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Natural Convection in an Array of Vertical Channelswith Two-Dimensional Heat Sources: Uniform and Non-Uniform Plate HeatingANA CRISTINA AVELAR a & MARCELO MOREIRA GANZAROLLI ba Aerospace Technical Center—CTA/IAE São José dos Campos, São Paulo, Brazilb State University of Campinas—UNICAMP Barão Geraldo—Campinas, São Paulo, BrazilVersion of record first published: 17 Aug 2010.

To cite this article: ANA CRISTINA AVELAR & MARCELO MOREIRA GANZAROLLI (2004): Natural Convection in an Array ofVertical Channels with Two-Dimensional Heat Sources: Uniform and Non-Uniform Plate Heating, Heat Transfer Engineering,25:7, 46-56

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Heat Transfer Engineering, 25(7):46–56, 2004Copyright C©© Taylor & Francis Inc.ISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457630490496273

Natural Convection in anArray of Vertical Channelswith Two-Dimensional HeatSources: Uniform andNon-Uniform Plate Heating

ANA CRISTINA AVELARAerospace Technical Center—CTA/IAE, Sao Jose dos Campos, Sao Paulo, Brazil

MARCELO MOREIRA GANZAROLLIState University of Campinas—UNICAMP, Barao Geraldo—Campinas, Sao Paulo, Brazil

An experimental and numerical analysis was performed to investigate the conjugatedconduction–convection heat transfer problem in an array of vertical, parallel plates forming openchannels with heated protruding elements attached to one of the walls. Uniform and non-uniformheating of the plates was analyzed. Both the distance between plates and power dissipated per platewas varied. In non-uniform heating in each plate and for a specified total heat generation rate, oneelement was electrically supplied with a different power level from the others, and the influence of thelocation of this element on the temperature distribution was investigated. The SIMPLEC algorithm,based on the finite volume method, was used to solve the pressure velocity coupling. Numerical andexperimental temperature profiles were compared and good agreement was observed.

INTRODUCTION

In the last decades, natural convection between ver-tical plates has been the focus of several studies dueto its application in various thermal systems, such asin the cooling of electronic components, chemical pro-cesses, and solar energy systems. The first work on nat-ural convection in open vertical channels was the exper-

The authors acknowledge to FAPESP for their financial support andCENAPAD-SP for computer support.

Address correspondence to Dr. Ana Cristina Avelar, Aerospace Tech-nical Center—CTA/IAE, Praca Marechal Eduardo Gomes, 50 Vila dasAcacias—CEP 12228-904, Sao Jose dos Campos, Sao Paulo, Brazil. E-mail:cristi@directnet.com.br

imental study of Elenbaas [1]. For the case of channelswith smooth plates, this heat transfer problem has beenstudied for several types of boundary conditions im-posed at the channel walls, such as uniform symmetricor asymmetric heated plates, one plate insulated andthe other with a discrete heated section, and two dis-crete heated plates. Many of them were reviewed byIncropera [2] and Peterson and Ortega [3], who carriedout comprehensive reviews on the thermal control ofelectronic equipment. The concept of partial heating,which is a particular case of non-uniform heating of theplates, has been explored in some studies. Wirtz andHaag [4] obtained experimental results for symmetricisothermal heated plates with an attached unheated entry

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Figure 1 Schematic view of experimental apparatus.

portion. Lee [5] numerically analyzed channels formedby isothermal or isoflux plates with unheated extensionsplaced near the entrance or exit of the channel. Theboundary layer approximation was used in the analysis.Campo et al. [6] reformulated the problem analyzed byLee [5] using the elliptic model for conservation equa-tions. There are also several works dealing with chan-nels with multiple protruding heat sources, mainly in thelast decade. Fujii et al. [7] numerically and experimen-tally analyzed the natural convection heat transfer to airfrom an array of vertical parallel plates with protrudingdiscrete and densely distributed heat sources. Behniaand Nakayama [8] performed numerical simulations onnatural convection considering the same geometry an-alyzed by Fujii et al. [7] for several values of substratethermal conductivity and channel width. Bessaih andKadja [9] carried out numerical simulations on turbu-lent natural convection cooling of three identical heatedceramic components mounted on a vertical adiabatic

Figure 2 Thermocouple position on the test plate.

channel wall. The effect on the cooling due to the spac-ing between components and the removal of heat im-posed on one of the components was investigated.

The present work is aimed at analyzing the con-jugated natural convection conduction heat transferproblem in an array of vertical channels with two-dimensional heat sources mounted on one of the platesurfaces. The situations of uniform and non-uniformheating of the plates was analyzed. In the situation ofnon-uniform heating, in each plate and for a specified to-tal heat generation rate, one element was provided witha different power level than the others, and the influ-ence of the location of this element on the temperaturedistribution was investigated. Numerical solutions wereobtained for the full elliptic two-dimensional Navier-Stokes equations using the SIMPLEC algorithm. Boththe distance between the plates and total power dissi-pated per plate were varied. The motivation for this workis that in spite of the great number of studies on thissubject, the problem of non-uniform heating in chan-nels with protruding heat sources is not usually ana-lyzed even though it is a common situation in electronicequipment.

EXPERIMENTAL ANALYSIS

A schematic view of the experimental apparatus isshown in Figure 1. An array of five fiberglass plateswas accommodated in a metallic structure that is usedin telecommunications devices and that allows somevariation of the distance between plates. The platesare numbered from 1 to 5 for convenience. Each platewas 365 mm high (l), 340 mm wide (w), and 1.5 mmthick (b), and had seven heat sources mounted onits surface. The protruding heat sources were con-structed from two aluminum bars 12.25 mm high,340 mm wide, and 6.125 mm thick, with one resis-tance wire inserted between them. The elements resultedwere screwed onto the fiberglass plates, and an equal

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Figure 3 Non-uniform heating cases.

Figure 4 Physical model and coordinate system.

spacing of 34.5 mm was adopted. The protruding heatsources were connected in a way so that any desiredpower level could be set to any given element, inde-pendently of the others. Power was supplied to the

Figure 5 Grid convergence analysis.

plates by regulated D.C. sources, and both sides of thechannels were closed to prevent lateral airflow. In or-der to reduce the radiation heat transfer influence, theheat sources were polished with diamond paste. The

Figure 6 Temperature excess profiles—uniform heating, d =2 cm.

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Figure 7 Temperature excess profiles—uniform heating, d =3.5 cm.

structure was maintained about 1 m from the groundand placed in a quiet room. Temperature measurementswere obtained by using 36 AWG type J calibrated ther-mocouples, a switch, and a digital thermometer. Specialcare was taken to embed the thermocouples in the alu-minum and the fiber glass surfaces. A very small holewas drilled in their surfaces, which was covered witha thin layer of thermal paste, and the thermocoupleswere fixed with epoxy adhesive. Experiments were per-formed varying the distance between the plates, and thetotal heat generation rate, QT , was set the same forall plates during the tests. The distance between platesranged from 2.0 to 4.0 cm, which corresponds to a ratio(L = l/d) between 9 and 18, and the total heat gener-ation per plate ranged from 20 to 60 W, correspondingto Rayleigh number values ranging from 1 × 104 to 8 ×105.

Figure 2 represents the test plate. The symbol (•)indicates the points in the protruding elements andplate surface where the temperature was measured. The

Figure 8 Temperature excess profiles—non-uniform heating,case 1, d = 2 cm.

Figure 9 Temperature excess profiles—non-uniform heating,case 1, d = 3.5 cm.

measurements were made on Plate 3. The air tempera-ture was also measured at the exit and the entrance ofthe channel formed by Plates 3 and 4. As indicated inFigure 2, in each element, the temperature was mea-sured at three points, and the average of these was takenas the element temperature. The temperature measure-ments were taken in steady-state condition, which wasobtained after about 1 h.

In the situation of non-uniform heating of the plates,one protruding heat source was electrically heated witha power level twice the value supplied to the other pro-trusions. Experiments and numerical simulations werecarried out setting the highest power to the protrudingheat sources 1, 4, and 7, as indicated in Figure 3.

Figure 10 Experimental values of temperature excess, d = 4 cm.

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Figure 11 Isotherms, d = 2.0 cm, QT = 25 W, uniform heating, um = 0.11 m/s.

NUMERICAL ANALYSIS

Figure 4 shows the physical model and coordinatesystem. An infinite number of plates were placed ina vertical parallel arrangement with equal spacing, d.Each plate has the same height, l, and thickness, b.On one surface of the plates, there were seven two-dimensional protruding heat sources mounted, sepa-rated by the distance sp. All heat sources had equaldimensions. The total heat generation in each plate wasset to be the same. The solution domain was chosen tobe the region bounded by the broken line in Figure 4.

The flow was assumed to be laminar, two-dimensional, and at a steady state. The air thermophysi-cal properties were assumed to remain constant, exceptfor the density in the buoyancy term of the momentumequation, which was assumed to follow the Boussinesqapproximation. The heat conduction in the plates andin the heat sources was taken into account. Uniformheat generation within the heat sources was admitted,while the radiation heat transfer among the plates andthe environment were not taken into account. The har-monic mean formulation suggested by Patankar [10]was used to handle abrupt variations in thermophysicalproperties, such as the thermal conductivity across theinterface of two different media. A unique form of themomentum equations was solved in the entire domain.To ensure zero velocities over the solid regions, a veryhigh value of the dynamic viscosity (1020) was used inthe momentum equations in these regions.

The governing equations are expressed in dimension-less form as follows:

∂U

∂X+ ∂V

∂Y= 0 (continuity equation) (1)

Figure 12 Isotherms, d = 2.0 cm, QT = 25 W, non-uniform heating, case 1, um = 0.116 m/s.

U∂U

∂X+ V

∂U

∂Y= −∂P

∂Y+

(Pr

Ra

)1/2(∂2U

∂X2+ ∂2U

∂Y 2

)

+ θ (momentum equation in X direction) (2)

U∂V

∂X+ V

∂V

∂Y= −∂P

∂Y+

(Pr

Ra

)1/2(∂2V

∂X2+ ∂2V

∂Y 2

)

(momentum equation in Y direction) (3)

U∂θ

∂X+ V

∂θ

∂Y=

(Pr

Ra

)1/2 ki

kair

(∂2θ

∂X2+ ∂2θ

∂Y 2

)

+ f x S∗ (energy equation) (4)

where f = 1 for the protruding heat sources and f = 0for the rest of the domain; ki is thermal conductivity ofthe correspondent region. i = 1, 2, and 3, for air, plate,and heating sources, respectively;

The source term, S∗, depends on the configurationbeing given by

S∗ = 2L

npt × XPT × YPT × √PrRa

(5)

for the case of uniform heating of the plates. In the caseof non-uniform heating of the plates, S∗ is given by

S∗ = 4L

(npt + 1) × XPT × YPT × √PrRa

(6)

for the protruding heat sources with the highest powerdissipation level, and by

S∗ = 2L

(npt + 1) × XPT × YPT × √PrRa

(7)

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Figure 13 Isotherms, d = 2.0 cm, QT = 25 W, non-uniform heating, case 2, um = 0.10 m/s.

for the other elements, where npt is the number of pro-truding heat sources, XPT and YPT are the dimension-less element dimensions.

The dimensionless variables in the above equationsare defined by

X = x

d, Y = y

d, L = 1

d, B = b

d,

Hp = h p

d, L p = l p

d(8)

θ = T − To

q ′′d/kair, U = u

uo, V = v

uo, P = p − ph

ρu2m

Ra = q ′′d4β

kairνairαair, Pr = ν

α

where q ′′ is defined based on the total surface area ofthe plate as

q ′′ = QT

2A= QT

2LW(9)

and the reference velocity, uo, is defined by

uo =(

d2gβq ′′

kair

)1/2

(10)

The boundary conditions are:

(X = 0): θ = V = 0; P = −0.5U 2m

(channel entrance) (11)

Figure 14 Isotherms, d = 2.0 cm, QT = 25 W, non-uniform heating, case 3, um = 0.096 m/s.

(Y = 0 and Y = B + 1) : U = V = 0

(channel walls) (12)

Y = l/d (channel exit):

∂U

∂X= ∂V

∂X= ∂θ

∂X= P = 0 (13)

A periodic boundary condition was imposed in re-lation to the temperature at the plate surface, i.e.,θ(X, 0) = θ(X, B +1). The pressure values at the chan-nel entrance and exit were obtained from potential flowtheory.

The governing equations were discretized using thecontrol volume formulation described by Patankar [10],where velocity control volumes are staggered regard-ing the pressure and temperature control volumes. Thecoupling of the pressure and velocity fields was treatedusing the SIMPLEC algorithm of Van Doormal andRaithby [11] with the power-law scheme. The conjugateproblem of conduction and convection was addressedby using the harmonic averaging thermal conductivityat the solid–fluid interfaces (see Patankar [10]). The pe-riodic boundary condition imposed with respect to thetemperature at the plate surface was handled by using theCyclic TriDiagonal Matrix Algorithm (CTDMA) fromPatankar et al. [12] to solve the discretized energy equa-tion. No specification of the wall temperature is requiredin this formulation. This hypothesis was considered tobe valid because the measured longitudinal tempera-tures (x direction) in the inner three plates are almost thesame. Negligible contact resistance between heat sourceand plate was assumed. This hypothesis was consid-ered because experimental measurements showed that

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Figure 15 Averaged Nusselt number in the heat sources, non-uniform heating, case 2.

the temperature difference between the protrusion’s topsurface and base was very small. The equations weresolved in a non-uniform grid crowded near the solidwalls.

Grid independence was checked by means of succes-sive mesh refinements in both directions. Figure 5 showsnumerical results in terms of dimensionless mean inletvelocity (Um) and dimensionless temperature (θ) at theseventh protuberance as a function of the number of gridpoints. The computations were performed for a uniformheating condition at Ra = 4.5E4 (Q = 10 W) and 2.5 cmspace between plates. Asymptotic convergence was ob-served, and the results from the 622 × 46 mesh werefound to be within 2% of the values from the 870 × 70mesh, with four times less cpu time. Hence, the 622 ×46 mesh was used in all subsequent calculations. For thedistances between plates larger than 2.5 cm, the numberof grid point in y-direction was increased proportionally.

The convergence of the iterative procedure was testedby the following criterion:

∣∣�ni, j − �n−1

i, j

∣∣max∣∣�n

i, j

∣∣max

≤ 5 × 10−6 (14)

where � stands for U , V , θ and the maximum residualin the continuity equation.

Figure 16 Streamlines between the channel entrance and the fourth protrusions, QT = 25 W, d = 2.5 cm.

RESULTS AND DISCUSSIONS

Figures 6 and 7 show experimental values of temper-ature excess profiles for the situation of uniform heatingof the plates for several values of power, and for the dis-tance between plates equal to 2.0 cm and 3.5 cm. Thetemperature excess, �T , is defined as the differencebetween the heat source temperature and the air inlettemperature, To.

Figures 8 and 9 show temperature excess profiles forcase 1 of non-uniform heating of the plates, for severalvalues of power, and for the distance between platesequal to 2.0 cm and 3.5 cm.

From Figures 6–9, it can be noticed that temperatureprofiles along the plate present slight shape variationswith the increase in the heat rate dissipated per plate.However, the temperature gradient along the plate de-creases with the increase of the distance between plates.This behavior was observed for all heating conditionsanalyzed.

From Figure 6 it can be observed that for the small-est plate spacing, 2.0 cm, the temperature profile is al-most linear from the second to the fifth element forall values of total heat generation rate. This behaviorwas described by Kelkar and Choudhury [13], who nu-merically investigated the periodically fully developednatural convection in a vertical channel with equallyspaced surface mounted heat generation blocks. Whenthis regime is established, the flow pattern in each mod-ule is the same, and the temperature difference be-tween corresponding points in adjacent modules is con-stant. On the other hand, for large values of plate toplate distance, the temperature profile approaches thatcorresponding to a vertical plate exposed to an infi-nite medium. The more flat temperature profile, dis-played in Figure 7 for distance d = 3.5 cm, can beexplained from the Nusselt number correlation for avertical wall subject to a surface condition of uniformheat flux Bejan [14] for which the temperature excessis proportional to the vertical distance raised to the1/5th power. This power law can be verified by present-ing the experimental values of the excess temperatureshown in Figure 7 in a logarithmic scale, as illustrated inFigure 10.

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Figure 17 Streamlines between the fourth and the last protrusions, QT = 25 W, d = 2.5 cm.

Figures 11 to 14 present isotherms, obtained fromthe numerical analysis, for the distance between platesequal to 2.0 cm and total heat generation rate of 25 W.Isotherms for the uniform heating in Figure 11 and thosefor the non-uniform heating conditions of the plates inFigures 12 to 14 are shown. The channel is displayed inthe horizontal direction for convenience.

From Figures 11 to 14, it can be noticed that theisothermal lines are crowded on the surfaces that areparallel to the main flow direction and near the upstreamcorner of the protrusions, meaning that the heat trans-fer rates are higher in these regions. In Figures 12 to14, it can also be observed that the isothermal linesare even more crowded around the protrusion with thehighest dissipation level, which means an intensifica-tion of the heat transfer rate in these elements. This in-tensification can be confirmed observing the numericalNusselt number profile based on the distance betweenplates and the air inlet temperature, as presented inFigure 15.

The same behavior was observed in all three cases ofnon-uniform heating and for all values of distance andpower simulated, making explicit the non-linear natureof the relation between the heat transfer rate and thetemperature difference in natural convection.

Comparing the air velocity values in the channel inthe three cases of non-uniform heating, a decrease inthe airflow rate as the most heated protuberance movestowards the channel exit can be noticed.

Figure 18 Numerical and experimental values of temperature ex-cess, uniform heating of the plates.

Figures 16 and 17 show streamlines for the samevalues of total heat generation rate and distance betweenplates.

It is observed that recirculation regions are formedbetween the protrusions and that the flow pattern aroundeach protrusion is almost the same in the region betweenthe second and fifth protrusions. No relevant variationin the flow pattern with heating conditions of the plateswas observed.

In Figures 18 and 19, numerical and experimentalvalues of dimensionless temperature excess for the valueof total heat generation rate per plate equal to 25 W andplate spacing equal to 2.0 cm are compared.

As can be noted from Figures 18 and 19, good agree-ment was observed between numerical and experimen-tal results. For distances of 2.0 and 2.5 cm, differ-ences of about 8% were observed, while for the greatestdistances of 3.0 and 3.5 cm, differences were around15%.

In Figures 20 and 21, the experimental temperatureexcess profiles for the three cases of non-uniform heat-ing of the plates are compared. In Figure 20, the tem-perature excess curves are shown for the distance of 2cm and power of 25 W, and in Figure 21 for the distanceof 3 cm and power of 40 W.

Comparing the temperature profiles for cases 2 and3, it can be noted that, for the first three heated sources,the curves are nearly coincidental. However, compar-ing cases 1 and 2, it can be noticed that the second and

Figure 19 Numerical and experimental values of temperature ex-cess, non-uniform heating of the plates, case 1.

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Figure 20 Temperature excess profiles, d = 2 cm, QT = 25 W.

third element temperatures are higher in case 1 than incase 2, and in comparing the curves for cases 1 and 3,it can be observed that first six element temperaturesare also higher than in case 1. This behavior was ob-served for all values of distance and power analyzed.Therefore, it can be concluded that the position of thehigher-powered element only slightly affects the tem-perature of the upstream elements. On the other hand,the thermal wake from the upstream elements inducesa temperature increase on the downstream elements.Comparing the maximum plate temperature value inuniform and non-uniform heating cases, a more ele-vated value in case 3 is observed, followed by case 2,with the lowest temperature excess value correspond-ing to the uniform heating situation. This was observednumerically and experimentally in all distances, ex-cept for the 2.0 cm plate spacing situation, where themaximum plate temperature value in the uniform heat-ing was higher than in case 1. Based on these obser-

Figure 21 Temperature excess profiles, d = 3 cm, QT = 40 W.

vations, in the case of temperature restriction and forhigh values of ratio L, positioning a higher dissipatingelement in the channel entrance region can be an ef-fective way of achieving higher power dissipation perplate than a uniform heating situation. These high valuesof the ratio L can occur, for instance, when the objec-tive is to maximize the heat generation per volume unitwhile the temperature is maintained below a prescribedlimit.

CONCLUSIONS

Natural convection heat transfer from an array of ver-tical, parallel plates, forming open channels containingheated protruding elements attached to one of the walls,was analyzed both numerically and experimentally. Theobtained results suggest the following conclusions.

The experimental results show that the temperatureexcess profiles present slight variations with the increasein the heat rate dissipated per plate. However, the tem-perature gradient along the plate becomes less accentu-ated with the increase in distance.

Good agreement between numerical and experimen-tal results was verified. The differences between numer-ical and experimental results ranged from 8–15%. Thehighest difference was verified for the highest values ofdistances between plates and power dissipated per plate.

For the smallest plate spacing, 2 cm, it was verifiedthat the flow is periodically fully developed betweenthe second and the fifth protruding heat source, i.e., theflow pattern is almost the same around each element,and the temperature variation between correspondingpoints in adjacent elements is constant. Increasing thedistance between plates causes longer lengths of un-heated fluid, approaching the condition of a plate in aninfinite medium.

In the non-uniform heating situation, it was observedthat the position of the most heated protuberance onlyslightly affects the temperature of the upstream ele-ments. On the other side, the thermal wake from theupstream elements causes a temperature increase on thedownstream protuberances. Comparing maximum tem-perature values on the plate in cases 1–3 and uniformheating, the lowest value was observed in the uniformheating situation, except for the 2.0 cm distance, wherethe uniform heating maximum temperature excess valuewas higher than in case 1. Therefore, when there are tem-perature constraints and the distance between the platesis small, positioning a higher dissipating element inthe channel entrance region can permit higher powerdissipation per plate compared to the uniform heatingsituation.

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NOMENCLATURE

A plate surface area, m2

b plate thickness, mcp specific heat at constant pressure of air,

J/kg Kd distance between plates, mD dimensionless distance between platesg gravitational acceleration, ms−2

h plate height, mh p protruding heat source height, mk air thermal conductivity, W m−1 K−1

kp plate thermal conductivity, W m−1 K−1

l channel height, mnpt number of protruding heat sources in each

platep pressure, N/m2

Pr Prandtl numberQ total heat transfer rate in one protruding

heat source, WQT total heat transfer rate, Wq ′′ heat flux, Wm−2

q ′′′ heat transfer density, Wm−3

Ra Rayleigh number (= q′′d4βg/kνα)s distance between protruding heat sources

in the plate, mS source term in the energy equation, W/m3

S∗ source term in the dimensionless energyequation

T temperature, KTo air temperature at the channel entrance, Ku air velocity in the x direction, m/sum mean air velocity in the channel entrance,

m/sU dimensionless velocity in the x directionUm dimensionless mean air velocity in the

channel entrancev air velocity in the y direction, m/sV dimensionless velocity in the y directionw plates width, mW dimensionless plates width, mx,y,z cartesian coordinates, mX, Y, Z dimensionless cartesian coordinatesXPT, YPT dimensionless protruding heat sources di-

mensions

Greek Symbols

α thermal diffusivity, m2s−1

β coefficient of volumetric thermal expan-sion, K−1

θ dimensionless temperature, = (T − TO )/(q′′d/kair)

� stands for U, V , θ , and the maximumresidual in the continuity equation

ρ density, kg m−3

ν kinematic viscosity, m2s−1

µ viscosity, kgs−1 m−1

ψ streamline

Subscripts

air airmax maximumo referencep protruding heat source

REFERENCES

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[2] Incropera, F. P., Convection Heat Transfer in Electronic Equip-ment Cooling, Journal of Heat Transfer, vol. 110, no. 43,pp. 1097–1111, 1988.

[3] Peterson, G. P., and Ortega, A., Thermal Control of ElectronicEquipment and Devices, Advances in Heat Transfer, vol. 20,pp. 181–314, 1990.

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[9] Bessaih, R., and Kadja, M., Turbulent Natural ConvectionCooling of Electronic Components Mounted on a VerticalChannel, Applied Thermal Engineering, vol. 20, no. 1, pp. 141–154, 2000.

[10] Patankar, S. V., Numerical Heat Transfer and Fluid Flow,Hemisphere Publishing Corporation, New York, 1980.

[11] Van Doormaal, J. P., and Raithby, G. D., Enhancements of theSimple Method for Predicting Incompressible Fluid Flows,Numerical Heat Transfer, vol. 7, pp. 147–163, 1984.

[12] Kelkar, K. M., and Choudhury, D., Numerical Prediction ofPeriodically Fully Developed Natural Convection in a VerticalChannel with Surface Mounted Heat Generating Blocks, In-ternational Journal of Heat and Mass Transfer, vol. 36, no. 5,pp. 1133–1145, 1993.

[13] Patankar, S. V., Liu, C. H., and Sparrow, E. M., Fully DevelopedFlow and Heat Transfer in Ducts Having Streamwise-Periodic

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Variations of Cross-sectional Area, ASME, Journal of HeatTransfer, vol. 99, pp. 180–186, 1997.

[14] Bejan, A., External Natural Convection, In Convection HeatTransfer, John Wiley & Sons, Inc., New York, pp. 202–204,1995.

Ana Cristina Avelar is a researcher at the In-stitute of Aeronautics and Space of the BrazilianAerospace Technical Center (CTA). She receivedher doctorate degree in thermal and fluid sciencesat State University of Campinas (UNICAMP),Brazil. Her thesis was on the effects of uniformand non-uniform heating on natural convectionin channels with protruding heating elements bymeans of experimental and numerical methods.Currently, she works with wind-tunnel testing intransonic and subsonic installations. Her research

interests are mainly related to natural convection and experimental methodsin heat transfer and fluid dynamics, particularly flow visualization.

Marcelo Moreira Ganzarolli is an associate pro-fessor at the Energy Department of the Facultyof Mechanical Engineering, State University ofCampinas (UNICAMP), Brazil. He teaches ther-modynamics, heat transfer, and computationalfluid dynamics. His research interests are mainlyconcerned to the following subjects: numericalheat transfer, natural and forced convection, andcooling of electronic equipment. He is currentlyinvolved in the numerical simulation of rotaryregenerators and natural convection in inclined

plates. He has published technical articles in conference proceedings andjournals. He is a member of the Brazilian Society of Mechanical Sciences(ABCM).

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