Measurements of heat dissipation from miniaturized vertical rectangular fin arrays under dominant natural convection conditions

ORIGINAL

Filino Harahap Æ Herry Lesmana

IKT Arya Sume Dirgayasa

Measurements of heat dissipation from miniaturized vertical rectangularfin arrays under dominant natural convection conditions

Received: 13 April 2005 / Accepted: 26 October 2005 / Published online: 22 December 2005� Springer-Verlag 2005

Abstract Experiments have been conducted to investi-gate the effects of miniaturizing the base platedimensions of vertically based straight rectangular finarrays on the steady-state heat-dissipation perfor-mance under dominant natural convection conditions.The miniaturization process was initiated from asquare-based array of 49·49 mm2 (maximum basearea) and terminated at a square-based array of25·25 mm2 (minimum base area) with rectangular-based arrays of varying intermediate areas in between.Two inter-fin separation distances of 3 and 11 mmwere used. The effect of base plate orientation on theheat-dissipation performance was studied throughcomparison of present results with those of an earlierwork, in which the arrays were miniaturized with thebase horizontally oriented. A correlation for minia-turized vertically based straight rectangular fin arrays,which employed the fin length as the prime geometricparameter, has been presented on the basis of theexperimental conditions of this investigation.

List of symbols

A Total heat transfer surface area of a fin array, m2

Ab Fin array base plate area, m2

Gr ð¼ gbhL3=m2Þ; Grashof number, see (16)Gr¢ ð¼ gbhS3ef�kH=kfmtgðS2=LHÞ0:5=m2Þ; modified

Grashof number, see (5)–(13)h Average heat-dissipation coefficient, Wm�2K�1

H Fin height, see Fig. 3 and Table 1I Measured dc current of main heater, Ak Air thermal conductivity, see (5)–(13) and defini-

tion of Gr¢kfm Fin material thermal conductivity, see (5)–(13)

and definition of Gr¢

L Fin length, see Fig. 3 and Table 1n Number of fins in an array, see Table 1Nu Average Nusselt numberPr Prandtl numberQ Steady state power input to the main heater, WRa Rayleigh numberRa* ð¼ Ra � S=LÞ; modified Rayleigh number, see

(2)–(4)S Inter-fin separation distance, see Fig. 3 and

Table 1t Fin thickness, see Fig. 3 and Table 1V Measured dc voltage across main heater, Vw Uncertainty intervalW Array base-plate width

Greek letters

b Air volumetric expansion coefficient, K�1

m Air kinematic viscosity, m2s�1

h Excess of the average fin base temperature overambient air temperature, K

Subscripts

(b) Non-dimensional parameter of the fin base, see (2)and (4)

(c) Non-dimensional parameter of the channelbetween two adjacent fins, see (2)

(f) Non-dimensional parameter of the fin flat, see (2)and (3)

f Properties evaluated at the film temperature, i.e.the average fin base temperature plus the ambientair temperature divided by two

h Dissipation coefficientL Non-dimensional parameter evaluated with the fin

length as the characteristic lengthl Non-dimensional parameter evaluated with half

of the fin length as the characteristic length

F. Harahap (&) Æ H. Lesmana Æ IKT A. S. DirgayasaMechanical Engineering Department,Thermal Engineering Laboratory, Institute of Technologyof Bandung (ITB), Jl Ganesha 10, 40132 Bandung, IndonesiaE-mail: [email protected]

Heat Mass Transfer (2006) 42: 1025–1036DOI 10.1007/s00231-005-0059-5

r Properties evaluated at the reference temperature,i.e. the ambient air temperature plus 0.62 multi-plied by the difference between the average fin basetemperature and the ambient temperature, see (1)

s Non-dimensional parameter evaluated with theinter-fin separation distance as the characteristiclength

o Properties evaluated at the average fin base tem-perature, see (2)–(4)

1 Introduction

Results of the measurements of steady state heat dissi-pation from a set of ten miniaturized horizontally basedrectangular fin arrays have recently been presented [1].In the present work the heat dissipation performancefrom the same set of miniaturized fin arrays but withboth the fin plates and the base plate vertically orientedis presented. This configuration has wide application asheat sinks for internally generated thermal energy ofmicroelectronic components, both under forced or nat-ural convection dominant modes. Cooling of compactand micro-sized electronic components in devices, suchas personal computers, requires the application of min-iaturized fins arrays. Applications under natural con-vection dominant modes are attractive because they arerelatively trouble free, noise free and economical com-pared to applications under the forced convection mode.

The literature [2–4, 6–18] lacks information on thesteady state heat dissipation performance of miniaturevertically based rectangular fin arrays. Generalized re-sults [4, 7, 13, 14] are only available for large fin arraysbased on experimental calorimetric data.

Pioneering experimental work in this area was carriedout by Starner and McManus [2]. They presented nat-ural convection heat transfer performance data for fourlarge rectangular fin arrays with the base vertically, at45� and horizontally oriented. The range of geometric-parameter variation was limited to inter-fin separationdistances (S) of 6.35 or 7.95 mm, and fin heights (H) of6.35, 12.70, 25.40, or 38.10 mm. Parameters kept

constant were the fin length (L), thickness (t), and thewidth of the base plate (W). Only one base-plate widthto the fin length ratio, W/L=0.5 (W=127 mm andL=254 mm), was employed. They concluded that rela-tive to the horizontally based, the use of the verticallybased orientation is the most favorable system forachieving high heat-transfer rates for arrays of the samegeometric dimensions and power input. Welling andWooldridge [3], who investigated large arrays withcomparable fin heights of H=6.35, 12.70, or 19.05 mm,confirmed the findings of Starner and McManus [2] forthe vertically based fin array orientation.

Chaddock [4] investigated the natural convectionheat transfer from 12 large vertically based fin arrays.Only one value of W/L, W/L=0.84 (W=214.38 mmand L=254 mm) was employed and the fin thickness(t=1.02 mm) was kept constant. The inter-fin separa-tion distance had a range of variation of S=5.84, 12.70,19.56, or 26.42 mm and the fin height, H=25.40, 63.50,or 101.60 mm. Based on a dimensional analysis Chad-dock [4] concluded that the same non-dimensionalparameters derived by Elenbaas [5] could be used togeneralize his experimental data, and proposed the fol-lowing correlation

Nus;r ¼ 0:112 Ras;r �SL

� �0:534

1� e�129=ðRas;r � S=LÞh i0:284

:

ð1Þ

Aihara [6, 7] viewed the heat transfer surface of astraight rectangular fin array as comprised of three maincomponents, i.e. the fin plate, the base plate, and the finplate edges. He conducted investigation of the naturalconvection from each component surface separately andcombined results in an empirical formula. The range ofthe inter-fin separation distance for L=195 mm andH=55 mm (L/H=3.55) kept constant, were S=7.0, 7.2,9.0, 13.1, or 20.0 mm; and for L=195 mm andH=28 mm (L/H=6.96) kept constant, S=7.0, 9.0, or20.0 mm. The array width and fin thickness(t=1.95 mm) were kept constant. Aihara [7] expressedthe average Nusselt number for the entire surface of allthe isothermal open-channels making up a large verti-cally based fin array as

Table 1 Summary of codes, geometric parameters, and N7 tolerances of fin arrays tested

No Array code H (mm) t (mm) S (mm) W (mm) L (mm) W/L Ab·10�4(m2) n

1 S3 W/L 1.00 Ab,max 13.5±0.2 1±0.1 3±0.1 49±0.3 49 1.00 24.01 132 S3 W/L 0.67 13.5 1 3 33±0.3 49 0.67 16.17 93 S3 W/L1.48 13.5 1 3 49 33 1.48 16.17 134 S3 W/L 0.51 13.5 1 3 25±0.2 49 0.51 12.25 75 S3 W/L 1.96 13.5 1 3 49 25 1.96 12.25 136 S3 W/L 1.00 Ab,min 13.5 1 3 25 25 1.00 6.25 77 S11 W/L 1.00 Ab,max 13.5 1 11±0.2 49 49 1.00 24.01 58 S11 W/L 0.51 13.5 1 11 25 49 0.51 12.12 39 S11 W/L 1.96 13.5 1 11 49 25 1.96 12.25 510 S11 W/L 1.00 Ab,min 13.5 1 11 25 25 1.00 6.25 3

The emissivity of the fin surfaces is assumed to be 0.11 at 24�C [20]

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Nus;o;ðcÞ ¼hðcÞ � S

ko¼

Nus;o;ðfÞ þ ðS=2HÞNus;o;ðbÞ1þ ðS=2HÞ ; ð2Þ

where

Nus;o;ðfÞ ¼Ra�s;o24

1� e�ð32;7=Ra�s;oÞ3=4 1�ðS=HÞ=ðRa�s;oÞ

1=4½ �h i

ð3Þ

for S/H<0.2Ras,o* , and

Nus;o;ðbÞ ¼Ra�s;o70

1� e�ð57=Ra�s;oÞ2=3

h i

� 1� e�10ð57=4Ra�s;oÞ1=4

h i1=3ð4Þ

Aihara claimed that (2) universally correlates theexperimental data of Starner and McManus [2] andWelling and Wooldridge [3].

Experimental investigation on the dissipation of heatunder dominant natural convection conditions fromlarge straight rectangular highly polished duralumin finarrays was continued by Leung et al. [8–14]. The effectsof varying all geometric parameters, and the array baseplate orientation were comprehensively studied andcollated for generalization in [13]. The only parameterkept invariant was the array base-plate width at a valueof W=190 mm. The ranges of geometric parametersvaried were: 3 mm £ S £ 76 mm, L=150, 250, 375, or500 mm, H=13, 30, 60, or 90 mm, and t=1, 3, 6, 9, or19 mm. The overall average base plate to the ambienttemperature difference, however, was limited to twovalues only, i.e. h=20 or 40 K. Leung and Probert [12]concluded that the inter-fin separation distance S hasthe most profound effect on the rate of heat dissipationfrom fin arrays and used it as the characteristic lengthfor generalizing their data. This choice corroboratedthe generalization methods applied by Chaddock [4]and Aihara [7]. Based on sets of more than 300experimental data Leung and Probert [13] generalizedtheir data using non-dimensional parameters derivedfrom a similarity analysis similar to that was intro-duced by Harahap and McManus [15]. The resultingcorrelations were presented in [13] by an expression ofthe form

Nus;f ¼ CðGr0s;f PrfÞa ð5Þ

For the vertically-based, vertically-finned arrays,

C = 0.135 and a = 1/2, when ðGr0s;f PrfÞ � 250 ð6ÞC=0.423anda=1/3,when250<ðGr0s;f PrfÞ<106 ð7Þ

For the horizontally based, vertically finned arrays,

C = 0.116 and a = 1/2, whenðGr0s;f PrfÞ � 500 ð8ÞC=0.457 and a= 1/3, when 500< ðGr0s;f PrfÞ< 106

ð9Þ

When (6) and (8) are compared it can be seen that in thelow ðGr0s;f PrfÞ range the vertically based arrays wouldhave higher rates of heat transfer relative to their hori-zontally based counterparts, and from (7) and (9) theopposite holds in the high ðGr0s;f PrfÞ range. This con-clusion, as observed by Harahap and Setio [16], isactually applicable to a variation of W/L from 0.38 to1.27 because even as W was kept constant, Leung et al.[8–13] did vary L. However, this range does not includethe use of square-based fin arrays (W/L=1.00). Fin ar-rays with W/L=1.00 were later included in the study ofthe heat dissipation performance of miniaturized hori-zontally based rectangular fin arrays, which employed aW/L-range from 0.51 to 1.96 [1].

Recognizing that the use of rectangular fins of veryshort length (e.g. (�100 mm) and small fin height (e.g.less than 20 mm) is popular in the natural-convectivecooling of electronic components, Leung and Probert[14] carried out additional experiments using highlypolished duralumin fin arrays with small fin height.Experimental parameters employed were: H=10 or17 mm, S=3, 6, 9, 12, 15, 21, 30, or 45 mm,W=190 mm, L=150 mm (W/L=1.27), t=3 mm andh=20 or 40 K. Results were correlated using the sameform as expressed by (5). For the vertically based largearrays with small fin height,

C = 0.144, a = 1/2, when ðGr0s;f PrfÞ � 250 ð10ÞC = 0.490, a = 1/3, when 250 < ðGr0s;f PrfÞ < 106 ð11Þ

and for the horizontally based large arrays with small finheight,

C = 0.129, a = 1/2, whenðGr0s;f PrfÞ � 500 ð12ÞC = 0.468, a = 1/3, when 500 < ðGr0s;f PrfÞ < 106 ð13Þ

When the C values of (10)–(13) are compared to theircorresponding values in (6)–(9), it is seen that the small-fin-height arrays possess slight higher heat dissipationrates than their corresponding tall-fin-height arraycounterparts. Equations (10)–(13) show that for small-fin-height arrays the vertically based orientation exhibitshigher steady heat dissipation rates than those for thehorizontally based orientation for the entire range ofðGr0s;f PrfÞ. This conclusion corroborated a recentobservation by Guvenc and Yuncu [17] who conductedexperiments with vertical rectangular fin arrays andcompared results with those of the same arrays but withthe base horizontally oriented [18]. The fin array geo-metric parameters were [17]: H=5, 15, or 25 mm,4.50 mm £ S £ 58.75 mm; W/L and t were kept con-stant at 0.40 and 3 mm, respectively.

The ranges of W/L employed in some of the earlierwork for large vertical rectangular fin arrays [2, 4, 11, 14,17] have been highlighted, because the variation of thisparameter was found to have a profound effect on therate of heat dissipation performance of miniaturizedhorizontally based rectangular fin arrays [1]. Therefore it

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is expected that W/L will have the same effect on theperformance of the vertically based case.

2 Experimental set up and procedure

The experimental set up and method used in the presentwork is depicted in cross-sectional view in Fig. 1. Thesame experimental set up, described in detail in [1] forthe horizontally based configuration, was employed, butnow supported by a wooden box in the front and a labjack in the rear to realize the vertically based verticallyfinned array configuration. The wooden box, providingfront support to the set up, was open in the rear andprovided with a window in the front. This window,positioned at the center of the square (400·400 mm)front wall of the support wooden box, had dimensionsthat would allow a snug fit of the horizontally positionedouter-wooden box housing the rock wool insulationblanket wrapped around a smaller-inner box. This innerbox functioned as the horizontal support for theassembly made up of the fin array, the main heater, andthe guard heater. The lab jack provided the rear supportto ensure stable and accurate horizontal positioning ofthe outer-wooden box.

The miniaturized fin arrays tested were made fromAlcoa aluminum alloy 7050 (k=157.488 W/m K) andmanufactured by wire cutting to N7 tolerances of (±0.1,

±0.2, and ±0.3 mm for array dimensions in the rangesfrom 0 to 6, 6 to 30, and 30 to 120 mm, respectively (seeTable 1). As schematically appears in Fig. 2, pressureinstalled using screw connections and lubricated withheat transfer conductive silicon at the rear of the array’svertical base plate was the 6 mm thick front main-heateraluminum plate, which in turn was backed by the mainheater made of 0.1 mm diameter Ni-chrome wire woundaround a mica plate and sandwiched between to micasheets. These were backed by the rear main-heater alu-minum plate, also 6 mm thick, which was held togetherto the front main-heater plate with screw connections. A6 mm thick asbestos slab was glued to the rear surface ofthe rear main-heater plate to reduce heat loss. Toachieve an almost zero heat loss system, a guard heater,which was identical to the main heater, and also heldtogether with screw connections, was then glued to therear surface of the asbestos insulating slab. The edges onall sides of the main heater front plate was tapered at 45�to provide mounting surface for the small inner box,which housed and provide horizontal support to theassembly made up of the main-heater, the asbestos-slaband the guard-heater. The inner box also providedsupport for the thermocouple, main-heater, and guard-heater power supply leads. Finally, a 15 mm thick rockwool blanket was wrapped around the inner box andenclosed by the outer-wooden box to prevent heat lossto the sides of the set up. The front surface of the outer-wooden box and the front surface of the support woo-den box were made flush against and along the frontsurface of the fin-array base plate (see Fig. 2).

The main and guard heater circuits were independentand supplied with direct current by a two channel DCPower Supply Unit. The accuracy of current and voltageadjustments to each circuit was 0.005 A and 0.05 V,

Fig. 1 Cross-sectional view of the experimental set up for thevertically based vertically finned configuration. a front-supportwooden box, b lab jack, c outer-wooden box, d rock woolinsulation, e small inner-wooden box, f fin array, g main heater,h guard heater, and i table board

Fig. 2 Schematic of the experimental system and method of thepresent investigation. a integral fin array, b main-heater front plate,c main-heater, d main-heater rear plate, e asbestos slab, f guard-heater front plate, g guard-heater, h guard-heater rear plate, i smallinner wooden box, j rock wool insulation, k outer wooden box, l dcpower supply, m ammeter, n voltmeter, o thermocouple wires,p thermocouple port, and q computer and monitor

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respectively. Power to the main heater was determinedby measuring the current (I), and the voltage (V) acrossthe heater with multimeters possessing accuracies of±0.0005 A and ±0.005 V, respectively. The currentand voltage measured overall during the experimentsranged, respectively, from 0.171 to 0.540 A and from2.56 to 12.56 V.

The fin array average temperature was approximatedby the average of five temperature measurements at theback of the base plate. Five thermocouple beads wereinstalled distributively in grooves at the rear surface ofthe fin-array base plate, one at the center, two at theedges of the longitudinal axis passing through the center,one at the edge of the transverse axis passing through thecenter, and one at a corner of the rectangular shapedsurface. Measurements at these five points showed amaximum difference of less than 1�C during steady state,indicating that isothermal condition was achieved at thefin array base plate. This difference is relatively small(less than 5%) compared to the amount by which theoverall average base temperature exceeds that at theambient (h), which ranged from 20 to 50�C during theexperiments. The temperature excess range is limited bythe optimal operating temperature of microelectroniccomponents.

Teflon coated copper-constantan thermocouple wirewith a 0.254 mm diameter was used in conjunction witha data acquisition program to monitor, measure, andrecord temperatures.

The lay-out of thermocouple beads in grooves at therear surface of the rear main-heater plate and the frontsurface of the front guard-heater plate was exactly thesame as that for the fin-array base plate. By adjustingthe power input to the guard heater, the average tem-perature of the font surface of the front guard-heaterplate was maintained equal to the average temperatureof the rear surface of rear main-heater plate duringsteady-state operation of the system. This arrangementand the good insulation against heat loss sideways en-sure that all the power input to the main-heater wasdissipated through the fin array to the ambient sur-rounding. One thermocouple bead was used to measurethe ambient temperature following standard practice(ASHRAE Standard 41.1–1986).

The 16 thermocouple leads were connected to a16�connection terminal port to provide input throughan interface connection to the data acquisition card in-stalled in the personal computer employed in thisinvestigation.

The power input to the main and guard heaters werecontrolled by adjusting the current and voltage acrossthe heaters. Steady-state condition was consideredachieved when all temperature readings remained un-changed with time within 0.1�C of respective values asmonitored through the data acquisition program. Thiscondition was achievable within 1 h for each value of themain heater power input. During this condition, theequality of the average temperature of the rear surface ofthe rear main-heater plate and that of the front surface

of front guard-heater plate was ensured and maintained.Recording of the true rms readings of the two multim-eters were only made after steady state conditions weremonitored to prevail for at least 30 min. The steady-state power input to the main-heater or the rate of heatdissipation under dominant natural convection condi-tions through the fin array to the ambient (Q), wasdetermined as Q=IÆV. The completion of measurementprocedure for each steady-state power input to the sys-tem required around 2 h on the average. The averageheat-dissipation coefficient, under dominant naturalconvection conditions, was calculated as

h = I � V /A � h ð14Þ

where A is the total heat transfer area of a fin array

A ¼ W � L þ 2 � n � H � t þ 2 � n � H � L ð15Þ

A total of ten miniature straight rectangular fin ar-rays was tested in the present work. Figure 3 and Ta-ble 1 summarize the geometric parameters of fin arraystested, and the code assigned to each of them. Table 1also includes the N7 tolerances of array dimensions asappropriate.

The fin height (H) and thickness (t) were held con-stant and were chosen to have typical values encoun-tered in applications for cooling of motherboards incomputers, namely, H=13.5 mm and t=1 mm. Twovalues of the inter-fin separation distance were employed(3 or 11 mm). S=3 mm was chosen because this value istypical in electronic cooling, and S=11 mm corresponds

Fig. 3 Miniaturized vertically based straight rectangular integralfin array tested in the present work: the geometric parameters aredefined in the list of symbols, see also Table 1

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closely to the optimal inter-fin separation distance sug-gested by Leung and Probert [12–14].

3 Uncertainty analysis

The second-power method [19] was applied to determinethe uncertainty interval of measured dissipation coeffi-cients. As can be expected, the maximum uncertainty ofthe set of dissipation coefficients of a fin array occurredwhen the electrical heating input is lowest. The variablesand the maximum uncertainty intervals of the dissipa-tion coefficient of all arrays tested are summarized inTable 2.

Variables affecting the uncertainty interval of the heatdissipation coefficient are I, V, W, L,H, t, and h [see (14)and (15)]. The uncertainty intervals of I and V wereestimated to be the same as the accuracies of respectivemeasuring instruments, respectively, ±0.0005 A and±0.005 V. The tolerances of manufacture of each array(see Table 1) were used as estimates for the uncertaintyintervals of W, L, H, and t. For all arrays H and t werekept constant, thus the uncertainty intervals areH=13.5±0.2 mm and t=1±0.1 mm. The uncertaintyinterval of the temperature excess was assumed to be±0.5 K.

From Table 2 it is seen that the largest maximumuncertainty of the dissipation coefficient is associatedwith array S3W/L 1.0 Ab,max at 3.25%. The main sourceof error was due to the uncertainty interval of themeasurement of the temperature excess, which propa-gates to over 95% of the uncertainty of the dissipationcoefficient.

4 Results and discussion

The variation of the steady-state heat-dissipation rateper unit base area (Q/WL) with the temperature excess(h) for different W/L ratios is presented in Figs. 4 and 5,respectively, for the set of arrays with S=3 mm andS=11 mm (near optimal). In both figures the heat-dis-sipation rate per unit base area is seen to increase withthe temperature excess and reduction of the base platearea, i.e. with miniaturization of the array. The two

figures also show that the dissipation rate of arrays withvalues of W/L>1.0 is consistently higher relative to thatof their counterparts of the same base area but withvalues of W/L<1.0. The same features were observed in[1] for the case of the horizontally based arrays.

A cross-plot of Q/WL versus W/L was derived fromFigs. 4 and 5, for h=20 and 40 K, is presented in Figs. 6and 7. Figure 6 shows that the heat-dissipation rate perunit base area of arrays with S=3 mm is consistentlylower than that for S=11 mm (nearly optimal), for allvalues of W/L if h=20 K, and if h=40 K only for W/Lvalues larger than 0.67. For the range 0.51<W/L<0.67if h=40 K, a trend reversal occurs; the steady-state heat-dissipation rate of arrays with S=3 mm is seen toovershoot the values for arrays with the nearly optimalinter-fin separation distance (S=11 mm). For the case ofthe horizontally based arrays this trend reversal wasobserved in [1] to occur for both values of h (20 and40 K) but at the high end of W/L range (1.72<W/L<1.96). Figure 6 also reveals that, for both h=20and 40 K, miniaturization of the base of arrayswith S=3 mm and S=11 mm would increase the heat-dissipation rate when the array possessing W/L=1 andAb,max was employed as the basis of comparison. Theincrease of the heat dissipation rate with miniaturizationof the base plate was found to be more pronounced forhorizontally based fin arrays in [1] than that is observedfor the present case. Alternatively, if the array with W/L=1 and Ab,min was employed as the basis of compari-son, as presented in Fig. 7, it is seen that, both for S=3and 11 mm, that array exhibits the greatest heat-dissi-pation rates be it at h=20 K or at h=40 K. In Fig. 7 theexistence of two steady-state heat dissipation rate min-ima could be inferred from available data for S=3 mm.For arrays with S=11 mm the existence of similar min-ima could not be inferred from available data because ofthe absence of data for base W/L values in mid range of0.5<W/L<1.0 and 1.0<W/L<2.0. Figure 7 shows thatthe heat-dissipation rates of the arrays with S=3 mm areconsistently below those of the arrays with S=11 mm forthe whole range of W/L at both h=20 K and h=40 K,except at one point where W/L=0.51 and h=40 K,where the opposite was measured. For the horizontallybased arrays of [1], two points were found where the heatdissipation rate of arrays with S=3 were measured

Table 2 Maximum uncertainty intervals of the heat dissipation coefficients of arrays tested

No Array code I (A) V (V) W (mm) L (mm) n h (K) h (W/m2 K) wh/h (%)

1 S3 W/L 1.00 Ab,max 0.231 6.04 49±0.3 49 13 15.44 4.53 3.252 S3 W/L 0.67 0.171 7.07 33±0.3 49 9 17.70 4.96 2.833 S3 W/L1.48 0.192 7.73 49 33 13 20.04 5.47 2.524 S3 W/L 0.51 0.318 3.87 25±0.2 49 7 20.56 5.61 2.445 S3 W/L 1.96 0.277 4.86 49 25 13 21.43 6.07 2.366 S3 W/L 1.00 Ab,min 0.285 2.56 25 25 7 19.18 6.87 2.647 S11 W/L 1.00 Ab,max 0.204 9.46 49 49 5 19.62 10.75 2.568 S11 W/L 0.51 0.298 3.89 25 49 3 19.97 11.01 2.519 S11 W/L 1.96 0.299 4.74 49 25 5 21.61 13.85 2.3410 S11 W/L 1.00 Ab,min 0.298 2.94 25 25 3 20.72 15.47 2.44

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higher than that for arrays with S=11 mm, i.e. at W/L=1.0 (square-base, Ab,min) and at W/L=1.96, for bothh=20 K and h=40 K. Thus, for the vertically basedorientation, Fig. 7 suggests that a considerable increaseof heat-dissipation rate per unit base area could beachieved if arrays with square bases (W/L=1.0) areminiaturized, and that the arrays with S=11 mm (nearoptimal) would always lead. This conclusion could not bearrived at from data of the experiments of Leung et al.[8–14] because arrays withW/L=1were excluded in their

program. The comparisons of the features revealed fromFigs. 6 and 7 with those observed for the arrays with thehorizontally based orientation of [1], suggest that theeffect of the parameter W/L on the heat-dissipation rateperformance as array bases were miniaturized is similarfor both cases, but relatively less pronounced for thevertically based case of the present study.

In Fig. 8 the average heat-dissipation coefficient isseen to increase with the temperature excess and thereduction of the array base area, i.e. miniaturization of

Fig. 4 Variation of the steady-state heat-dissipation rate perunit base plate area with thetemperature excess for therange of W/L from 0.51 to 1.96and for the set of six arrays withS=3 mm

Fig. 5 Variation of the steady-state heat-dissipation rate perunit base plate area with thetemperature excess for therange of W/L from 0.51 to 1.96and for the set of four arrayswith S=11 mm

Fig. 6 Variation of the steady-state heat-dissipation rate perunit base area with theparameter W/L at h=20 and40 K for arrays with S=3 and11 mm employing Q/WL at W/L=1.0 Ab,max as the reference

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the arrays. For arrays of the same base area the averageheat-dissipation coefficients of arrays with W/L>1.0 areconsistently higher than that for arrays with W/L<1.0,for both S=3 mm and S=11 mm. However, the mag-nitude of this trend is seen in Fig. 8 to be only slight forarrays with S=3 mm. For the same base area, theaverage heat-dissipation coefficient for the arrays withthe near optimal inter-fin separation distance(S=11 mm) is seen in Fig. 8 to be consistently higherthan those for arrays with S=3 mm. The same trendwas also observed in [1] for the horizontally based case.For arrays with square base plates, Fig. 8 shows thathalving the size of the sides or reduction of the base areato almost a quarter, is seen to result, on the average inthe range 20 K<h<40 K, in an increase of the averageheat-dissipation coefficient by a factor of 1.55 for arrayswith S=3 mm, and by a factor of 1.43 for arrays withS=11 mm. The corresponding increases for the hori-zontally based case of [1] were 1.84 and 1.46, respec-tively. This comparison shows that although having thesame trend, the effect of base area reduction, on theincrease of the average heat-dissipation coefficient is lessfor square-based vertical fin arrays when compared tothat for the horizontally based arrays. The average heatdissipation coefficients presented in Fig. 8 account forthe contributions of both the natural convection and

radiation modes of heat transfer. These coefficients areof direct practical interest because in practical applica-tions, the objective is to maximize heat dissipation ratesrather than to be too concerned on the mode of heattransfer. Assuming a total fin surface emmisivity of 0.11[20], estimates of the radiation contribution to the totalheat dissipation rate at h around 50 K, were made andfound to be highest for the array S3 W/L 1.00 Ab,max at4.78% and lowest for S11 W/L 1.00 Ab,min at 2.33%.Because the radiation component is less than 5% it canbe assumed that the contribution of the natural con-vection mode is the dominant one for the low absolutevalues of the average array-base temperatures involvedin the experiments.

In Fig. 9 the average heat dissipation coefficients ofthe vertically based fin arrays of the present study arecompared to those of their corresponding counterpartsof the horizontally based case presented in [1]. Thecomparison is on an array per array basis and as afunction of the temperature excess. The lowest sets ofdata points for array S3 W/L 1.00 Ab,max show a dis-cernable trend, even as differences are perhaps within theuncertainty interval limits of the data, that the coeffi-cients of the vertically based orientation (red triangles)are higher than those for the horizontally based case(black triangles) in the interval 20 K<h<50 K. This

Fig. 7 Variation of the steady-state heat-dissipation rate perunit base area with theparameter W/L at h=20 and40 K for arrays with S=3 and11 mm employing Q/WL at W/L=1.0 Ab,min as the reference

Fig. 8 Variation of the averageheat-dissipation coefficient withthe temperature excess as the finarrays were miniaturized withthe base vertically orientedfrom W/L 1.0 Ab,max toW/L=1.0 Ab,min for S=3 and11 mm

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trend is seen in Fig. 9 to apply also for two arrayspossessing Ab=16.17·10�4 m2, namely S3 W/L 1.48and S3 W/L 0.67. These observations are in agreementwith the trend on the effect of base orientation of largefin arrays [13] as have been inferred earlier throughcomparison of (6) and (8), and for large arrays withsmall fin height [14] by comparing (10) and (12), both for(Gr¢s,f Prf)<250 and the findings of [17], also for largearrays. In contrast, the three arrays with S=3 mmhaving their base areas smaller than Ab=16.17·10�4 m2, namely S3 W/L 0.51, S3 W/L 1.96, and S3W/L 1.0 Ab,min, can be seen in Fig. 9 to exhibit lowervertically based array heat-dissipation coefficients thanthose for their respective corresponding arrays with thebase horizontally oriented. The above observations forarrays with S=3 mm suggest that as the base area of thearrays were miniaturized then at some base area valuebetween Ab=16.17·10�4 and Ab=12.25·10�4 m2, irre-spective of the value of W/L in the range 0.51 £ W/L £ 1.96, a reversal of the effect of the base plate ori-entation has occurred. For arrays having S=11 mm(near optimal), however, a somewhat different patternemerged from the cluster of experimental data pointspresented at the top part of Fig. 9. Three arrays havingthe value of W/L‡1.0, i.e. S11 W/L 1.0 Ab,max, S11 W/L1.96, and S11 W/L 1.0 Ab,min, exhibit higher verticallybased array dissipation coefficients than those for theirrespective corresponding arrays with the base horizon-tally oriented. The trend of the effect of base plate ori-entation of these arrays run counter to the trend forlarge fin arrays [13] as reflected from the comparison of(6) and (8) for (Gr¢s,f Pr)>500, but is in-line with thetrend for large arrays with small fin height [14] as couldbe inferred from (10) and (12) for (Gr¢s,f Pr)>500 andalso in agreement with the conclusion of [17], also forlarge fin arrays. One array, S3 W/L 0.51, whilst havingthe same base plate as that for S11 W/L 1.96, is seen toexhibit higher horizontally based array dissipationcoefficients than those for its corresponding array butwith the base vertically oriented. These observationssuggest that the divide for reversal of the effect of the

base orientation for the arrays with the inter-fin sepa-ration distance near optimal (S=11 mm) is a value ofW/L somewhere within the range 0.51<W/L<1.0 irre-spective of the value of base area in the range6.25·10�Æ4 m2 £ Ab £ 24.01·10�4 m2. The patterns justidentified could be viewed as further elaboration of theconclusion stated in [13]: ‘‘when there is a choice forvertical fins to protrude from a base whose surface iseither horizontal or vertical, the effects of other geo-metric parameters of the fin array (e.g. length) will dic-tate which orientation will lead to the highest heat-transfer rates being achieved’’.

5 Generalization of experimental observations

In Figs. 10–12 the experimental data of the present workwas generalized using the methods applied by Chaddock[4], Aihara [7], and Leung and Probert [14], respectively,and compared with respective correlation equations.Figure 10 shows that the correlation equation proposedby Chaddock [4] for large fin arrays (1), could not cor-relate the generalized data of miniaturized fin arrays ofthe present work. Similarly, it is seen in Fig. 11 thatgeneralization of the present data with the non-dimen-sional parameters proposed by Aihara [7], did not cor-relate with (2), which was plotted using its componentequations, (3) and (4). In Fig. 12 the generalization ofthe present data with the non-dimensional parametersproposed by Leung and Probert [14] for large arrayswith small fin height is seen to show adequate results asfar as the correlation of the set of data for the minia-turized arrays with S=11 mm (near optimal) by (11) isconcerned. However, the set of data for arrays withS=3 mm failed to be adequately correlated by (10). Inthis connection, it has to be recalled that the experi-mental set up and procedure applied in the present worksimulated those of Leung et al. [8–11]. In sum, if athermal design engineer were to predict the average heatdissipation coefficient of miniaturized fin arrays, theproposed correlations applicable for large arrays, both

Fig. 9 Comparison of thevariation of the heat dissipationcoefficients with thetemperature excess for thevertically based configuration(red symbols) with theircorresponding counterpartsassociated with the horizontallybased configuration (samesymbols, black) from Ref. [1] onan array per array basis

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for the vertically based [4, 7, 13, 14] and the horizontallybased [13, 14, 16] fin array configurations, should not beapplied directly to calculate the average Nusselt number.

Relying on the non-dimensional parameters derivedin [15], which used the fin length as the characteristicdimension instead of the inter-fin separation distance,and recognizing that the flow pattern of the verticallybased fin array configuration, as visualized in [6], differsfrom that for the horizontally based configuration, asvisualized in [18], the data for miniaturized verticalrectangular fin arrays of the present study was general-ized and presented in Fig. 13. The resulting correlationequation is

NuL;f ¼ 3:350 ðRaL;fÞ0:153ðL=W Þ0:121ðS=HÞ0:605 ð16Þ

for the interval 20·104<RaL,f<5·105. The dependenceof NuL;f on (L/W)0.121 for the vertically based array isweaker than that for the horizontally based array case[1], for which Nul;f is proportional to (L/W)0.620.

6 Conclusions

The results of measurements of steady-state heat dissi-pation from a set of ten miniaturized horizontal

rectangular fin arrays have recently been reported [1].The effect of miniaturizing the array base on the rate ofheat dissipation per unit base area and the average heat-dissipation coefficient was studied and generalizedexperimental data has been compared to those for largefin arrays. In the present work, the same set of minia-turized fin arrays is investigated along the same line asthat was applied by Harahap et al. [1], but with the baseplate vertically oriented. This approach allowed delin-eation of the effect of the base plate orientation on theheat-dissipation performance of the miniaturized arrays.

Miniaturizing vertical rectangular fin arrays byreducing the base area had the effect of increasing, boththe steady-state heat-dissipation rate per unit base areaand the average heat-dissipation coefficient. Consider-able increase of the heat-dissipation rates per unit basearea could be achieved if arrays with square base platesare miniaturized, irrespective of the value of the inter-finseparation distance. These conclusions are consistentwith those observed by Harahap et al. [1] when the arrayminiaturization was effected with the base plate hori-zontally oriented.

As the arrays were miniaturized with the base verti-cally oriented, the average heat-dissipation coefficient ofarrays with near optimal inter-fin separation distance(S=11 mm) was consistently higher than those for

Fig. 11 Comparison ofgeneralized data of the presentwork with the correlationequation proposed by Aihara[7] for large vertical rectangularfin arrays, (2), which wasplotted from its components (3)and (4), with propertiesevaluated at the filmtemperature. Note that S/H=0.22 applies for six arrayswith S=3 mm and S/H=0.82applies for four arrays withS=11 mm

Fig. 10 Comparison ofgeneralized data of the presentwork with the correlationequation proposed byChaddock [4] for large verticalrectangular fin arrays, (1), withproperties evaluated at the filmtemperature

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S=3 mm, if the base area is the same. For arrays withsquare base plates, reduction of the base area to almost aquarter resulted in a higher increase of the average heat-dissipation coefficient for arrays with S=3 mm than theincrease for arrays with the near optimal inter-fin sepa-ration distance (S=11 mm). While this trend is the sameas that had been observed by Harahap et al. [1] for ar-rays with the base horizontally oriented, the verticallybased case of this study exhibited less difference of theincrease of the dissipation coefficients.

For arrays with non-square base plates of the samearea, orientation of the length of the fins parallel to theshorter side of the base plate (W/L>1.0) consistentlyresulted in higher average heat-dissipation coefficientsthan if the length of fins were oriented parallel to thelonger side of the base plate (W/L<1.0). This trend isalso the same as that had been observed by Harahapet al. [1] for arrays with the base horizontally oriented.However, the gain in the increase of the heat-dissipationcoefficient of arrays having the same non-square baseareas with W/L>1.0 over that with W/L<1.0 is mostpronounced if the inter-fin separation is near optimal(S=11 mm) and the base is vertically oriented.

Based on the comparison of the average heat-dissi-pation coefficients of the vertical rectangular fin arraysof the present study with those for the correspondingvalues for the horizontally based case of Harahap et al.[1] on an array per array basis, the experimental data

features the emergence of two patterns pertaining to theeffect of the base plate orientation on the heat-dissipa-tion performance. For arrays with S=3 mm, irrespec-tive of the value of W/L within its range covered in thisstudy and in [1], a value of the base area is the divide forreversal of the effect of base orientation from the verti-cally based fin arrays exhibiting better performance thanthe horizontally based to the vertically based fin arraysexhibiting lower performance than the horizontallybased. For arrays with S=11 mm (near optimal), irre-spective of the value of the base area, it is a value of theparameter W/L that determines that divide.

Relevant correlations proposed for large fin arrayswere found not applicable for generalization of experi-mental data on heat dissipation from miniaturized ver-tical rectangular fin arrays of the present study. Onecorrelation equation, which relied on the generalizationapproach of [15], and the flow pattern associated for thevertically based vertically finned configuration visualizedby Aihara [6], has been presented on the basis of theexperimental conditions of the present work.

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Fig. 12 Comparison ofgeneralized data of the presentwork with the correlationequations proposed by Leungand Probert [14] for largevertical rectangular arrays withsmall-fin-height, (10) and (11);properties were evaluated at thefilm temperature

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