Energy Conversion and Management 51 (2010) 2449–2456

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Optimum geometric arrangement of vertical rectangular fin arraysin natural convection

Xiaohui Zhang *, Dawei LiuSchool of Energy, Soochow University, Suzhou, Jiangsu, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 August 2009Accepted 5 May 2010Available online 12 June 2010

Keywords:Fin arraysSpacingNatural convectionConstructal theory

0196-8904/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.enconman.2010.05.009

* Corresponding author. Tel.: +86 512 61080967; faE-mail address: xhzhang@suda.edu.cn (X. Zhang).

The optimal spacing between isothermal laminar natural convection plates cooled by air for maximalheat transfer has been determined analytically and numerically. It is found that the optimal plate’s allo-cation spacing is different from the conventional way, where the boundary layers of the plates mergeearly, it is the distinguishing feature of the outlet velocity that causes an enhancement of heat transfer.Further comparison results verify that our expression and numerical results and Rohsenow expressionare very consistent. In the general condition, an optimized spacing expression of the maximum heattransfer density for a given volume is derived. It may be expected that the heat transfer enhancementcan be accomplished by inserting multi-scale plates into the boundary layer according to the constructaltheory.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

One of the most important aspects of electronic equipmentmanagement has always been recognized to be the dissipation ofthe heat produced in the electronic components, which is neededto avoid overheating of the apparatus.

During the years the electronic equipment cooling problem hasbecome even more crucial, as a consequence of the continuousevolution reached by the electronics industry year by year, creatingapparatus even more compact in their dimension. Consequently,the quantity of heat to be dispersed is high per unit area. Withthe above in mind it is clear the importance to be given to the opti-mization of cooling system of electronic equipment.

The heat transfer to the external ambient atmosphere by theelectronic apparatus can be obtained mainly by using the mecha-nisms of the heat transfer by forced convection, natural convectionand by radiation heat transfer.

Finned surfaces in natural convection provide a significantimprovement in efficiency (increasing the heat flux) compared toun-finned ones. Both the fin thickness and the spacing betweenfins are determining factors that cannot be neglected in optimizingthe total fin array heat transfer, as they influence both the effi-ciency of individual fins and the number of fins that can be accom-modated in a given area. A number of studies have attemptedfinding the optimal configuration for a vertical fin array.

Earlier studies on searching for the optimal spacing betweenparallel plates under natural convection cooling can be found in

ll rights reserved.

x: +86 512 65111907.

Refs. [1–3]. In these works the model is restricted to plates of rect-angular cross-section, Vollaro [1] developed the optimum values ofthe fins spacing as a function of parameters which feature in theconfiguration, in Ref. [2] the case studied is that of fins whose tem-perature varies with distance from the base, it is found that theoptimum fin width and spacing depend upon the fin thicknessand its thermal conductivity. The optimal spacing between effi-cient isothermal fins is found in Ref. [3].

Recently, Bejan and his coworkers have developed in a series ofpapers on the ‘constructal’ theory (see e.g. Refs. [4–8]) to solve thisproblem. Bejan [5,8,9] analyzed it using scale analysis and theintersection of asymptotes and deduced an approximate result.According to this method, the flow configuration is free to morphin the pursuit of maximal global performance under global con-straints. The resulting optimal (constructal) configuration is de-duced, not assumed. It is the winner of the competition in whichall the eligible configurations are simulated and compared. Oneclass of constructal-design configurations are the optimal internalspacing determined for heat-generating volumes cooled withchannels, staggered plates, and pin fins, based on these, an opti-mized multi-scale flow structure that achieves even higher levelsof heat transfer rate density, under the same constraints as the sin-gle-scale structures optimized in Bejan work.

Further examination to the optimal spacing results under samecondition between Ref. [3] and Refs. [5,8,9], 60% deviation occurs.Seldom study gives a thorough understanding.

In this paper, this problem is studied in line with the generationof shape and structure for achieving maximal performance throughtheoretically analysis and numerical simulation, without the sim-plification, and a new clear result for solving the optimal spacing

Nomenclature

D spacingD dimensionless spacingg gravitational accelerationGr Grashof numberk location thermal conductivityL length of plateNu Nusselt numberp pressureRa Raleigh numberP non-dimensional pressurePr Prandtl numberQ heat transfer rateT temperatureu, v velocity componentsU, V dimensionless velocityW volume widthx, y Cartesian coordinatesX, Y dimensionless coordinates

Greek symbolsa thermal diffusivityb coefficient of thermal expansiond velocity boundary-layer thicknessdT thermal boundary-layer thicknessq density of the fluidm kinematic viscosity of the fluidH dimensionless temperatureSubscripts0,1 generation of multi-scale architecturef fictitious velocitymax maximalopt optimizationR reference1 reference value at great distance from a body

2450 X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456

is obtained. By comparing the exact result with the approximateresult deduced by Bejan [10], a deviation is found, this deviationis due to the assumption that the downstream boundary layersdoes not affect. Theoretical solution and numerical correlation con-sistent with exact solution obtained from Ref. [3].

2. Problem description

Let us consider the problem of finding the ‘best’ shape of a part(unit-cell) of a long three dimensional thermal fin (see Fig. 1). Thebase of the fins attached to a system has constant temperature. Theface opposite the base is exposed to an ambient temperature. Thetop and bottom faces will have small surface areas compared to theconvective boundary and therefore we neglect the heat transferthrough these surfaces. We can thus reduce the problem to atwo-dimensional setup for the cross-section (see Fig. 2). A fin made

Fig. 1. Axonometric projection view.

of pure aluminum, highly conductive, axial heat conduction in thefins can be neglected.

The goal of this analysis to search for the optimal configurationfor finned plate (with rectangular and vertical fins) to be cooled asshown in Fig. 1. Consider the two parallel plates system shown inFig. 2, where a fluid (air) is contained between two vertical platedseparated by the distance D, the vertical isothermal plates generateheat and the stack is cooled by natural convection with singlephase fluid temperature.

3. Analytical solution

We choose the x-coordinate along the plate and the y-coordi-nate perpendicular to the plate. The heated surface consisted of0.015 m height plates in the span wise direction maintained at auniform temperature 348 K. The ambient temperature is 298 Kand physical properties are evaluated at the mean temperature.

For Pr = 1 fluid, the thickness in the velocity and thermal fieldsof the boundary layer is d = dT.

The boundary expression for the boundary-layer thickness is[11]:

d ¼ 3:93 � x � 0:952þ Pr

Grx � Pr2

� �1=4

ð1Þ

Fig. 2. Configuration of problem and the coordinate system.

X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456 2451

The velocity profile [11] is derived as:

u ¼ Vf

dy 1� y

d

� �2ð2Þ

where Vf is a fictitious velocity that is function of the plate height.So we obtain from Eq. (1), for x = L = 15 cm

d ¼ 1:088 cm

According to Bejan [8], the optimal spacing D is such that thethermal boundary layers merge at the top of the structure.

D ¼ 2d ¼ 2:176 cm

Substituting for x = 15 cm from Eq. (1) into Eq. (2) gives outletvelocity function:

u ¼ 149;0446y3 � 32;434y2 þ 176:5y ð3Þ

The boundary layers of the two plates upstream region mergetogether as the spacing of two plates decrease, which can be gottenby the superposition of velocity function, u, for the two plates. Aplot of velocity superposition as a function of plate-to-plate spac-ing is given in Fig. 3. Notice that the two peaks of the curve(Fig. 3a) evolves as one peak of the curve (Fig. 3d) during the de-crease of the spacing. There exists a critical spacing where the pro-file of the superposition of velocity function appears flat in themiddle of the space for the two boundary layers, here the criticalspacing is 0.00145 m, as shown in Fig. 3c.

Now let us manipulate Eq. (2) to determine the critical spacing,the maximization condition is: du

dy ¼ 0, which gives the two loca-tions y1, y2, so

Dopt ¼ y1 þ y2 ¼ �4V=d2

3V=d3 ¼43

d ð4Þ

When plate-to-plate spacing is twice bigger than boundarylayer, i.e. D > 2d, the two sides of plates boundary cannot merge,therefore the two plates does not in any way interfere with theboundary layer flow each other; with decreasing spacing,43 d < D < 2d, the location where the peak of velocity inside one sideboundary is unaffected by another side’s velocity profile, the twovelocity peaks profile have not been changed, therefore the heattransfer is unaffected by another plate. If the plate-to-plate spacingis less than the value given by Eq. (4), D < 4

3 d, the two sides bound-

Fig. 3. Superposition of velocity fun

ary layers merge and become one peak buoyancy flow, the velocityprofile changes radically. It is expected that heat transfer charac-teristics of the natural convection boundary lay drastically changeby controlling fluid motions in the layer. So, the spacing of D ¼ 4

3 dis the critically spacing for enhanced heat transfer.

Importing Eq. (2) into Eq. (4) leads to:

Dopt ¼43

dðLÞ � 7:11 � L � ðGrÞ�14 ð5Þ

The dimensionless form is:

Dopt ¼Dopt

L¼ 7:11ðGrÞ�

14 ð6Þ

According to scale analysis, the thermal boundary layer of aplate with laminar natural convection flow and Pr = 1 has a thick-ness of order:

dT / L � Gr�1=4 ð7Þ

By setting dT / D in Eq. (7),

DL/ Gr�1=4 ð8Þ

which anticipates very well the theoretical correlation: Eq. (6).Note that the isothermals are more closely near the plate sur-

face, indicating a higher temperature gradient in that region. Whenthe velocity boundary layers have not merged as one peak velocityprofile, the two peaks of velocity occupy the region of high temper-ature gradient, while the lower temperature gradient area occursin the minimum velocity area. The temperature fields and velocityfields coordinate each other, the heat transfer is enhanced. Whenthe space decreases, one peak velocity superposition result ap-pears, the peak of velocity occurs in lower temperature gradient re-gion, poor contribution for heat transfer, the higher temperaturegradient region are not effective to heat transfer derived fromthe buoyant flow, these can be verified furthermore in the follow-ing part.

4. Numerical verification

In the second phase of this study, we optimize the scale struc-ture numerically. The physical configuration and the computational

ction when both sides nearing.

Fig. 4. (a) Interferometer photo of heating vertically plate in natural convection[11] and (b) computation.

Fig. 5. Relationship between heat transfer and fin spacing for L = 0.15 m.

2452 X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456

domain are shown in Fig. 2. In this study, the investigations arecarried out through the variation of plates distance D.

In the present model, the flow is simulated as a two-dimen-sional phenomenon with the following assumptions or simplifica-tions: (a) the fluid (air) is Newtonian, incompressible and the flowis laminar and (b) the temperature difference TW � T1 is small, sothat the effect of temperature on fluid density is expressed ade-quately by the Boussinesq approximation.

Next, we consider the following dimensionless variables:

X ¼ xL

; Y ¼ yL

; U ¼ uUR

; V ¼ vUR

; P ¼ p

qU2R

; H

¼ T � T1TW � T1

; Pr ¼ ma

; Ra ¼ gbL3DTma

where the reference velocity is defined as UR = (RaPr)1/2a/L.The governing equations, that express the conservation of mass,

momentum and energy in the fluid domain, become:

@U@Xþ @V@Y¼ 0 ð9Þ

U@V@Xþ V

@V@Y¼ � @P

@Y1

ðRaPrÞ1=2 þPr

ðRaPrÞ1=2 � r2V ð10Þ

U@U@Xþ V

@U@Y¼ � @P

@X1

ðRaPrÞ1=2 þPr

ðRaPrÞ1=2 � r2U þH ð11Þ

U@H@Xþ V

@H@Y¼ 1

ðRaPrÞ1=2 � r2H ð12Þ

where r2 = @2/@X2 + @2/@Y2.No-slip condition is imposed on all the plates for the velocities,

and thermal boundary conditions are at constant temperature,H = 1 for the plates.

The entrance plane, V = 0 and @U/@X = 0, as required by massconservation, and isothermal fluid H = 0.

The velocity and temperature boundary conditions on the outletplane are @U/@X = @V/ @X = @H/ @X = 0.

The average Nusselt number is given below:

Nu ¼Z 1

0

@T@YjY¼0 dX þ

Z 1

0

@T@YjY¼D=L dX

Eqs. (9)–(12) are solved using a finite volume method (FVM) ona staggered grid system [12]. In the course of discretization, QUICKscheme is adopted to deal with convection and diffusion terms. Theequations from the discretization of Eqs. (9) and (12) are solved bythe line-by-line procedure, combining the tri-diagonal matrix algo-rithm (TDMA) and successive over-relaxation iteration (SOR) andthe Gauss–Seidel iteration technique with additional block-correc-tion method for fast convergence. The SIMPLER algorithm [12] isused to treat the coupling of the momentum and energy equations.Pressure-correction and velocity-correction schemes are imple-mented in the model algorithm to arrive at converged solutionwhen both the pressure and velocity satisfy the momentum andcontinuity equations. The convergence criterion is that the maxi-mum residual of all the governing equations is less than 10�6.

Non-uniform staggered grid system is employed with densergrids clustering near the plate so as to resolve the boundary layerproperly. Test runs are performed on a series of non-uniform gridsto determine the grids size effects for the Rayleigh numbers 104,105 and 106 at grid systems 300 � 40, 150 � 20 and 75 � 10,respectively. For each calculation case, a grid independent resolu-tion is obtained. The maximum difference in average Nusselt num-ber between grid (75 � 10) and grid (150 � 20) is 5%, the differencein average Nusselt number between grid (150 � 20) and grid

(300 � 40) is less than 0.2%, so the 150 � 20 non-uniform gridsare used.

The developed computational model is also validated againstexperimental work of Ref. [11]. An interferometer indicates linesof constant density in fluid flow fields, for a gas in free convectionthese line of constant density are equivalent to lines of constanttemperature. The computationally obtained temperature pattern(Fig. 4b) is compared with the density contour visualization inFig. 4a [11]. It is seen that the model adequately predicts the tem-perature patterns obtained in the visualizations.

The variations of the heat transfer rate Q with respect to theplate-to-plate spacing D is shown in Fig. 5 by numerical simulationfor Pr = 0.7 and Ra = 1.1 � 107. Further inspection of this figure re-veals that there are two different evolutions of the heat transferwith the spacing distance. Indeed, the slope of heat transfer ratedecay is important with decreasing spacing distance from 1.5 cmto 0.7 cm. According to conventional way [8], the optimal platespacing is such that the thermal boundary layer merge at the topof the structure, D = 2d = 2.176 cm, Fig. 6 gives the temperature

Fig. 6. Temperature fields for D = 0.022 m and L = 0.15 m.

Fig. 7. Relationship between heat transfer and fin spacing for different length ofplates.

X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456 2453

fields, it is indeed that the thermal boundary layer touches whenspacing is 2.2 cm, while the numerical simulation shows that, forD > 1.5 cm, the heat transfer rate decrease slowly and leads tosome weak gradients. We note that the numerical models predictthe location of the maximum heat transfer is D = 1.45 cm, whichis consistent with the prediction by Eq. (4), Dopt ¼ 4

3 d ¼ 1:45 cm.The same procedure is performed for three different Rayleigh

number. Fig. 7 shows the variation of heat transfer rate Q withspacing distance D for different plate length L. The curves in theFig. 7 show the same trend as Fig. 5, but the maximum points incurves are different.

The temperature profile has a similar shape: two patterns ofboundary behavior emerge with the spacing decrease slowly,which is shown in Fig. 8, but the temperature in the middle ofthe region increases as the spacing closer.

Then, we examine how the peak value umax varied by fixing thelength of plates and changing the value of spacing distance D. It hasbeen shown from Fig. 9: the two maximum velocity peak locationsexist to the edge of the boundary near the two plates. Withdecreasing the spacing from 0.022 m to 0.01 m, one peak velocityboundary layer evolution occurs. The maximum vertical velocityshifts toward the center region position.

Consider a case where the spacing is 2.0 cm, corresponding toD > (4/3)d, and two peaks of velocity profile through the space isexperienced, as shown in Fig. 9a. As the space is lowered to1.0 cm, corresponding to D < (4/3)d, the anticipated numericalvelocity profile is shown in Fig. 9d.

The heat transfer variations may be evidenced by a comparisonbetween the velocity profile and temperature profile as indicatedin Fig. 10. The deviations from the velocity profiles as shown inFig. 10 are the results of the fact that the two higher velocities lo-cate in the bigger temperature gradient region for D = 2.0 cm,while, the higher velocity locates in the smaller temperature gradi-ent region for D = 1.0 cm. The vertical velocity distributions arevery similar to what boundary layers solution predicts (see Fig. 3).

Rohsenow [3] presented an optimal isothermal flat plates spac-ing expression for entrance flow condition and fully developedflow condition:

D0opt ¼ 5:428cp �q2gaðTw � T1Þ

lkL

� ��0:25

ð13Þ

In order to verify the accuracy of the method of solution: Eq. (4),and the numerical computations, the data are compared with re-sults of Rohsenow expression and classical way, shown in Table 1.

Data obtained from Eq. (4) is presented in Table 1, along withthe numerical data and Rohsenow equation, it can be seen thatthe relative discrepancies between these data are all less than5.7%, agreement among the Eq. (4) and numerical data and theRohsenow equation is good. The outlet velocity distribution foroptimal spacing, namely D = 1.45 cm, is presented by numericalsimulation in Fig. 11, which is in agreement with an exact theoret-ical prediction as shown in Fig. 5. This critical situation is the loca-tion of the boundary layer evolution process where the two peaksvelocity profile evolves to one peak velocity profile.

This comparison gives the strong evidence that the presentnumerical solutions are close to those of precedent author Rohse-now, but a deviation from Bejan’s result [5,8], the reason is that thedetailed velocity fields and temperature fields are neglected andthe development of boundary layers at the downstream end ofthe blades do not affect each other, it is stressed the importanceof trying the simplest method [5,10], though it is powerful in con-ceptual study.

5. Numerical optimization of geometry for heat transfer ratedensity

In practice, the volume, amounts of materials and weight areadditional constraints. In this section, we propose to fix the baseboard square, WL = constant (see Fig. 1). The global figure of meritof the designed flow structure is the total heat transfer rate packedin the given square. The allocation of the plate’s intervals withmaximum heat transfer density corresponds to the best de-sign.There are two design types, A (long and less blades) and B(short and more blades). Then, attention is focused on the heattransfer rate density. The optimal spacing is founded by fixing Pra-ndtl number, and then by varying D and the Rayleigh number fordifferent length of blades in given square WL = 225 cm2. Thenumerical simulation results for the optimal design are reportedin Fig. 12.

Fig. 8. Upper outlet temperatures changing with different fin spacing.

Fig. 9. Upper outlet velocities changing with different fin spacing.

Table 1A comparison among the values obtained from the numerical and classical data andvalues predicted by the correlation (4).

Spacing (cm)

L (cm) Rohsenow Simulation 43 d 2d

15 1.3713 1.45 1.45 2.1812 1.2969 1.35 1.37 2.0610 1.2392 1.25 1.31 1.97

7 1.1334 1.15 1.19 1.80

2454 X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456

If the plate is long, for example, the length is 15 cm, blades canseldom fill in the fixed square, the boundary layers merge early, theoverall heat transfer is small.

If the plate is short, for example, the length is 7 cm, the un-heated region is large and the volume occupied by working fluid(the boundary layers) is small, but much more blades can fill inthe unheated region by fixing square, so the heat transfer is big.The maximal heat transfer density means optimal packing suchthat flow regions cannot contribute to global performance areeliminated. Thus, comparing four curves for different length of

(a) D=0.02m (b) D=0.01m

Fig. 10. Comparative analysis of relations between temperature and velocity ofupper outlet (fin length is 15 cm, fin spacing are 2.0 cm and 1.0 cm).

Fig. 11. Velocity of upper outlet under the condition of fin length is 15 cm, finspacing is 1.45 cm.

Fig. 12. The effect of the different fin spacing on the heat transfer density.

Fig. 13. Insertion of smaller plate in the unused flow.

X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456 2455

plates in Fig. 12, it can be expected that a decrease of plate lengthdisplaces the position of maximum to the left. The optimal spacingdetermined in this manner are summarized in Table 2, they arecorrelated by the following expression:

DQ ;opt �12

Dopt ¼23

d ð14Þ

In view of Eq. (5), the dimensionless expression is

DQ ;opt

L¼ 3:55ðGrÞ�

14 ð15Þ

An alternative way beyond the single length scale is multi-scaleconstructal flow geometry [8] to maximal heat transfer, and in par-ticular, the optimal spacing expression is Eq. (4) not the expres-sion: 2d followed by Ref. [8] which has been justified in theprevious section. Constructal theory is also applied to each newgeneration of blades, and this method leads to the optimal spacingbetween blades and the optimal lengths scales.

First, let L0, is the length of the plate, the optimal allocation ofplates can be get based on Eq. (4). Then, the volumetric density

Table 2Relationships between maximal heat transfer density and optimized fin spacing.

L(cm)

Optimal spacing corresponding the maximal heattransfer (cm)

Eq. (5), Dopt

(cm)

15 0.8 1.4512 0.7 1.3710 0.7 1.31

7 0.6 1.19

of heat transfer can be increased by inserting heat transfer surfacesin the flow regions that contain unheated fluid, for example, a newplate of length L1 is inserted in the center of the entrance to origi-nal channel, as shown in Fig. 13. From Eq. (4):

2� 43

dðL1Þ ¼43

dðL0Þ ð16Þ

Combining Eq. (16) and Eq. (1) with Pr � 1 gives:

L1 ffi1

16L0 ð17Þ

It is shown that Eq. (17) is same as Ref. [8].Either one plate can be added or two and three plates can be in-

serted, the constructal flow geometry method [8] is valid, but theequi-partion spacing is determined by Eq. (4). The extension ofabove discussion to insert much more plates case is very straight-forward, and not be discussed here for simplicity.

The prediction of the optimized spaces discussed above (i.e. Eqs.(4) and (14)) is the same if the plate temperature and ambient tem-perature change.

6. Conclusions

We conclude as follows from the theoretical and numericalresults.

(a) The optimal plate-to-plate spacing theoretical expression isderived by the natural convection boundary layer theory, itis found the optimal spacing is 4

3 d, where the substantial

2456 X. Zhang, D. Liu / Energy Conversion and Management 51 (2010) 2449–2456

increase of heat transfer is caused by a coordination of thetemperature and the superposition of velocity. The scaleanalysis is performed to verify the optimal spacing theoret-ical expression.

(b) The numerical simulation is performed in a number of con-figurations in order to determine the effect of spacing on theheat transfer, the optimized results consistent with theoret-ical solutions.

(c) Comparison of the data obtained from Eq. (4), alongwith the numerical data and Rohsenow equation showsthat the relative discrepancies between these data areall less than 5.7%. Agreement among the Eq. (4) andnumerical data and the Rohsenow equation is good, adeviation from Bejan’s results is found which is thecause of the deviation in the approximate analysis, theapproximation is due to the assumption that boundarylayers does not affect significantly the downstreamdevelopment.

(d) In a fixed two-dimensional volume, in order to get heattransfer enhancement, the shorter length of plate is prefer-ence, the optimal vertical plate’s interval in fixed volume isalmost half of that without the constraint.

(e) By inserting multi-scale plates in the boundary layer, theheat transfer enhancement could be conducted effectivelyand the intervals of the multi-scale plates depend on theheight of the plates.

Acknowledgment

This work was supported by the Jiangsu Planned Projects forPostdoctoral Research Funds of China (0901017B). The financialsupport from the Preliminary Research Grants of Soochow Univer-sity (Q3108951) to X.Zhang is gratefully acknowledged.

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