heat and in low-aspect-ratio rectangular with staggered walls*

International Journal of Rotating Machinery1998, Vol. 4, No. 4, pp. 283-291Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in India.

Heat Transfer and Friction in a Low-Aspect-RatioRectangular Channel with Staggered Slit-Ribbed Walls*

JENN-JIANG HWANG a,] and TONG-MIIN LIOU b

Department of Mechanical Engineering, Chung-Hua University, Hsinchu, Taiwan 300, ROC,"Department of Power Mechanical Engineering, National Tsing-Hua University, Hsinchu, Taiwan 300, ROC

(Received 20 May 1997;In finalform 2 July 1997)

Fully developed heat transfer and friction in a rectangular channel with slit-ribbed walls areexamined experimentally. The slit ribs are transversely arranged on the bottom and topchannel walls in a staggered manner. Effects of rib open-area ratio (/3-- 24%, 37%, and46%), rib pitch-to-height ratio (Pi]H= 10, 15 and 20), and Reynolds number (10,000_Re <_ 50,000) are examined. The rib height-to-channel hydraulic diameter ratio is fixed atH]De 0.081. It is disclosed that the heat transfer coefficient for the slit-ribbed channel ishigher than that for the solid-ribbed channel, and increases with rib open-area ratio. Resultsalso show that the friction factor for the slit-ribbed channel is significantly lower than thatfor the solid-ribbed one. Moreover, the ribs with larger open-area ratios in a higher flowReynolds number condition could give the better thermal performance under the constantfriction power constraint. Roughness functions for friction and heat transfer are furtherdeveloped in terms of rib and flow parameters.

Keywords." Slit rib, Rib open-area ratio, Thermal performance, Roughness function

INTRODUCTION

Applying rib-turbulators on the surfaces of highheat flux devices is widely considered because itprovides an abundant heat transfer enhancement.The use of solid ribs has been explored for applica-tions such as gas-cooled nuclear reactors and gasturbine blade cooling. The ribs break up the viscoussublayer and promote local wall turbulence that, inturn, increase the heat transfer from the rough aswell as smooth surfaces. Inevitably, the enhance-

ment in heat transfer accompanies a higher pressuredrop penalty of the fluid flow. Therefore, manyinvestigations have been conducted toward estab-lishing an optimal rib geometry which gives thebest heat transfer performance for a given pumpingpower. Relevant geometric parameters discussedin previous works are channel aspect ratio (Hanet al., 1978), rib angle-of-attack (Han et al., 1985),rib spacing (Liou and Hwang, 1992a), rib height(Liou and Hwang, 1992b), rib shape (Liou andHwang, 1993), and relative arrangement of ribs

* This paper was originally presented at ISTP-9.Corresponding author. Tel.: 886-35374281 Ext. 8334. Fax: 886-35373771. E-mail: [email protected].

283

284 J.-J. HWANG AND T.-M. LIOU

(Lau et al., 1991). This paper concerns the effect ofslit rib-turbulators on the heat transfer and frictionin a rectangular channel. Slit ribs are of greatinterest since they provide a substantially lower dragforce in comparison with solid ribs and still provideenhanced heat transfer coefficient in comparisonwith the smooth-walled duct. Three main issues areprovided in this study. First, experiments are con-ducted to examine the effect of replacing conven-tional solid rib-turbulators by slit rib-turbulatorson the heat transfer and friction in a rectangularchannel. Then, by using the average heat transferand friction data the thermal performance compar-ison is made between the solid- and slit-ribbedchannels under the constant pumping power condi-tion. Finally, roughness correlations of the frictionand heat transfer for turbulent flows in a rectan-gular channel with slit ribs are developed. Based onthe law-of-the-wall similarity and the applicationof the heat-momentum transfer analogy (Dippreyand Sabersky, 1963), the geometrically non-similar

roughness parameters rib pitch-to-height ratio

Pi/H and angle-of-attack c have been respectivelycorrelated into the roughness functions for turbu-lent flow in a tube (Webb et al., 1971) and thatbetween parallel plates (Han et al., 1985) withrepeated solid-type ribs. In the current work, thisconcept will be extended to correlate the data forturbulent flow in a rectangular channel withrepeated slit roughness by taking into account theparameter/3.

DESCRIPTION OF EXPERIMENT

The horizontal low-speed wind tunnel, as shown inFig. 1, is operated in suction mode. Air is drawninto the settling chamber from the temperaturecontrolled laboratory room, traverses the bell-likecontraction, flows through the slit-ribbed channel,a flow straightener, a flow meter, and then isexhausted by a 3 hp blower. The test channel is

Staggered

..)1

Chamber Therocouple

SupplyPressureHybrid

co er T’ransduce

InverterDevices Computer

FIGURE Schematic drawing of flow system and experimental apparatus.

HEAT TRANSFER AND FRICTION 285

1200mm long and has an aspect-ratio of 4:1(40 mm 160 mm for Y-Z plane). The hydraulic(equivalent) diameter (De) of the test channel is6.4 mm. The upper and lower horizontal walls ofthe test section are heated, and the two verticalwalls (Plexiglas plates) are thermally insulated. Theconfiguration of the heat transfer walls is shownschematically in Fig. 2(a). Each heat transfersurface is made of a 3-mm-thick aluminum platewith a 0.18-mm-thick thermofoil. The aluminumplate is highly polished to minimize emissivity,hence radiative losses. In addition, to prevent con-ductive heat losses, a 6-mm-thick fiberglass boardand a 20-mm-thick balsa wood are adhered uni-formly on the back side of the heated plate. Thealuminum slit ribs are glued staggeringly on theupper and lower channel walls in a requiredarrangement. The rib angle-of-attack is 90 Thinlayers of thermally conducting epoxy cement(electrically insulated) serve to ensure good contact

between the heat transfer surfaces and ribs.Aluminum ribs are used in this study for their highthermal conductivity and machinability.

Thirty-nine copper-constantan thermocouplesare located between X/De 12-14 for wall surfacetemperature measurements. As shown in Fig. 2(a),for the baseline experiments (Pi/H= 10), thechannel wall between ribs and the rib top wall areinstrumented with ten and three thermocouples,respectively. The junction-beads (about 0.15 mm)of the thermocouples are carefully embedded intothe walls, and then ground flat to ensure that theyare flush with surfaces. In addition, one and fivethermocouples are placed at the test channel inletand exit, respectively, for the airflow temperaturemeasurements. The temperature signals are trans-ferred to a hybrid recorder, and subsequently sentto a PC-486 computer via a multi-I/O interface. Thepre-processing of the raw data is done in a built-in BASIC program where the non-dimensional

39 ThermocouplesSlit Ribs(5.2mm thick)

Aluminum Plate(3mm)Thermofoil(O. ! 8mm)Fiberglass Plate (6mm)Balsa good (20ram)

(b) Section

FIGURE 2 (a) Sketch of configurations of test section. (b) Dimensions of the slit rib.


parameters are calculated. Two pressure tapssituated at X/De-6 and 17 measure the staticpressure differences for frictional losses. A micro-differential transducer is connected to each pressuretap. The measured pressure signal is subsequentlyamplified to a digital readout. As shown inFig. 2(b), a total of nine slit-slots is perforatedthrough the aluminum rib (5.2-mm-thick) for therequired open-area ratio. The open-area ratio ofthe slit rib is defined as/3- (n. w. h)/(W. H), wheren is the number slit-slots distributed on the rib, w,the width of the slot, h, the height of the slots, W,the channel width, and H, the rib height. The ribopen-area ratios investigated are 24%, 37%, and46%. The other variable parameters investigatedand compared are: rib pitch-to-height ratio, Pi/H10, 15, and 20; and flow Reynolds number 10,000 <Re <_ 50,000.

DATA REDUCTION

The convection heat transfer coefficient of theheated surface is presented in terms of the averageNusselt number Nu, which is defined as

Nu h. De/kf Qconv" DelIA. (Tw Tb)" kf](Q Qloss)" DeI[A. (Tw rb). ks]. (1)

Qconv represents the net heat transfer rate fromthe ribbed wall to the coolant, and is calculated bysubtracting the heat loss (Qloss) from the suppliedelectrical power (Q). The electrical power generatedfrom the thermofoil is determined from the mea-sured thermofoil resistance and the current throughthe thermofoil on each surface. It is also checked bymeasuring voltage drop across each thermofoil. Theeffect of the temperature variation on the localthermofoil is calculated to be small and negligible.Thus, the thermofoil ensured a nearly uniform heatflux on each surface of the test duct. The heat losscan be estimated by the following equation:

Qloss-Qc+Qr/Qa. (2)

Qc is the conductive heat loss from the back sides ofthe heated plates (ribbed walls) and the verticaladiabatic plates (smooth walls) to the environment,and is estimated to be less than 3% and 6%,respectively, for the range of Reynolds numberstested. The radiative heat loss from the roughenedsurface to its surroundings, Qr, is evaluated by adiffuse gray-surface network (Siegel and Howell,1981), and is less than 0.5% of the total electricalpower input. Qa is the axial conductive heat lossthrough upstream and downstream ends of theheated plates and is estimated to be less than 1%of the total electrical power input for Reynoldsnumber larger than 10,000. The energy conservationis further checked by comparing the total net heattransfer rate from the test channel to the cooling air(Qconv) with the cooling air enthalpy rise along thetest channel. Results show that only 5% in discre-pancy is found between these two methods. To placethe results on a common basis the heat transfer area(A) in Eq. (1) serves as the projected area of thecorresponding ribless wall. The wall temperature,Tw is the average value of the outputs of the 39thermocouples and could be read from the hybridrecorder directly. The bulk mean air temperature(Tb) is calculated from the measured channel inletand outlet air temperature by assuming a linear risealong the test channel. As estimation by the methodof Kline and McClintock (1953) the maximumuncertainty of the Nusselt number is less than6.6% for Reynolds number larger than 10,000.The friction factor of the periodic fully developed

flow is calculated from the pressure drop across thetest duct and the mean velocity of the air andexpressed as

f-[(-AP/AX). De]/(p. U2/2), (3)

where the pressure gradient, AP/AX, is evaluatedby taking the ratio of the pressure difference andthe distance of two pressure taps. The maximum

uncertainty off is estimated to be less than 7.3%for Reynolds number greater than 10,000 by theuncertainty estimation method of Kline andMcClintock (1953).


The friction data for turbulent flow in a rectan-gular duct with two opposite ribbed walls can becorrelated by the modified equations (Han, 1988;Liou and Hwang, 1993):

R(H+) -(f/8)-l/2+2.51n{(2H/De)[2W/(W/ B)]} + 2.5,

(4)

where

H+ (H/De) Re (f/8) 1/2. (5)

Similarly, heat transfer data for fully ,developedturbulent flow in a rectangular duct with two oppo-site ribbed walls can be correlated by the followingequations (Han, 1988; Liou and Hwang, 1993):

G(H+,Pr) (f/8)l/2/St-+- 2.51n(2H/De) + 2.5.

(6)

In the present data reduction program, Eqs. (4)-(6) are used to calculate the friction roughnessfunction R(H +) and the heat transfer roughnessfunction G(H +, Pr).

EXPERIMENTAL RESULTS ANDDISCUSSION

Average Heat Transfer Coefficientsand Friction Factors

Figure 3 gives the average Nusselt number ratioof the slit-ribbed walls as a function of Reynoldsnumber, where Nus is obtained from the Dittus-Boelter correlation. The solid line is the results ofthe solid-ribbed wall (Hwang and Liou, 1995).It can be observed from this figure that for all ofthe rib open-area ratios investigated the heattransfer augmentation (Nu/Nus > 1) is achieved.The average Nusselt number ratios for both theslit- and the solid-ribbed walls decrease withincreasing Reynolds number. It is also found thatthe increment of the average Nusselt number ratiosfor the slit-ribbed wall is higher than that of the

4

3-

2-

? 24% 37% 46?;0 37% 37%Pi/H 10 10 10 15 20

Solid-ribbed wall, Pi/H= 10 12]

Re10 -4

FIGURE 3 Average Nusselt number ratio vs. Reynoldsnumber.

solid-ribbed wall, and increases with increasing therib open-area ratio. For a given Reynolds number,the level of the heat transfer augmentation of theslit-ribbed wall is about 8%, 20%, and 35% higherthan that of the solid-ribbed wall, respectively, for/3= 24%, 37%, and 46% at Pi/H= 10. The higherheat transfer enhancement accompanied by the slit-ribbed wall may be due to the combined factors ofthe higher turbulent mixing of multijet-streamsafter the slit rib and the higher heat transfer surfacearea as flow through the slit ribs. It is interesting tonote that the slit rib has about 2.3-fold heat transfersurface area as compared with the solid rib.

Figure 4 illustrates the effects of rib open-arearatio and flow Reynolds number on the frictionfactor for the slit-ribbed geometry. The rib height isfixed at H/De=O.081. The pressure drops acrossthe test channel are measured by the unheated flowconditions. It is seen from this figure that the fric-tion factor for the slit-ribbed channel is higher thanits counterpart for the smooth channel (dashed-line, Blasius correlation), but lower than that for thesolid-ribbed channel (dotted line, Hwang and Liou,1995). It is almost independent of the Reynoldsnumber. In comparison with the results of thechannel flows with solid ribs, the values of f are


0.1

0.01

24% 37% 46% 37% 37%10 10 10 15 20

/fs= 0.316Re-0"25

/2:0, H/De=O.081, Pi/H: 01121042,A =0, H/De=0. 0,1Haln[1

Rexl0 -4

FIGURE 4 Friction factor vs. Reynolds number.

approximately 84%, 72%, and 60% for fl= 24%,37%, and 46%, respectively, in the range of theReynolds number investigated. As expected, for a

given Reynolds number the friction factor decreaseswith increasing fl because of the less cross-sectionalblockage for ribs with the larger ft. A comparisonof the present data with those in the previous workis also made in Fig. 4. In Han’s work (Han, 1984),solid ribs with H/De 0.042 were arranged on thetwo opposite channel walls. The channel blockageis largely similar to that of the present slit-ribbedchannel with H/De 0.081 and fl 46%. It is foundthat a good agreement is achieved between thepresent and Han’s works.

Thermal Performance

In practical applications, the thermal performancebased on unit pumping (or blowing) power is widelyconsidered. Although, the heat transfer could beenhanced with ribs applying on the channel wall,the pressure loss is inevitably increased. To keep-the fluid pumping power constant, the flow velocitymust be reduced. Subsequently the average Nusseltnumber is reduced. Fortunately, this reduction cansometime be overcome by heat transfer enhance-ment which is given by rib-turbulators. The relation-ship of the ribbed and smooth channels for the

2.0

1.0

Pi/H

C>

24% 37% 46% 37% 37%10 10 10 15 20

OO

S01id-ribbed channel[12] "N,, II ,, ... ,l iI

i0

Rex 10-4

FIGURE 5 Thermal performances of the ribbed channelsunder constant pumping power condition.

same pumping power has been described in detailin Hwang and Liou (1995). The pumping powerrequired to feed the fluid through the channel is

proportional to f. Re3. Thus in Fig. 5 the perfor-mance shown by the ratio of Nu/Nu’ is plottedagainst (f/fs) 1/3. Re (denotes as Re* in Fig. 5),where Nu is the average Nusselt number for asmooth channel with the flow rate at which thepumping power is the same as that obtained in aribbed channel. Figure 5 shows that the improve-ment in Nusselt number ratio of the slit-ribbedchannel is more pronounced than that of the solid-ribbed channel (solid line, Hwang and Liou, 1995),especially for the larger rib open-area ratio. Atlower Reynolds number both the slit- and solid-ribbed geometries perform better than those athigher Reynolds number. Therefore, the usage ofthe slit ribs with the large rib open-area ratios in thelow Reynolds number range is recommended.

Roughness Functions

For the results of the rectangular channel with slitturbulence promoters to be most useful for


designers, general correlations are required forboth the heat transfer and friction over a widerange of rib and flow parameters. According to thefriction similarity law derived in Eq. (4), the mea-sured average friction factor, the rib height-to-channel hydraulic diameter ratio, channel aspectratio, and Reynolds number could be correlatedwith the friction roughness function R(H +). In thiswork, both the geometrically non-similar param-eters, Pi/H and/3, are taken into account, where/3has not been considered in the past. As shown inFig. 6(a), the correlation of the friction roughnessfunction R can be written as

R 3.25[(Pi/H)./101"53(1 -/3) -’81 (7)

The deviation of Eq. (7) is 5% for 95% of the datashown in Fig. 6(a). Note that R in the aboveequation is highly dependent on Pi/H and /3but independent of H+, which implies that the

6

3O

(a)

Pi/H2470 37% 46% 37% 37%i0 10 I0 15 20

,q.()

/R/(Pi/H/i0} =3.2, /=018,13]

(b),q.(,o)

z

lOO ffoo

+

FIGURE 6 Roughness friction and heat transfer parametersvs. roughness friction function.

average friction factor keeps a constant value as theReynolds number is varied. Figure 6(a) furthershows the comparison of the present friction resultswith the previous correlations of the solid-ribbedgeometries (Webb et al., 1971; Hwang and Liou,1995). Basically, these correlations are largely iden-tical with regardless of the effect of the rib open-area ratio. The values of R given in Eq. (7) for theslit-ribbed geometry is significantly higher thanthose of the solid-ribbed geometry, which confirmsthat the lower pressure drop is accompanied by theslit-ribbed geometry. After R is correlated experi-mentally from Eq. (7), the friction factor can bepredicted by the following procedure. As /3 andPi/H are given, the roughness friction function isdetermined by Eq. (7), and subsequently the frictionfactor can be predicted by Eq. (6) for a given ribheight and channel aspect ratio.

Similarly, in accordance with the heat transfersimilarity law derived in Eq. (5), the measuredaverage Nusselt number, the average friction factorf, and R could be correlated with the heat transferroughness function G(H+,Pr). For a Prandtlnumber of 0.703 of the present study, the correla-tion of G shown in Fig. 6(b) can be represented by

G 2.8(H +)"35(1 )-o.81. (8)

The deviation of Eq. (8) is 8% for 95% of the datashown in Fig. 6(b). The equation shows a signifi-cant dependence of G on H+ and/3. After the heattransfer roughness function is correlated experi-mentally from Eq. (8), the average heat transfercoefficient can be predicted by combining Fig. 6(b),and Eqs. (4), (6), and (8) for a given H/De, Pi/H,/3, and Re.The above heat transfer and friction correlations

are valid for the slit-rib geometry with the param-eters ranges 10_<Pi/H<20, 24%</3<46%,10,000 _< Re <_ 50,000, c 90, W/B- 4.0, H/De0.081, and Pr-0.703. These correlations may beuseful in the design of the related devices such asheat exchangers, electric cooling package, andturbine blade cooling channel.


SUMMARY AND CONCLUSIONS

The effect of slit ribs on the fully developed heattransfer and friction factor in a low-aspect-ratiorectangular channel has been examined experimen-tally. The slit ribs are attached staggeringly on theceiling and bottom walls of the channel. As com-pared with the conventional solid-ribbed geometry,the slit-ribbed geometry yields a higher heat transferenhancement and pays a lower pressure droppenalty at the same rib height and spacing. Thefriction factor decreases as the rib open-area ratiois increased, whereas the opposed trend is true forthe average Nusselt number. Performance compar-isons reveal that the slit with larger rib open-arearatios in a higher Reynolds number range couldperform the superior heat transfer enhancementunder the same pumping power condition. Basedon the law of the wall similarity and the applicationof heat-momentum transfer analogy, general fric-tion and heat transfer correlations have beendeveloped by taking into account of rib spacing,open-area ratio, and flow Reynolds number, wherethe geometrically non-similar parameter, fl, hasnot been considered in the past.

NOMENCLATURE

ABDefisHhh

Nu

Nbts

Nu

heat transfer areaheight of the channelhydraulic diameterfriction factor for the ribbed ductfriction factor for the smooth ductrib heightslit heightaverage heat transfer coefficientair conductivityaverage Nusselt number for periodicallyfully developed ribbed duct flow, hDe/kfaverage Nusselt number for the smoothduct at the same flow rate as therib-roughened ductaverage Nusselt number for the smoothduct at the same pumping power as therib-roughened duct.

n slit-slot numberPi rib pitchPr Prandtl numberP pressureQ electrical power supplied by thermofoilQconv net convective heat transfer from the wallQlossReRe*StTTbTwUWW

XYZ

total heat lossReynolds number, U. De/uif/L) 1/3. ReStanton number, Nu/(Re. Pr)temperature of airbulk mean temperature of airlocal wall temperatureaverage through flow velocity without ribsrib (or channel) widthslit-slot widthaxial coordinatestreamwise coordinatespanwise coordinate

Greek Symbols

perforated rib open-area ratioair density

Subscripts

b bulk means smoothw wall

Acknowledgements

Support for this work was provided by the NationalScience Council of the Republic of China undercontract No. NSC 85-2212-E-216-003.

References

Dipprey, D.F. and Sabersky, R.H., 1963, Heat and momentumtransfer in smooth and rough tubes at various Prandtlnumbers, Int. J. Heat Mass Transfer, 6, 329-353.

Han, J.C., Glicksman, L.R. and Rohsenow, W.H., 1978, Aninvestigation of heat transfer and friction for rib-roughnesssurfaces, Int. J. Heat Mass Transfer, 21, 1143-1156.

Han, J.C., 1984, Heat transfer and friction in channels with twoopposite rib-roughened walls, Trans. ASME, Journal ofHeatTransfer, 106, 774-781.

Han, J.C., Park, J.S. and Lei, C.K., 1985, Heat transferenhancement in channels with turbulence promoters, Trans.ASME, J. Heat Transfer, 107, 628-635.


Han, J.C., 1988, Heat transfer and friction characteristics inrectangular channels with rib turbulators, Trans. ASME, J.Heat Transfer, 110, 321-328.

Hwang, J.J. and Liou, T.M., 1995, Effect of permeable ribs onturbulent heat transfer and friction in a rectangular channel,Trans. ASME, J. Turbomachinery, 117, 265-271.

Kline, S.J. and McClintock, F.A., 1953, Describing uncertain-ties on single-sample experiments, Mechanical Engineering,57, 3-8.

Lau, S.C., McMillin, R.D. and Han, J.C., 1991, Turbulent heattransfer and friction in a square channel with discrete ribturbulators, Trans. ASME, J. Turbomachinery, 113, 360-366.

Liou, T.M. and Hwang, J.J., 1992a, Turbulent heat transfer andfriction in periodic fully developed channel flows, Trans.ASME, J. Heat Transfer, 114, 56-64.

Liou, T.M. and Hwang, J.J., 1992b, Developing heat transferand friction in a rectangular ribbed duct with flow separationat inlet, Trans. ASME, J. Heat Transfer, 114, 565-673.

Liou, T.M. and Hwang, J.J., 1993, Effects of ridge shapes onturbulent heat transfer and friction in a rectangular channel,Int. J. Heat Mass Transfer, 36, 931-940.

Siegel, R. and Howell, J.R., 1981, Thermal Radiation HeatTransfer, 2nd ed., McGraw-Hill, New York.

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