. 2. Rectangular ducts of aspect ratios other than 1.0 may be applied to the internal cooling passage near the trail- ing edge of a turbine blade and in the cooling passage in 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 heat conduction parameter (Kp), four pieces of stainless-steel film heater of 0.01-mm thickness were separately lined with 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 are presented as follows.
1. Due to rotation, the Coriolis-induced cross streams impinge directly toward the pressure side, then cause a return flow which carriers already heated, relatively qui- escent fluid from the pressure side and side walls to near the suction side.Therefore, the Coriolis-induced cross streams create additional mixing to the main flows and enhance the heat transfer rate, especially at the pressure side.Also, the enhancement of heat transfer rate is gradually attenuated with increasing through-flow Reynolds number because of the effect of the turbulence becoming progressively larger than that induced by rotation.
2. The rotation number, effect of the Coriolis-induced cross streams on the forced main flows, performs an im- X/Dh-17.5 portant parameter to the internal convective heat trans- fer for radially rotating duct flows and a higher value can be obtained by either increasing the rotational speed or decreasing the through-flow Reynolds number.The higher the rotation number, the more intensified the strength of the Coriolis-induced cross streams and the more noticeable the enhancement of the heat transfer rate.
3. For high aspect ratio narrow duct, the short side Cori- olis force gradient is large, but the Coriolis-induced cross streams are weakened by viscous force over the longer side walls.Therefore, due to the combination of these two effects, the heat-transfer enhancement on the pressure side for 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 Re-14,000 P.S. of the stabilizing effect of the vortex motion on the main flow disturbances. 4. Varying the difference of the outlet-to-inlet air bulk temperature while other operating parameters were held constant during the experiment, the increasing rotational buoyancy 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 for higher Reynolds numbers.These phenomena can be found for buoyancy-induced opposing flows in a vertical heated tube: the buoyancy effect decreases the laminar heat trans- fer rate but increases the turbulent heat transfer rate, and this trend is more pronounced for AR 0.5 and 2.0 than that for AR 1.0._l'"l'" ''"":ll '"'"' 1'"'":', Re=lO,O00    (c) Effects of Ra* on Nu/Nuo for the cases of aspect ratios of (a) 0.5, (b) 1.0, and (c) 2.0.

FIGURE 4
FIGURE 4 Wall and coolant temperature variations along the test section in the case of AR 1.0.

FIGURE 5
FIGURE 5 Comparison of Nu in laminar and turbulent flow regime for AR 1.0.

FIGURE 8
FIGURE 8 Variations of Nun/Nu0 from Ro for the cases of different aspect ratio.
FIGURE9 (b) thermal conductivity of air, W/(m C) wall heat conduction parameter kwt/(kairDh) thermal conductivity of wall, W/(mC) actively heated length of duct, mm mach number Nusselt number hxOh / kair   Nusselt number for nonrotating condition Nusselt number for rotating condition pressure side FIGURE 9

U
t-x/Dh)[(Tw,x Tb,x)/Tb,x] Pr rotational buoyancy parameter Raf/Re through-flow Reynolds number pUoDhllZ rotational Reynolds number pflD/l Rotation number Reu/Re 2Dhl Uo S.S. suction side wall thickness, mm bulk temperature of air, C inlet air bulk temperature, C outlet air bulk temperature, C air bulk temperature at local position x, C recovery temperature of air, C duct wall temperature at x, C mean air velocity, m/s air dynamic viscosity, kg/ms air density, Kg/m rotational speed, rad/s the local heat transfer coefficient AuthorYear