a numerical study of natural convection heat transfer in fin

Research ArticleA Numerical Study of Natural Convection HeatTransfer in Fin Ribbed Radiator

Hua-Shu Dou1 Gang Jiang2 and Lite Zhang1

1The Province Key Laboratory of Fluid Transmission Technology Faculty of Mechanical Engineering and AutomationZhejiang Sci-Tech University Hangzhou 310018 China2Huadian Electric Power Research Institute Hangzhou 310030 China

Correspondence should be addressed to Hua-Shu Dou huashudouyahoocom

Received 6 March 2014 Accepted 11 April 2014

Academic Editor Zhijun Zhang

Copyright copy 2015 Hua-Shu Dou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper numerically investigates the thermal flow and heat transfer by natural convection in a cavity fixed with a fin array Thecomputational domain consists of both solid (copper) and fluid (air) areas The finite volume method and the SIMPLE schemeare used to simulate the steady flow in the domain Based on the numerical results the energy gradient function 119870 of the energygradient theory is calculated It is observed from contours of the temperature and energy gradient function that the position wherethermal instability takes place correlates well with the region of large119870 values which demonstrates that the energy gradientmethodreveals the physical mechanism of the flow instability Furthermore the effects of the fin height the fin number and the fin shape onthe heat transfer rate are also investigated It is found that the thermal performance of the fin array is determined by the combinedeffect of the fin space and fin height It is also observed that the effect of fin shape on heat transfer is insignificant

1 Introduction

Natural convection heat transfer from finned surfaces hasreceived much attention in both numerical simulations andexperimental modeling it is adopted extensively in variousindustrial devices such as gas cooled nuclear reactors auto-mobile aerospace vehicles and electronic systems Nowa-days as electronic equipment tends to be large as well asminiature removing heat rapidly from the equipment ismuchmore desirable than before since the thermal efficiencyof heat removal from the equipment can impact the life-spanof the equipmentThere are mainly two ways to enhance heattransfer from heat-generating electronic equipmentThe firstconvenient method is to cool the electronic equipment byblowing air at a moderate velocity and the second way isto use fin arrays Since the forced convection approach hasinherited problems and has to bear extra running cost moreandmore researchers focus on designing optimum fin arrayswhich can provide moderate heat transfer rates if designedproperly

In early stage Starner and McManus [1] made someexperiments by inserting four different rectangular fin arrays

into a vertical 45-degree plate and a horizontal platerespectivelyThey obtained some free natural convection datawhich were widely used in the subsequent investigationsHarahap and McManus [2] obtained more detailed data byinvestigating free convection heat transfer from horizontalfin arrays using the schlieren shadowgraph technique Jonesand Smith [3] applied a Mach-Zender interferometer in theirexperiments to study the variation of local heat transfercoefficient for isothermal vertical fin arrays on a horizontalbase With a fixed fin height 119871 = 254mm the experi-mental data shows that the overall heat transfer coefficientdepends strongly on the fin space but weakly on the finheight Rammohan Rao and Venkateshan [4] experimentallyinvestigated the interaction of free convection and radiationin a horizontal fin array The most important conclusionmade in their researchwas that therewas amutual interactionbetween free convection and radiation and hence a simplisticapproach based on additivity of radiation and convectionheat transfer calculated independently based on isothermalsurfaces was unsatisfactory Yuncu and Anbar [5] also madesome experiments to research the effects of the fin space thefin height and the temperature difference on heat transfer

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 989260 13 pageshttpdxdoiorg1011552015989260

2 Mathematical Problems in Engineering

with a fixed fin height and fin thickness They found thatthere was an optimum fin spacing which was not related tothe temperature difference However optimum fin spacingwas inversely proportional with the fin height Mobedi andYuncu [6] numerically investigated the steady state naturalconvection heat transfer in longitudinally short rectangularfin arrays on a horizontal base They observed two types offlow patterns For the fin arrays with narrow fin spacingair could only enter into the channel from the end regionsHowever for the fin arrays with wide fin spacing the air wasalso entrained into the channel from the region between thefins turned 180 degrees at the base and thenmoved up alongthe fin surface while it flowed into the central part of thechannel

More recently lots of researchers tried to enhance heattransfer rate of fin arrays using fin arrays Arquis and Rady[7] investigated natural convection heat transfer and fluidflow characteristics from a horizontal fluid layer with finnedbottom surface and observed that the number of convectioncells between two adjacent fins is a function of the values ofthe fin height and Rayleigh number Liu [8] considered anoptimum design problem for the longitudinal fin arrays witha constant heat transfer coefficient in a fuzzy environmentwhere the grid requirements to strictly satisfy the total finvolume and array width and maximize the heat dissipationrate are softened Harahap et al [9] conducted some exper-iments to investigate the effects of miniaturizing the baseplate dimensions of vertically based straight rectangular finarrays on the steady state heat dissipation performance underdominant natural convection conditions They found thatthe relevant correlations proposed for large fin arrays werenot applicable to the experimental data obtained from theminiaturized vertical rectangular fin arrays SubsequentlyHarahap et al [10] conducted concurrent calorimetric andinterferometric measurements to investigate the effect thatthe reduction of the base plate dimensions has on the steadystate performance of the rate of natural convection heattransfer fromminiaturized horizontal single plate-fin systemsand plate-fin arrays Their conclusions suggested that the finheight119871 and the fin number are the prime geometric variablesfor generalization Dogan and Sivrioglu [11] experimentallyinvestigated mixed convection heat transfer and the resultsobtained showed that the optimum fin spacing which yieldedthe maximum heat transfer is 119878 = 8-9mm and the optimumfin spacing depends on the value of Ra Azarkish et al[12] used a modified genetic algorithm to maximize theobjective function which is defined as the net heat transferrate from the fin surface for a given height Their resultsshow that the number of the fins is not affected by the finprofile but the heat transfer enhancement for the arrayswith optimum fin profile is about 1ndash3 percent more thanthat for the arrays with conventional fin profiles Giri andDas [13] numerically performed laminar mixed convectionover shrouded vertical rectangular fin arrays attached to avertical base They found that the drop in pressure defect forforced convection induced velocity formixed convection andoverall Nusselt number for mixed convection are correlatedwell with governing parameters of the considered problemWong and Huang [14] made some three-dimensional (3D)

numerical simulations to investigate the effect of fin parame-ters on dynamic natural convection from long horizontal finarraysThey observed that the optimumfin spacing decreasessignificantly with the fin height and increases slightly with thefin length Furthermore their observations agreed well withthe results reported in the literature

All the above experimental and numerical researchfocused on optimum design of the fin arrays in order toimprove heat transfer rate Almost all the related factorsresult in laminar convection involving low values of heattransfer coefficients However it is well known that the heattransfer rate will be prompted when the base flow loses itsstability and flows in a turbulent manner [15] Althoughsome optimum fin arrays design can obtain a satisfying heattransfer rate which resulted from flow instability the physicalmechanism of flow instability of natural convection is stillnot fully understood Recently Dou et al [16ndash24] suggestedan energy gradient method which can reasonably reveal thephysical mechanism of flow instability Dou and Phan-Thien[16] described the rules of fluid material stability from theviewpoint of energy field They claimed that the instability ofnatural convection could not be resolved by Newtonrsquos threelaws for the reason that a material system moving in somecases is not simply due to the role of forces This approachexplains the mechanism of flow instability from physics andderives the criteria of turbulence transition Accordinglythis method does not attribute Rayleigh-Benard problem toforces but to energy gradient It postulates that when the fluidis placed on a horizontal plate and is heated from below thefluid density in the bottom becomes low which leads to anenergy gradient 120597119864120597119910 gt 0 along 119910-coordinate Only when120597119864120597119910 is larger than a critical value will the flow becomeunstable and then fluid cells of vorticities will be formedMore recently Dou et al [25] applied the energy gradientmethod to natural convection and the results from numericalsimulations accord well with those predicted based on thecriteria originated from energy gradient method

This study is focused on the research of effects of fin arraysparameters on convection heat transfer coefficient Then theenergy gradient method is employed to reveal the physicalmechanism of flow instability and explain the reason why theoptimum fin arrays can result in better heat transfer rate

2 Computational Geometry andNumerical Procedures

21 Computational Geometry The computational geometryis shown in Figure 1Here the geometry is simplified from the3D solid of GH-4 ribbed radiator model [26] The simplifiedcavity in this study is a two-dimensional (2D) square inwhich the length of the square cavity is 250mm and theorigin of the coordinates is at the lower left corner of thecavity The fin arrays are fixed at the hot bottom of the cavitywith an equal distance Here 119878 is the fin space 119867 is the finheight and 119867 and 119878 are variable In addition 119891

119905means the

thickness of the fin arrays which is fixed at 2mm in this studyand can be neglected by comparing to the fin height andthe length of the cavity while 119887

119905means the thickness of the

bottom plate which is equal to 3mm

Mathematical Problems in Engineering 3

Sym Sym

S

H

Cold top

Hotbottom

ft

bt

Figure 1 Numerical geometry

The computational geometry is divided into two areaswhich contains a solid area and a fluid area The solid areaincludes the fin arrays and the base plate in which thematerial is copper while the rest of the geometry is the fluidarea where the air fills in it All the walls of the cavity are solidand no-slipThe wall style of the left and the right walls of thecavity is symmetric The top wall of the cavity is given a fixedtemperature of 300K and the base of the plate is given a fixedtemperature of 360K Furthermore the fluid area is given aninitial temperature of 300K

22 Governing Equations For the problem described in theprevious section the development of natural convectionin the fluid area is governed by the following continuityequations two-dimensional steady Navier-Stokes equationsand an energy equation The heat transfer in the solid area isgoverned by a heat conduction equation All these equationsare based on Boussinesq approximation and listed as follows

120597119906

120597119909+

120597V120597119910

= 0

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588

120597119901

120597119909+ 120592(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

= minus1

120588

120597119901

120597119910+ 120592(

1205972V

1205971199092+

1205972V

1205971199102) + 119892120573 (119879 minus 119879

0)

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910= 119896(

1205972119879

1205971199092+

1205972119879

1205971199102)

1205972119879

1205971199092+

1205972119879

1205971199102= 0

(1)

where 119909 and 119910 are the horizontal and vertical coordinates 119905is the time 119879 is the temperature 119901 is the pressure 119906 and V

are the velocity components in the 119909 and 119910 directionsrespectively 119892 is the acceleration due to gravity 120573 is thecoefficient of thermal expansion 120588 is the fluid density 119896 isthe thermal diffusivity and 120592 is the kinematic viscosity

23 Numerical Algorithm The governing equations (1) areimplicitly solved using a finite-volume SIMPLE scheme withthe QUICK scheme approximating the advection term Thediffusion terms are discretized using central differencingwith a second-order accuracy The discretized equationsare iterated with the specified underrelaxation factors Inaddition it should be noted that the flow is steady

In this study the mean value of the local Nusselt numberalong the top cold wall is calculated for measuring heattransfer performance as follows

Nu =1

119871int

119871

0

(

10038161003816100381610038161205971198791205971199101003816100381610038161003816119910minus119871

Δ119879119871)119889119909 =

1

Δ119879int

119871

0

10038161003816100381610038161003816100381610038161003816

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910minus119871

119889119909 (2)

Here 120582 is the thermal conductivity Δ119879 is the temperaturedifference and 119888

119901is the specific heat capacity

3 Grid Independent Test

In order to examine the independence of the gird sizeto the computing result three different sized meshes areconstructed Here grid (a) is uniformly meshed with Δ119909 =

0004mm grid (b) is uniformly meshed with Δ119909 =

0002mm and grid (c) is uniformly meshed with Δ119909 =

0001mmFigure 2 shows the temperature contours in three dif-

ferent mesh sizes It can be quantitatively observed that thetemperature contours are almost the same in three differentmesh sizes by making a comparison Figure 3 shows thetemperature versus iterations at the same monitor pointwith the three different mesh sizes It can be found thatthe temperature difference is getting smaller as the dif-ference of the mesh size is getting smaller Furthermorefor the two finer meshes the difference of the steady-statetemperature is negligible In what follows the medium(or fine) mesh resolution is adopted for all the calcula-tions

4 Application of Energy Gradient Method

41 Energy Gradient Method It is observed in Figure 2 thatthe base flow loses its stability by forming vorticities andby moving with a wave form which is helpful to enhanceheat transfer However the physical mechanism of the flowinstability of natural convection is still not fully understoodEnergy gradientmethodwill be applied in natural convectionto explain the physical mechanism of flow instability in thisstudy


Grid (a)

(a)

Grid (b)

(b)

Grid (c)

(c)

Figure 2 Temperature contours with three different meshes grid (a) Δ119909 = 0004mm grid (b) Δ119909 = 0002mm and grid (c) Δ119909 = 0001mm

Dou et al [16ndash24] suggested an energy gradient methodwhich can be treated as the criteria of flow instability And thecriteria of flow instability can be expressed as follows

119865 =Δ119864

Δ119867=

(120597119864120597119899) (2119860120587)

(120597119867119904120597119904) (120587120596

119889) 119906

=2

1205872119870

119860120596119889

119906=

2

1205872119870V1015840119898

119906lt Const

(3)

where

119870 =120597119864120597119899

120597119867119904120597119904

(4)

Here 119906 is the streamwise velocity of main flow 119860 is theamplitude of the disturbance distance 120596

119889is the frequency of

the disturbance V1015840119898

= 119860120596119889is the amplitude of the disturbance

of velocity and 120587 is the circumference ratio Furthermore 119865is a function of coordinates which expresses the ratio of theenergy gained in a half-period by the particle and the energyloss due to viscosity in the half-period 119870 is a dimensionless

field variable (function) andmeans the ratio of the transversalenergy gradient and the rate of the energy loss along thestreamline 119867

119904is the loss of the total mechanical energy per

unit volumetric fluid along the streamline for finite heightHere 119864 = 119901+ 12 120588119881

2 expresses the total mechanical energyper unit volumetric fluid 119904 is along the streamwise directionand 119899 is along the transverse direction

42 Application of Energy Gradient Method in Natural Con-vection In the present study we will use the energy gradientfunction119870 to reveal the physical mechanism of flow instabil-ity in natural convection The fluid fills in a 2D simplified finradiator model and the base flow is initially stationary Basedon the energy gradient method and the particular conditionof the base flow the energy gradient function 119870 of naturalconvection can be expressed as follows

119870 = radic(120597119864

120597119909)

2

+ (120597119864

120597119910)

2

(5)

It should be noted that the influence of the gravitationalenergy is neglected in terms of the total mechanical energy in


0 500 1000 1500 2000 2500 3000 3500 4000300

305

310

315

320

325

330

Tem

pera

ture

(K)

Iterations

Grid (a)Grid (b)Grid (c)

Figure 3 Temperature profile calculated with three different gridmeshes grid (a) Δ119909 = 0004mm grid (b) Δ119909 = 0002mm and grid(c) Δ119909 = 0001mm

120597E120597y

120597E120597x

y

K

xo

Figure 4 Calculation of 119870

this study since the fluid is air then 119864 sim 1199010is obtained con-

sequently the energy gradient function119870 of transient naturalconvection can be written as 119870 = radic(120597119901

0120597119909)2+ (1205971199010120597119910)2

which is shown in Figure 4 Here 1199010represents the total

pressure

5 Results and Discussions

51 Physical Mechanism of Flow Instability Patterson andImberger [15] announced that the flow instability couldenhance heat transfer rate of natural convection and the heattransfer rate increases with the addition of the intensity offlow instability In the following the physical mechanism offlow instability will be discussed using the energy gradientmethod

Figures 5(a) and 5(b) show the contours of velocityand total pressure respectively with iterations of 4000 Thefollowing observations can be made in Figure 5(a) Firstlythe velocity in the whole flow field distributes symmetricallyalong the vertical center line Secondly the magnitude ofvelocity near the left and right walls and in the center of

the cavity is relatively large while the velocity between theadjacent fins is very weak This large velocity difference inthe cavity is due to the variation of the heat transfer modesThe reason why the velocity near the left and right walls andin the center of the cavity is large is that the heat flux isfocused in these areas and moves downward driven by thebuoyancy However the heat flux between the adjacent finsis rare and the buoyancy can be neglected attributed to thesmall temperature difference between adjacent fins hence thevelocity between adjacent fins is very weak Thirdly thereare two blue areas where the velocity is very weak abovethe fin arrays This is due to the convection of heat alongthe symmetrical wall and these blue areas are not affectedevidently by convection

Similarly the findings from Figure 5(b) can be summa-rized as follows Firstly the total pressure decreases graduallyfrom the top to the bottom The fluid flow and the heatflux along the symmetrical wall move upward driven by thebuoyancy and concentrates near the top wall of the cavitythus the total pressure difference will be produced Secondlytwo circulating regions occur symmetrically above the finarraysThis is due to the circulation flow of the heat flux alongthe symmetrical walls Thirdly the total pressure gradient isirregular above the fin arrays while the total pressure gradientbetween the adjacent fins is regular The circulation flow ofheat flux above the fin arrays which results in the irregulardistribution of total pressure gradient The regular gradientdistribution between the fins is attributed to the symmetricalmovement of heat flux along the fin arrays surfaces and theweak buoyancy between the fins

Figures 6(a) and 6(b) show the contours of temperatureand the contours of the value of 119870 at the iterations of4000 respectively It can be seen in Figure 6(a) that the baseflow downstream of the fin loses its stability forming smallvorticities and by moving in a wave form It can be seen inFigure 6(b) that the value of119870 in the red area and the yellowarea is much higher than that in other areas In themeantimethere exist two symmetrical green areas with high value of119870 on the trajectory of the heat flux circulation Based onthe criteria of flow instability of the energy gradient methodthere is a critical value 119870

119888 above which the flow will lose its

stability Thus the flow instability could most likely occur inthe red yellow and green areasMaking a further comparisonbetween Figures 6(a) and 6(b) it is easy to observe thatthe positions where instabilities take place are in accordancewith the area with the higher value of 119870 This phenomenonindicates that the application of the energy gradient methodin the simplified mode of ribbed radiator is reliable and theenergy gradient method can reveal the physical mechanismof flow instability of natural convection

Moreover it can be found that the regions with highvelocitymagnitude accordwell with the areas with large valueof119870 Consequently the flow instability of natural convectionhas an instinct affection on the velocity of the fluid flow

52 Effect of Fin Height In order to investigate the effectof fin height on heat transfer rate here two groups of finarrays are chosen for comparison In the first group all the


(a) Velocity contours (b) Total pressure contours

Figure 5 Velocity contours and total pressure contours119867 = 65mm

(a) Temperature contours (b) 119870 contours

Figure 6 Validation of energy gradient method 119867 = 65mm

fin spaces are fixed at 61mm and the fin height changes from25mm to 105mm with an equal increment of 20mm In thesecond group all the fin spaces are fixed at 26mm and the finheight changes over the same range in the first group In thefollowing section the Nusselt number is selected to representthe heat transfer rate

Figure 7 shows the temperature contours and contoursof the value of 119870 with different fin height where the finspace is fixed at 61mm Figure 8 shows the temperaturecontours and contours of the value of 119870 with different finheight where the fin space is fixed at 26mm Table 1 showsthe Nusselt number in different numerical cases It is easyto observe the following results by comparing Figures 7and 8 and Table 1 Firstly Figures 7 and 8 show that theareas where flow instabilities occur in temperature contoursaccord well with the region where the value of 119870 is verylarge in contours of the value of 119870 These results validatethe criteria of flow instability based on the energy gradient

method which in turn demonstrates that the application ofenergy gradient method to natural convection is reasonableSecondly it is found that the intensity of flow instabilitychanges dramatically in Figure 7 when the fin height changesfrom 45mm to 65mmwith a fixed fin space of 61mm It alsomeans that the heat transfer rate will be increased fiercelyHowever it is found in Figure 8 that the flow instability variesin a very small range which means that the variation ofheat transfer rate is limited At the same time it is foundfrom the second row of Table 1 that the variation range ofNusselt number is notable while the Nusselt number alterswithin a limited extent detected from the third row of Table 1This accordance between flow instability and heat transferrate validates the hypothesis made by Patterson and Imberger[15] The heat transfer rate increases with the addition of theintensity of flow instabilityThirdly it is found in Figure 7 thatthe heat flux moves right by jumping over the fins when thefin height is 25mm and 45mm respectively whereas the flux


Table 1 Effect of fin height119867 and 119878 mean fin height and fin space respectively

Nu 119867 = 25mm 119867 = 45mm 119867 = 65mm 119867 = 85mm 119867 = 105 mm119878 = 61mm 192 197 235 235 234119878 = 26mm 237 238 238 239 239

T temperature contoursH = 25mm

K K contours H = 25mm

(a)



(b)



(c)

T temperature contoursH = 85 mm


(d)



(e)

Figure 7 Temperature contours and contours of the value of 119870 with different fin heights where fin space is fixed at 61mm (a) 119867 = 25mm(b) 119867 = 45mm (c) 119867 = 65mm (d) 119867 = 85mm and (e) 119867 = 105mm



(a)



(b)



(c)



(d)



(e)



Table 2 Effect of fin space 119867 and 119878 mean fin height and fin space respectively

Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243

T temperature contours

(a)


(b)


(c)

T temperature contoursT temperature contours

(d)


(e)


(f)


(g)


(h)

Figure 9 Temperature contours with a fixed fin height of 45mm and the fin space changes from 82mm to 232mm characterized by theincrease of fin number (119899) (a) 119899 = 2 (b) 119899 = 3 (c) 119899 = 4 (d) 119899 = 5 (e) 119899 = 6 (f) 119899 = 7 (g) 119899 = 8 and (h) 119899 = 9

moves upward along the symmetric wall when the fin heightis 65mm 85mm and 105mm respectively It is observedin Figure 8 that the heat flux moves upward symmetricallyin all the numerical cases The difference of flow behavior isjointly affected by the fin height and the fin space When thefin height is small the heat flux jumps over the fins drivenby the buoyancy When the fin height is small the buoyancybetween the fins is weak and it is not very easy for the heatflux to jump over the fins In addition when the fin spacereduces characterized by the increase of the fin number thedifficulty for the heat flux jumps over the fins will increaseMoreover the temperature difference between adjacent finsis very weak which results in a negligible buoyancy betweenadjacent fins Thus the decrease of the fin space reduces thedriven force which is needed for the heat flux to jump overthe fins

It is summarized from the above discussion that theflow behavior is dominated by the combined influences offin space and fin height which affects the heat transferrate ultimately The influences on heat transfer rate can beexplained with the energy gradient theory

53 Effect of Fin Space Two sets of numerical cases areselected to study the effect of fin space on flow instabilityor heat transfer rate In the first set of numerical cases thefin height is fixed at 45mm and the fin space decreasescharacterized by the increasing of the fin number In the

second set of numerical cases the fin height is fixed at85mm and the fin space decreases characterized by the samepattern in the first set Figures 9 and 10 show the temperaturecontours and contours of the value of 119870 for the first set ofnumerical cases described above in which all the fin heightsare given as 45mm Figures 11 and 12 show the temperaturecontours and contours of the value of 119870 for numerical casesin which all the fin heights are fixed at 85mm Table 2 lists theNusselt number of different numerical cases

Comparing Figures 9ndash12 and Table 2 some observationscan be obtained Firstly when the fin number is 2 thestrongest intensity of flow instability occurs which possessesthe most efficient heat transfer rate The heat flux focusesin the center of the cavity and it moves upward along thevertical center line of the cavity Once the largely extractedheat flux loses its stability and spreads randomly the intensityof flow instability is much stronger than any other patterns offlow instability Secondly Figures 9 and 10 indicate that theintensity of flow instability is very weak when the fin numberis 3 However the intensity difference of flow instability canbe neglected when the fin number is larger than 3 Thisresult can be validated by the data in Table 2 When thefin number is 3 the heat flux moves right by jumping overthe fins and focuses in the adjacent fins near the right wallHence little heat flux moves upward which results in a weakintensity of flow instability However the heat fluxes in allthe cavities move upward symmetrically which results in


K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)

Figure 10 Contours of the value of 119870 with a fixed fin height of 45mm and the fin space changes from 82mm to 232mm characterized bythe increase of fin number (119899) (a) 119899 = 2 (b) 119899 = 3 (c) 119899 = 4 (d) 119899 = 5 (e) 119899 = 6 (f) 119899 = 7 (g) 119899 = 8 and (h) 119899 = 9


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)


a negligible heat transfer rate when the fin number is morethan 3 Thirdly it is easy to find that the intensity of flowinstability changes slightly when the fin number is more than2 by comparing Figures 11 and 12 and Table 2 This can beattributed to the effect of fin space When the fin space islarge the convection between adjacent fins plays a leadingrole and it can trigger flow instability Meanwhile more heatflux focuses between adjacent fins and little heat flux movesupward thus the flow instability will be restricted When

the fin space is small the heat conduction between adjacentfins plays a leading role and it restricts flow instabilityAt the same time more heat flux moves upward whichresults in stronger flow instability The change of the finspace results in the change of specific weight of the heatconduction and heat convection between adjacent fins aswell as the change of the accumulation of heat flux betweenadjacent fins At last the intensity of flow instability changesslightly


K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)


Model-(A)-T temperaturecontours

(a)

Model-(B)-T temperaturecontours

(b)

Model-(C)-T temperaturecontours

(c)

Model-(D)-T temperaturecontours

(d)

Model-(A)-K K contours

(e)

Model-(B)-K K contours

(f)

Model-(C)-K K contours

(g)

Model-(D)-K K contours

(h)

Figure 13 Temperature contours and contours of the value of 119870 in different fin shape with a fixed fin space of 82mm

Consequently in addition to the validation of criteria offlow instability based on the energy gradientmethod anothertwo conclusions can be summarized as follows (1)The cavitypossesses the most efficient heat transfer rate when the finnumber is 2 (2) The change of the fin space results in thechange of the specific weight of heat conduction and heatconvection as well as the accumulation of heat flux betweenadjacent fins hence the heat transfer rate changes slightly

54 Effect of Fin Shape In this part the effect of the finshape on flow instability or heat transfer rate is discussedThe

straight fin arrays are chosen as the base group as model-(A)shown in Figure 13 The shapes of models-(BndashD) shown inFigure 13 are all different from each other and a fixed fin spaceof 26mm is chosen for all cases The fin arrays in model-(B)are curved for one piece in the same direction with a radius of70mm The fin arrays in model-(C) are curved for one piecewith a radius of 70mmwhich are fixed symmetricallyThefinarrays in model-(D) are curved for four pieces with a radiusof 10mm

Figures 13ndash15 show the temperature contours and con-tours of the value of119870when the fin space is 82mm 484mm



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)


and 295mm respectively Figure 16 shows the heat transferrate obtained with four different fin shapes and fin space ischaracterized by the fin number

It is easy to get the following results by comparing Figures13ndash15 to Figure 16 Firstly it is found in Figure 13 that thelargest intensity of flow instability occurs in model-(A) andthe intensity difference in other models can be neglectedThese results accord well with the data in Figure 16 Thisis attributed to the effect of fin shape which results in thedifferent form of flow instability Convection occurs mainly

in the center of the cavity and spreads around and producesa large intensity of flow instability However the heat fluxesin models-(BndashD) move upward symmetrically and producerelatively small flow instability Thus this result is validatedin Figure 16 which shows that the most efficient heat transferrate occurs in model-(A) when the fin space is 82mmSecondly it is found in Figure 14 that the flow instabilityin model-(C) is very weak and the intensity variation offlow instability in other models is very small This heattransfer rate difference is due to the bent direction of the fins


2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)

Figure 16 Nusselt number versus fin space characterized with finnumber in different fin shape

The symmetrically bent fins in model-(C) will restrict themovement of the heat flux above the fin arrays thus reducethe intensity of flow instability The fins in other models areeither curved in the same direction or are straight which hasa limited effect on the variation of the heat transfer modes(ie convection versus conduction)Thus the intensity differ-ence of flow instability is not distinct Comparing theseresults to the data in Figure 16 it is easy to make a conclusionthat the symmetrically bent fins can reduce the heat transferrate when the fin space is fixed at 484mm Thirdly it isobserved in Figure 15 that the intensity of flow instabilityin four different fin shapes is almost the same which isconfirmed in Figure 16 The heat convection in adjacentfins is very weak when the fin space is 295mm and itis not easy for the weak heat convection to trigger flowinstability

Some observations can be summarized as follows bycomparing Figures 13ndash15 to Figure 16 Firstly the model withstraight fins possesses the most efficient heat transfer rateSecondly the model with symmetrically bent fins can restrictheat transfer rate when the fin space is 484mm Thirdly theheat transfer rate is dominated by the fin space not by the finshape when the fin space is 295mm

6 Conclusions

In this paper the convection and heat transfer rate behaviorsof fin arrays are studied based on the criteria of flow instabilityderived from energy gradient method All the numericalprocedures are based on steady Navier-Stokes equations andBoussinesq approximation The effect of fin height fin spaceand fin shape on heat transfer rate and flow instability areexaminedThe physicalmechanism of flow instability and therelationship between flow instability and heat transfer rate areanalyzed Conclusions can be drawn as follows

(1) The areas of flow instabilities in temperature contoursand the regions of high value of 119870 in contours ofthe value of119870 accord well which reveals the physicalmechanism of flow instability

(2) The fin height and the fin space jointly dominate theflow behavior of the base flow and thus affect the heattransfer rate

(3) For a given fin height the change of the fin spaceresults in the change of the accumulation and move-ment of the heat flux between adjacent fins as wellas the relative magnitude of heat conduction andheat convection between adjacent fins hence the heattransfer rate changes slightly

(4) The effect of the fin shape on heat transfer is distinctwhen the fin space is large

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by the Science Foundation of ZhejiangSci-Tech University (ZSTU) under Grant no 11130032241201and the special major project of science and technologyof Zhejiang Province (no 2013C01139) The authors wouldlike to thank Associate Professor Chengwang Lei at TheUniversity of Sydney and Dr Feng Xu in Beijing JiaotongUniversity for the helpful discussionsThe authors also thankthe support fromHuadian Electric Power Research Institute

References

[1] K E Starner and H N McManus ldquoAn experimental investi-gation of free convection heat transfer from rectangular fin-arraysrdquo Journal of Heat Transfer vol 85 no 3 pp 273ndash278 1963

[2] F Harahap and H N McManus ldquoNatural convection heattransfer from horizontal rectangular fin arraysrdquo Journal of HeatTransfer vol 89 no 1 pp 32ndash38 1967

[3] C D Jones and L F Smith ldquoOptimum arrangement of rect-angular fins on horizontal surfaces for free convection heattransferrdquo Journal of Heat Transfer vol 92 no 1 pp 6ndash10 1970

[4] V Rammohan Rao and S P Venkateshan ldquoExperimentalstudy of freeconvection and radiation in horizontal fin arraysrdquoInternational Journal of Heat and Mass Transfer vol 39 no 4pp 779ndash789 1996

[5] H Yuncu and G Anbar ldquoAn experimental investigation onperformance of rectangular fins on a horizontal base in freeconvection heat transferrdquo Heat and Mass Transfer vol 33 no5-6 pp 507ndash514 1998

[6] M Mobedi and H Yuncu ldquoA three dimensional numericalstudy on natural convection heat transfer from short horizontalrectangular fin arrayrdquoHeat and Mass Transfer vol 39 no 4 pp267ndash275 2003

[7] E Arquis and M Rady ldquoStudy of natural convection heattransfer in a finned horizontal fluid layerrdquo International Journalof Thermal Sciences vol 44 no 1 pp 43ndash52 2005


[8] F Liu ldquoA fuzzy approach to the convective longitudinal fin arraydesignrdquo International Journal ofThermal Sciences vol 44 no 3pp 211ndash217 2005

[9] F Harahap H Lesmana and A S Dirgayasa ldquoMeasurementsof heat dissipation from miniaturized vertical rectangular finarrays under dominant natural convection conditionsrdquo Heatand Mass Transfer vol 42 no 11 pp 1025ndash1036 2006

[10] F Harahap H Lesmana and P L Sambegoro ldquoConcurrentcalorimetric and interferometric studies of steady-state naturalconvection from miniaturized horizontal single plate-fin sys-tems and plate-fin arraysrdquo Heat and Mass Transfer vol 46 no8-9 pp 929ndash942 2010

[11] M Dogan and M Sivrioglu ldquoExperimental investigation ofmixed convection heat transfer from longitudinal fins in ahorizontal rectangular channelrdquo International Journal of Heatand Mass Transfer vol 53 no 9-10 pp 2149ndash2158 2010

[12] H Azarkish S M H Sarvari and A Behzadmehr ldquoOptimumdesign of a longitudinal fin array with convection and radiationheat transfer using a genetic algorithmrdquo International Journal ofThermal Sciences vol 49 no 11 pp 2222ndash2229 2010

[13] A Giri and B Das ldquoA numerical study of entry regionlaminar mixed convection over shrouded vertical fin arraysrdquoInternational Journal of Thermal Sciences vol 60 pp 212ndash2242012

[14] S Wong and G Huang ldquoParametric study on the dynamicbehavior of natural convection from horizontal rectangular finarraysrdquo International Journal of Heat andMass Transfer vol 60no 1 pp 334ndash342 2013

[15] J Patterson and J Imberger ldquoUnsteady natural convection in arectangular cavityrdquo Journal of Fluid Mechanics vol 100 no 1pp 65ndash86 1980

[16] H S Dou and N Phan-Thien ldquoInstability of fluid materialsystemsrdquo in Proceedings of the 15th Australasian Fluid Mechan-ics Conference The University of Sydney Sydney AustraliaDecember 2004

[17] H S Dou ldquoViscous instability of inflectional velocity profilerdquoin Proceedings of the 4th International Conference on FluidMechanics Tsinghua University Press amp Springer 2004

[18] H Dou ldquoMechanism of flow instability and transition toturbulencerdquo International Journal of Non-LinearMechanics vol41 no 4 pp 512ndash517 2006

[19] H S Dou ldquoThree important theorems for flow stabilityrdquoin Proceedings of the 5th International Conference on FluidMechanics Tsinghua University Press amp Springer 2007

[20] H Dou and B C Khoo ldquoMechanism of wall turbulence inboundary layer flowrdquo Modern Physics Letters B vol 23 no 3pp 457ndash460 2009

[21] H Dou and B C Khoo ldquoCriteria of turbulent transition inparallel flowsrdquoModern Physics Letters B vol 24 no 13 pp 1437ndash1440 2010

[22] H Dou and B C Khoo ldquoInvestigation of turbulent transition inplane couette flows using energy gradient methodrdquoAdvances inApplied Mathematics and Mechanics vol 3 no 2 pp 165ndash1802011

[23] H Dou B C Khoo and K S Yeo ldquoEnergy loss distributionin the plane Couette flow and the Taylor-Couette flow betweenconcentric rotating cylindersrdquo International Journal of ThermalSciences vol 46 no 3 pp 262ndash275 2007

[24] H Dou B C Khoo andK S Yeo ldquoInstability of Taylor-Couetteflow between concentric rotating cylindersrdquo International Jour-nal of Thermal Sciences vol 47 no 11 pp 1422ndash1435 2008

[25] H Dou G Jiang and C Lei ldquoNumerical simulation andstability study of natural convection in an inclined rectangularcavityrdquoMathematical Problems in Engineering vol 2013 ArticleID 198695 12 pages 2013

[26] Y Gao A heat transfer study on the horizontal rectangular finarrays for the LED street lamp GH-4 [MS thesis] ChongqingUniversity 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with a fixed fin height and fin thickness They found thatthere was an optimum fin spacing which was not related tothe temperature difference However optimum fin spacingwas inversely proportional with the fin height Mobedi andYuncu [6] numerically investigated the steady state naturalconvection heat transfer in longitudinally short rectangularfin arrays on a horizontal base They observed two types offlow patterns For the fin arrays with narrow fin spacingair could only enter into the channel from the end regionsHowever for the fin arrays with wide fin spacing the air wasalso entrained into the channel from the region between thefins turned 180 degrees at the base and thenmoved up alongthe fin surface while it flowed into the central part of thechannel

More recently lots of researchers tried to enhance heattransfer rate of fin arrays using fin arrays Arquis and Rady[7] investigated natural convection heat transfer and fluidflow characteristics from a horizontal fluid layer with finnedbottom surface and observed that the number of convectioncells between two adjacent fins is a function of the values ofthe fin height and Rayleigh number Liu [8] considered anoptimum design problem for the longitudinal fin arrays witha constant heat transfer coefficient in a fuzzy environmentwhere the grid requirements to strictly satisfy the total finvolume and array width and maximize the heat dissipationrate are softened Harahap et al [9] conducted some exper-iments to investigate the effects of miniaturizing the baseplate dimensions of vertically based straight rectangular finarrays on the steady state heat dissipation performance underdominant natural convection conditions They found thatthe relevant correlations proposed for large fin arrays werenot applicable to the experimental data obtained from theminiaturized vertical rectangular fin arrays SubsequentlyHarahap et al [10] conducted concurrent calorimetric andinterferometric measurements to investigate the effect thatthe reduction of the base plate dimensions has on the steadystate performance of the rate of natural convection heattransfer fromminiaturized horizontal single plate-fin systemsand plate-fin arrays Their conclusions suggested that the finheight119871 and the fin number are the prime geometric variablesfor generalization Dogan and Sivrioglu [11] experimentallyinvestigated mixed convection heat transfer and the resultsobtained showed that the optimum fin spacing which yieldedthe maximum heat transfer is 119878 = 8-9mm and the optimumfin spacing depends on the value of Ra Azarkish et al[12] used a modified genetic algorithm to maximize theobjective function which is defined as the net heat transferrate from the fin surface for a given height Their resultsshow that the number of the fins is not affected by the finprofile but the heat transfer enhancement for the arrayswith optimum fin profile is about 1ndash3 percent more thanthat for the arrays with conventional fin profiles Giri andDas [13] numerically performed laminar mixed convectionover shrouded vertical rectangular fin arrays attached to avertical base They found that the drop in pressure defect forforced convection induced velocity formixed convection andoverall Nusselt number for mixed convection are correlatedwell with governing parameters of the considered problemWong and Huang [14] made some three-dimensional (3D)

numerical simulations to investigate the effect of fin parame-ters on dynamic natural convection from long horizontal finarraysThey observed that the optimumfin spacing decreasessignificantly with the fin height and increases slightly with thefin length Furthermore their observations agreed well withthe results reported in the literature

All the above experimental and numerical researchfocused on optimum design of the fin arrays in order toimprove heat transfer rate Almost all the related factorsresult in laminar convection involving low values of heattransfer coefficients However it is well known that the heattransfer rate will be prompted when the base flow loses itsstability and flows in a turbulent manner [15] Althoughsome optimum fin arrays design can obtain a satisfying heattransfer rate which resulted from flow instability the physicalmechanism of flow instability of natural convection is stillnot fully understood Recently Dou et al [16ndash24] suggestedan energy gradient method which can reasonably reveal thephysical mechanism of flow instability Dou and Phan-Thien[16] described the rules of fluid material stability from theviewpoint of energy field They claimed that the instability ofnatural convection could not be resolved by Newtonrsquos threelaws for the reason that a material system moving in somecases is not simply due to the role of forces This approachexplains the mechanism of flow instability from physics andderives the criteria of turbulence transition Accordinglythis method does not attribute Rayleigh-Benard problem toforces but to energy gradient It postulates that when the fluidis placed on a horizontal plate and is heated from below thefluid density in the bottom becomes low which leads to anenergy gradient 120597119864120597119910 gt 0 along 119910-coordinate Only when120597119864120597119910 is larger than a critical value will the flow becomeunstable and then fluid cells of vorticities will be formedMore recently Dou et al [25] applied the energy gradientmethod to natural convection and the results from numericalsimulations accord well with those predicted based on thecriteria originated from energy gradient method

This study is focused on the research of effects of fin arraysparameters on convection heat transfer coefficient Then theenergy gradient method is employed to reveal the physicalmechanism of flow instability and explain the reason why theoptimum fin arrays can result in better heat transfer rate

2 Computational Geometry andNumerical Procedures

21 Computational Geometry The computational geometryis shown in Figure 1Here the geometry is simplified from the3D solid of GH-4 ribbed radiator model [26] The simplifiedcavity in this study is a two-dimensional (2D) square inwhich the length of the square cavity is 250mm and theorigin of the coordinates is at the lower left corner of thecavity The fin arrays are fixed at the hot bottom of the cavitywith an equal distance Here 119878 is the fin space 119867 is the finheight and 119867 and 119878 are variable In addition 119891

119905means the

thickness of the fin arrays which is fixed at 2mm in this studyand can be neglected by comparing to the fin height andthe length of the cavity while 119887

119905means the thickness of the

bottom plate which is equal to 3mm


Sym Sym

S

H

Cold top

Hotbottom

ft

bt




120597119906

120597119909+

120597V120597119910

= 0

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588

120597119901

120597119909+ 120592(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

= minus1

120588

120597119901

120597119910+ 120592(

1205972V

1205971199092+

1205972V

1205971199102) + 119892120573 (119879 minus 119879

0)

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910= 119896(

1205972119879

1205971199092+

1205972119879

1205971199102)

1205972119879

1205971199092+

1205972119879

1205971199102= 0

(1)





Nu =1

119871int

119871

0

(

10038161003816100381610038161205971198791205971199101003816100381610038161003816119910minus119871

Δ119879119871)119889119909 =

1

Δ119879int

119871

0

10038161003816100381610038161003816100381610038161003816

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910minus119871

119889119909 (2)












Grid (a)

(a)

Grid (b)

(b)

Grid (c)

(c)



119865 =Δ119864

Δ119867=

(120597119864120597119899) (2119860120587)

(120597119867119904120597119904) (120587120596

119889) 119906

=2

1205872119870

119860120596119889

119906=

2

1205872119870V1015840119898

119906lt Const

(3)

where

119870 =120597119864120597119899

120597119867119904120597119904

(4)











119870 = radic(120597119864

120597119909)

2

+ (120597119864

120597119910)

2

(5)



0 500 1000 1500 2000 2500 3000 3500 4000300

305

310

315

320

325

330

Tem

pera

ture

(K)

Iterations



120597E120597y

120597E120597x

y

K

xo




0120597119909)2+ (1205971199010120597119910)2


pressure
























(a)



(b)



(c)



(d)



(e)




(a)



(b)



(c)



(d)



(e)




Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sym Sym

S

H

Cold top

Hotbottom

ft

bt




120597119906

120597119909+

120597V120597119910

= 0

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588

120597119901

120597119909+ 120592(

1205972119906

1205971199092+

1205972119906

1205971199102)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

= minus1

120588

120597119901

120597119910+ 120592(

1205972V

1205971199092+

1205972V

1205971199102) + 119892120573 (119879 minus 119879

0)

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910= 119896(

1205972119879

1205971199092+

1205972119879

1205971199102)

1205972119879

1205971199092+

1205972119879

1205971199102= 0

(1)





Nu =1

119871int

119871

0

(

10038161003816100381610038161205971198791205971199101003816100381610038161003816119910minus119871

Δ119879119871)119889119909 =

1

Δ119879int

119871

0

10038161003816100381610038161003816100381610038161003816

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910minus119871

119889119909 (2)












Grid (a)

(a)

Grid (b)

(b)

Grid (c)

(c)



119865 =Δ119864

Δ119867=

(120597119864120597119899) (2119860120587)

(120597119867119904120597119904) (120587120596

119889) 119906

=2

1205872119870

119860120596119889

119906=

2

1205872119870V1015840119898

119906lt Const

(3)

where

119870 =120597119864120597119899

120597119867119904120597119904

(4)











119870 = radic(120597119864

120597119909)

2

+ (120597119864

120597119910)

2

(5)



0 500 1000 1500 2000 2500 3000 3500 4000300

305

310

315

320

325

330

Tem

pera

ture

(K)

Iterations



120597E120597y

120597E120597x

y

K

xo




0120597119909)2+ (1205971199010120597119910)2


pressure
























(a)



(b)



(c)



(d)



(e)




(a)



(b)



(c)



(d)



(e)




Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Grid (a)

(a)

Grid (b)

(b)

Grid (c)

(c)



119865 =Δ119864

Δ119867=

(120597119864120597119899) (2119860120587)

(120597119867119904120597119904) (120587120596

119889) 119906

=2

1205872119870

119860120596119889

119906=

2

1205872119870V1015840119898

119906lt Const

(3)

where

119870 =120597119864120597119899

120597119867119904120597119904

(4)











119870 = radic(120597119864

120597119909)

2

+ (120597119864

120597119910)

2

(5)



0 500 1000 1500 2000 2500 3000 3500 4000300

305

310

315

320

325

330

Tem

pera

ture

(K)

Iterations



120597E120597y

120597E120597x

y

K

xo




0120597119909)2+ (1205971199010120597119910)2


pressure
























(a)



(b)



(c)



(d)



(e)




(a)



(b)



(c)



(d)



(e)




Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500 2000 2500 3000 3500 4000300

305

310

315

320

325

330

Tem

pera

ture

(K)

Iterations



120597E120597y

120597E120597x

y

K

xo




0120597119909)2+ (1205971199010120597119910)2


pressure
























(a)



(b)



(c)



(d)



(e)




(a)



(b)



(c)



(d)



(e)




Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















(a)



(b)



(c)



(d)



(e)




(a)



(b)



(c)



(d)



(e)




Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


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(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


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(h)








(a)


(b)


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(a)


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2 3 4 5 6 7 8 9

225

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235

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Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Nu 119878 = 82 119878 = 61 119878 = 484 119878 = 40 119878 = 34 119878 = 295 119878 = 26 119878 = 232

119867 = 45mm 276 197 237 227 230 232 238 242119867 = 85mm 295 236 238 228 231 235 239 243


(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


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(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)





K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










K K contours

(a)

K K contours

(b)

K K contours

(c)

K K contours

(d)

K K contours

(e)

K K contours

(f)

K K contours

(g)

K K contours

(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)








(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)



(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)






2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 3 4 5 6 7 8 9

225

230

235

240

245

250

255

Nus

selt

num

ber

Fin number

Model-(A)Model-(B)

Model-(C)Model-(D)




6 Conclusions








Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









a numerical study of natural convection heat transfer in fin

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