Discussion on the Effect of a Fluid Domain Around Fins and Grid

Discretization in Buoyancy-driven Convection

Sven De Schampheleire, Kathleen De Kerpel, Bernd Ameel, Özer Bağcı, Michel De Paepe

Department of Flow, Heat and Combustion Mechanics, Ghent University – UGent, Ghent,

Belgium

Address correspondence to Sven De Schampheleire, Department of Flow, Heat and Combustion Mechanics, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium. E-mail: Sven.DeSchampheleire@ugent.be.

Phone Number: +32 9 264 33 55. Fax Number: +32 9 264 35 75.

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Abstract

In the numerical study of heat sinks it is known that a sufficient amount of fluid domain should be

added at each side of the heat sink. However, the question in this context is: what can be defined

as sufficiently far away from the heat sink? Different authors use different sizes of the

computational domain around the heat sink. In this work the impact of the size and location of

the fluid domain on the calculated heat transfer coefficient is investigated. Three fin row types

are studied: a rectangular, an interrupted rectangular and an inverted triangular fin row. First,

the influence of adding fluid domain to the sides of the heat sink is studied. A large decrease of

the heat transfer coefficient on both sides and bottom is observed. Next, the influence of adding

fluid domain on both the top and the sides is studied. For the rectangular fins the impact on the

lumped heat transfer coefficient is +12% compared to the case without any fluid domain added.

For the inverted triangular fin shape no net effect is observed on the lumped heat transfer

coefficient. So the impact of adding fluid domain depends on fin shape that is investigated. For

the sides only a small amount of fluid needs to be added, while for the fluid domain on top of the

heat sink, 130% of the equivalent fin height is found as a good option to simulate the fin in

computational fluid dynamics.

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INTRODUCTION

Focus on buoyancy-driven heat transfer

Electronic components are omnipresent in everyday products e.g. in cars, computers and

power converters. It is important to reduce the electrical energy used by the electronic

components and limit the chance of electronics failure. The main causes of electronics failure

are elevated operating temperatures, over-voltage and moisture [1]. The failures caused by

elevated operating temperatures can be avoided by proper cooling of the electronic components.

The main failure mechanics caused by elevated operating temperatures are diffusion in the

semiconductor materials, creep of the bonding materials and chemical reactions [2]. To keep the

temperature level of these components low, one option is to use heat sinks in buoyancy-driven

heat transfer. In this heat transfer mode, the air flow is caused by buoyancy forces. A heat sink

generally consists of a substrate with fins to increase the heat transferring surface area. In open

literature many studies on different fin shapes for heat sinks in natural convection are available.

One heat sink in buoyancy-driven heat transfer outperforms another when for the same energy

dissipation the flow resistance is lower, in this case the temperature of the heat sink will be

lower.

The flow patterns induced by heating in natural convection can be understood intuitively.

Hot fluid generally has a lower density and therefore rises (flow against gravity) while cold fluid

with higher density tends to move downwards (flow with gravity). Heat sinks that operate in

buoyancy-driven heat transfer can generally only be used at relatively low heat transfer rates as

the base plate temperature typically has to be kept below 70°C. For higher heat transfer rates heat

sinks in forced convection can be used. The fin shapes for the latter type of heat sink are more

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complex to manufacture e.g.: the bonded finned, the single finned and/or the skived finned heat

sinks [3]. Hence, heat sinks made to operate in natural convection imply a lower investment,

maintenance and operating costs.

This work focuses on low-end electronics cooling with base plate temperatures lower

than 100°C.

Work done towards heat sinks in buoyancy-driven heat transfer

A large amount of work has been done towards heat sinks in buoyancy-driven heat

transfer. In fact, studies found in open literature mainly deals with the following three methods:

analytical [4], experimental [5] and numerical [6].

To study buoyancy-driven heat transfer analytically, certain assumptions are made in

order to simplify the solution and clearly define the problem. Harper and Brown [7] were the

first to state assumptions to solve a one-dimensional steady state case. These assumptions are

extended by Gardner [8]. However, these assumptions do not correspond to real fin conditions,

which can lead to oversimplification. In several cases 2-D conduction effects can become

significant and a 2-D solution is necessary e.g. in case of varying heat transfer coefficients on fin

faces, non-uniform base temperature, space-varying heat transfer coefficient, anisotropic

materials [9].

One of the first experimental studies on buoyancy-driven heat transfer between parallel

plates was performed by Elenbaas [10] in 1942. Most research found in open literature on this

topic is quite diffuse, due to the many parameters involved in buoyancy-driven heat transfer:

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a) Characterization of the heat sink geometry. The heat sink material has to be reported.

Another important parameter is the thickness of the fin itself, which is not always

reported [11].

b) Characterization of the test rig. In order for the experiments to be repeatable, the test rig

has to be described properly. A discussion on this topic can be found in De

Schampheleire et al. [12].

c) Emissivity of the material. Most of the time, the authors are neglecting radiation [13].

They assume that radiation is only 5% of the total heat transfer rate in finned surfaces for

a temperature difference of about 40K [14]. However, Sahray et al. [15] found that for

dense pin fin heat sinks the radiative contributions were up to 45% of the total heat

transfer rate. The same was observed by Sparrow and Vemuri [16] in their study on the

orientation and radiation in pin fins. They found that fractional contributions of radiation

to the combined heat transfer were generally in the 25%-40% range with larger

contributions for the smaller temperature differences between base plate and

environment.

Ideally, the fin material has to be polished to reach an emissivity value around 0%. This

is the only possible way to compare correlations with experimental results or other

correlations. However, this is practically not feasible. Instead one should measure the

emissivity and report correlations without the radiative influence [17].

d) Inclination angle. The impact of the inclination angle is very dependent on the fin shape

[18].

e) Enclosure. The enclosure in which the heat sink is placed of course has a big impact on

the performance [12].

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All these dependencies make it quite difficult to optimize and test one specific heat sink

for all these parameters. To limit the amount of necessary experiments, some work is also done

on numerical simulation of heat sinks in natural heat transfer with the help of Computational

Fluid Dynamics (CFD). In any CFD calculation, it is necessary to choose a computational

domain around the heat sink. Much like placing an enclosure around the heat sink impacts the

obtained experimental results, it can be expected that the choice of the computation domain will

also influence the obtained simulation results. As CFD is the topic of current paper, the

methodology for CFD calculations is discussed further in the following section.

NUMERICAL WORK ON HEAT SINKS IN BUOYANCY-DRIVEN HEAT TRANSFER

This work focusses on applications with low Rayleigh numbers, hence turbulent flows

will not be considered here. Due to this focus, only studies considering Rayleigh numbers far

below the critical Rayleigh number of 109 [19] will be discussed in this section. The Rayleigh

number is defined by

RaL=gL3 β (T b−T o)

ν α (1)

The characteristic length L in Eq. (1) is dependent on the fin shape and the inclination

angle under which it was tested. Table 1 gives a non-exhaustive overview on some of the

numerical work done on heat sinks with simple fin shapes in buoyancy-driven heat transfer and

low Rayleigh numbers. It is clear that different authors use different sizes of the computational

domain. The purpose of Table 1 is to give an overview of the possible sizes; so other authors

who also use extra fluid domains are e.g. [20-22] amongst many others.

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Ahmadi et al. [23] studied natural heat transfer in vertically-mounted rectangular

interrupted fins. The authors used a 2D numerical model. They simulate only one fin row and use

symmetrical boundary conditions. Extra fluid domain (or a calming section as the authors call it)

is implemented before the interrupted fins, however, no fluid domain is implemented after the

last fin. No indication is given how wide this extra fluid domain actually is. All calculations

where performed for steady state with no turbulence model. The fluid is modelled through the

Boussinesq approximation. According to Ref. [24] this approximation can only be used when the

changes in actual density are small, specifically the Boussinesq approximation is only valid when

β ( T−T0 )≪1. In the work of Ahmadi et al. [23] the studied temperature difference varies

between 50K and 90K. For an operational temperature of 20°C, the outer values for β range

between 3 × 10-3 1/K and 2.68 × 10-3 1/K [25]. Consequently β ( T−T0 ) varies between 0.15 and

0.24.

Dialameh et al. [14] studied rectangular fins of a short length. The thickness of the fins

varied between 3 mm and 7 mm, while the length of the heat sink was always smaller than 50

mm. Only half of one fin row was simulated in 3D, and only one fin was taken into account, the

other side was open. The authors also added an extra fluid domain at the side and on top of the

heat sink. Sideways this extra fluid domain measured ¾ times the length of the fin, while on top

9 times the fin height was added. According to the authors, radiation was negligible. The fluid

properties were assumed to be constant, except for the density, which was a function of the

temperature.

Shen et al. [26] studied rectangular fins in 3D by simulated the half of the complete heat

sink (other part is symmetrical). A calming section was provided at the sides, top and bottom of

the heat sink. One time the length of the fin L was added to the side and 5 times the height of the

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fin H was added as an extra fluid domain on top of the heat sink. No turbulence model was used

and radiation was taken into account. Air was modelled as an incompressible ideal gas.

Mehrtash and Tari [27] and Tari and Mehrtash [28] simulated a complete heat sink. The

authors placed the heat sink and insulating material in a square box of 3 m. At least 6 times the

length of the heat sink was added sideways of the heat sink, while ± 116 times the height of the

heat sink was added as an extra fluid domain on top. A non-conformal mesh of 2.8M cells was

used, as the authors mention this in the paper, but it is unclear on how the mesh itself was

generated. For their calculations they used a zero order turbulence model. This is a turbulence

model based on the mixing length hypothesis [24]. Air was modelled as an ideal gas.

Dogan et al. [11] compared a rectangular shape with five different fin shapes. For this,

the authors simulated one fin row with a fin at each side similar to the geometry shown in Figure

1. This approach was different in comparison with the work by Dialameh et al. [14] where only

one fin is taken into account and the other end was open. No extra fluid domain was provided

before or after the heat sink, as can be seen in Figure 1. The finest discretisation used consisted

of 1 mm cells. No turbulence model was used and radiation was included.

As can be seen from this detailed review even in most recent papers there is quite some

ambiguities and variability on how heat sinks or fin rows should be studied numerically with

CFD. In this paper, the case of Dogan et al. [11] will be taken as a starting point. The value of

this work is however not limited to the work of Dogan et al. [11]. In this relatively recent

publication several fin shapes are compared. Similar fin shapes will be considered in this work.

In open literature it is stated that the fluid domain has to be ‘far’ enough from the heat sink itself,

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similar to the Saint-Venant principle in mechanics [29]. However, the question here is: what is

far enough? Over different sources in open literature there is hardly any consensus. As shown in

Table 1 the extra fluid domain that is added to the side of the heat sink varies between 0 and 6

times the length of the fin, while the amount of extra fluid domain that is added on top of the

heat sink varies between 0 and 116 times of the height of the fin. In this paper, the influence of

the fluid domain on the thermal results will be discussed and the dependency of this parameters

on the studied fin shape. In essence the influence of the location of the boundary conditions is

discussed. Three different fin shapes will be studied. Only the horizontal orientation will be

discussed.

In addition to the fluid domain around the fin structure, also the effect of grid

discretization on the uncertainty of the simulations are discussed in this work.

NUMERICAL CALCULATIONS

Used geometry

The used reference geometry is based on the case simulated by Dogan et al. [11] except

that in this paper only half of the heat sink is simulated. Three different fin shapes were tested.

The fin shapes are shown in Figure 2. In this figure, only half of the fin rows are shown, as their

symmetry boundary condition is already implemented in this figure. The symmetry plane is

indicated by a dashed rectangle in Figure 2. The other boundary conditions will be explained in

the next Section. In this work, the considered fin shapes are:

a) A rectangular fin shape (Fig. 2(a)) with a length (L) of 127 mm, a fin spacing (S) of 6.4

mm and a height (H) of 38 mm.

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b) An interrupted fin shape (Fig. 2(b)), all characteristics are the same as for the rectangular

fin shape, except that in the middle of the fin 7 mm is cut away. It seems valuable to

study this interrupted fin design since the simulations by Dogan et al. [11] show that the

temperature in the middle of this fin was approximately the base plate temperature.

c) An inverted triangular fin shape (Fig. 2(c)) with a length (L) of 127 mm, a fin spacing (S)

of 6.4 mm and a height (H) of 76 mm.

The heat sink dimensions discussed in the work of Dogan et al. [11] are similar to the

heat sinks dimensions discussed in the overview given in the paper of Tari and Mehrtash [28].

However, Dogan et al. [11] did neither specify the thickness of the base plate, nor did

they specify the thickness of the fin itself. Although the thickness of the base plate is not of high

importance, the thickness of the fin is: the boundary condition at the base plate is expressed as a

heat flux in W/m². Hence, if the fin thickness increases, the power sent to the base plate increases

accordingly. Comparing the temperature contours provided in Dogan et al. [11] with the

temperatures in our simulations, a good match was found for a fin thickness of 3 mm. Therefore,

a fin thickness of 3 mm will be assumed in this work. The thickness of the base plate is taken to

be 2 mm [28].

In this work the influence of the fluid domain on the thermal results will be assessed.

More specifically, the variation of the average heat transfer coefficient on each of the fin surfaces

will be determined. For all of the tested fin shapes, 8 different fluid domains were generated. In

Table 2 the simulated geometries for each fin shape are shown. ‘Side 5 mm’ for example means

that 5 mm of fluid domain is added to the side of the fin shape, while ‘Top 10 mm’ means that 10

mm of fluid domain is added to the top of the fin shape. The geometry ‘Mesh-top20mm’ means a

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fin row with 20 mm of fluid domain on each of the sides and 20 mm of fluid domain on top of

the fin. In total 9 different meshes are studied for each of the studied fin shapes. For the

rectangular fin shape, geometry ‘Mesh-top20mm’ is shown in Figure 3.

The choice of mesh and flow domain which is added to the sides is partly based on the

computational power that is needed to calculate the results. From the case ‘Mesh-top50mm’ on,

the authors have decided to keep the fluid domain to the sides at 20 mm. However, test cases

show that adding 30 mm and 50 mm to the sides of the ‘Mesh-top50mm’ case does not give rise

to different results compared to the case with 20 mm at the sides. Hence, in order to limit the

necessary computational power, only 20 mm is added to the sides. Likewise, increasing the

height of the fluid domain at the top further to 200 mm did not lead to different results from the

100 mm case.

In Figure 4, an illustration is made of the fin surfaces that will be discussed in this work.

The illustration is made for the rectangular fin shape. However, for the other fin designs similar

illustrations can be made.

Governing equations and boundary conditions

For each cell the mass (Eq. (2)), momentum (Eq. (3)) and energy equation (Eq. (4)) for

the fluid are solved under steady-state condition together with the energy conduction equation

for the solid (Eq. (5)). For the momentum term this is the so-called incompressible Navier-Stokes

equation in convective form [30].

∇ ∙ ( ρ v⃗ )=0 (2)

( v⃗ ∙∇ ) v⃗−ν∇2 v⃗=−1ρ0

∇ p+ g⃗ (3)

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v⃗∇ ( ρ0 T )= μPr

∇ ² T (4)

∇2T=0 (5)

Next to heat transfer and conduction, radiative heat transfer is also taken into account, as

was also the case for Ahmadi et al. [23], Shen et al. [26], Mehrtash and Tari [27], Tari and

Mehrtash [28] and Dogan et al. [11]. References [11, 27-28] are using the surface to surface

(S2S) model for radiation in the CFD package by Ansys®. This is also the model that will be

used in this work. The main assumption in this model is that any absorption, emission or

scattering of radiation by the fluid can be ignored. Therefore, only the energy exchange from one

surface to another has to be taken into account. This energy exchange depends on the sizes,

separation distance and orientation. This dependence is accounted for by a function of the

geometry: the view factor. This view factor is calculated automatically in the CFD software. The

used radiation model assumes all surfaces to be gray and diffuse, where the emissivity and the

absorptivity of a gray surface are independent of the wavelength. The emissivity was assumed to

be 1 in this work for sake of simplicity.

Furthermore, the pressure-velocity coupling in this work is done with the simple-

algorithm, while momentum and energy discretization is done second order upwind. Air is

modelled as an incompressible ideal gas. Therefore the operating density (ρ0) has to be given.

This operating density is taken at environment temperature (20°C). When the Boussinesq

approximation is not used, the operating density ρ0 appears in the body-force term in the

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momentum equations as ( ρ−ρ0 ) g, where ρ is calculated via the incompressible ideal gas law:

ρ=pop

RM w

T . In this equation, the operation pressure is held at its default value (101325 Pa).

In Figure 3 the applied boundary conditions are illustrated for a rectangular fin row. For

the two other fin shapes considered, the boundary conditions are exactly the same. Symmetrical

boundary conditions are used for surfaces ABCD, BCGF and ADHE. Surface EFIJ is assumed to

be adiabatic. ABJI is the heated surface. Unless otherwise stated, the boundary condition at the

heated surface is a fixed flux of 2250 W/m², in all the studied cases. Also note that the area of the

base plate is the same: 0.127 ∙0.0094 m ², in all cases. The power that is sent to the fin row is thus:

2.69W. This is exactly the same boundary condition as used by Dogan et al. [11]. Surface HGEF

is a so-called free surface where the gauge total pressure excluding hydrostatic pressure

differences is set to 0 Pa. For surface DCHG the gauge static pressure excluding hydrostatic

pressure differences is fixed at 0 Pa.

Mesh and uncertainty analysis

In order to be certain that the impact of changing the size of the computational domain

does not depend on the cell size, a grid discretization study is performed.

In this work the Roache’s grid convergence index (GCI) is used to obtain an estimate of

the grid discretization error [31]. The finest discretization tested has the following specifications:

Discretization of the thickness of the fin material: 0.09 mm per cell

Discretization of the fin height and adjacent fluid domain: 0.24 mm per cell

Discretization of the fin length and adjacent fluid domain: 0.25 mm per cell

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Discretization of the width of the fluid domain (fin spacing): 0.08 mm/cell

The coarse grid consists of cells which are a factor of 2 larger in each direction. The

results for the average heat transfer coefficient for both fin faces of the grids used are indicated

in Table 3. The uncertainty when calculating with the coarse grid is given by Eq. (6), while the

uncertainty for simulations with the finest grid can be determined through Eq. (7) [32]. r and p in

both equations stand for factor of refinement ( = 2) and order of convergence (also, = 2),

respectively. Since the order of convergence was not calculated from a third result but set equal

to the order of the numerical scheme, Roache [32] recommends using a safety factor of 3.

The largest relative uncertainty on the heat transfer coefficient is found on the right and

left side of the fin. When calculating with the finest grid the GCI is 1.8%, while for the coarse

grid this is 7.1%.

GCI coarse=3 ∙hcoarse−h fine

h fine∙ r p

r p−1 (6)

GCI fine=3 ∙hcoarse−h fine

h fine∙ 1

r p−1 (7)

The different fluid domains that are added on the sides and at the top of the simulated fin

shape are also discretized with the same discretization as mentioned above. There is an exception

for the larger fluid domains placed at the top of the heat sink, namely: ‘Mesh-50mmtop’ and

‘Mesh-100mmtop’. For these fluid domains triangular cells with a growth rate of 1.05 will be

used. In the remainder of this work the finest discretization will always be used.

RESULTS AND DISCUSSION

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In Figures 5 to 8 the following heat transfer coefficients are shown for the different meshes as

discussed in Table 2. The calculation of these heat transfer coefficients is done automatically in

the CFD software:

hsides. This is the heat transfer coefficient calculated as an area-averaged value on the

internal sides of the fin row (left and right) as 1A ∫

A

❑

hsides,i dA.

hbottom. This is the heat transfer coefficient calculated as an area-averaged value on the

bottom side of the fin row as 1A ∫

A

❑

hbottom , idA .

hwithradiation=Qtot

Afins ∙(T hot−T env ). This heat transfer coefficient is an average heat transfer

coefficient over all the available fin surfaces. If a fluid domain is added to e.g. the sides

of the fin row, the heat transfer coefficient of the frontal fins will also be taken into

account. Therefore hwithradiation is in fact a lumped heat transfer coefficient.

hwithradiation , lim ¿¿. This is the heat transfer coefficient only for the left, right and bottom side

of the fin, independent of which mesh (Table 2) being studied. This heat transfer

coefficient is an area-averaged value of hsides and hbottom.

All the results with their percentage difference compared to the base case (Mesh-Ref) are

reported in Table A.1 in the Appendix. First of all, the largest influence of the domain is

observed when the rectangular fin shape is compared to the inverted triangular one. The results

for the interrupted rectangular fin shape are very similar to those for the rectangular fin shape.

Locally the flow resistance in the middle of the interrupted fin is decreased due to the

interruption. However, as a single chimney pattern is observed in both the rectangular case and

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the interrupted case, no extra fluid will be drawn to the middle of the interrupted fin.

Furthermore, the region where the fin is interrupted has a symmetric boundary condition, this

makes it more difficult for air to penetrate from above the heat sink.

As can be seen from Figures 5 to 8, an increase in the height of the fluid domain on top of

the fin from 50 mm to 100 mm does not have any significant influence on the heat transfer

coefficients (the uncertainty on the fine grid is 1.77%, as shown in Table 3 and discussed in

Section ‘Mesh and Uncertainty analysis’). Higher and/or wider fluid domains result in no further

significant changes to the heat transfer coefficient. The results for a 200 mm high fluid domain

are reported in Table A.1 and depicted in Figures 5 to 8, however, they will not be discussed

further in this work as the boundary conditions for this grid are slightly different. When the fluid

height is increased, the unsteadiness in the flow will also increase. This makes it impossible for

the simulation to reach steady state, unless the fluid viscosity is increased. To be able to simulate

this case, the fluid viscosity was increased linearly with a factor 10 starting from a fluid height of

100 mm (well above the fluid motions of interest). By increasing the viscosity over the height,

the turbulences are damped and the solution converges.

First the influence of adding a fluid domain to the sides of the fin row on the previously

defined heat transfer coefficients will be discussed. In a latter Section the influence of adding a

fluid domain to both the sides and top of the fin row on the previously defined heat transfer

coefficients will be determined.

Effects of adding fluid at sides of fin row

Adding fluid domain at the sides of the fin material has a large impact on hsides and hbottom

as can be seen from Figures 5 and 6. The heat transfer coefficient on both left, right and bottom

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side of the fin row decreases by adding extra fluid domain. The heat transfer coefficient hsides

decreases by 18% for the rectangular and interrupted fins. The impact for the inverted triangular

fin is approximately the same: a relative decrease of around 20%. For the heat transfer

coefficient at the bottom side of the fin row, hdown, the impact is even larger: a decrease of around

28% is observed, independent of the type of fin row that is simulated. From Figure 9, for the

rectangular fin row without any fluid domain added, it can be observed that the velocity profile

at the inlet of the fin row (for the rectangular fin) is a single chimney pattern. This means that the

flow is drawn from one side and pushed to the top as can be seen from the streamlines in Figure

9. This is a typical pattern which is observed in high fins (higher than 5 mm) [33]. As shown in

Figure 9, the direction of the velocity is forced to be perpendicular to the inlet, as was also the

case in Dogan et al. [11].

However, with fluid domain around the side of the heat sink, Figure 10 shows that the

vector of the fluid velocity is not perpendicular anymore to the inlet of the fin, but instead enters

at an angle. This angle is negative for the upper half of the fin row, while it is positive for the

lower half referred to the orientation of the base plate. The fact that the lower half has a positive

angle is caused by the base plate: the incoming fluid heats up along the frontal fins, just before

entering the fin row. This is also explains why the impact on the heat transfer coefficient at the

bottom side of the fin row (hbottom) is, in all cases, larger compared to the impact on the heat

transfer coefficient at the sides (hsides).

The combined impact on hwithradiation , lim ¿¿, an area average of hbottom and hsides, is thus

strongly negative and independent of the kind of fin row as is derived from Figure 8. However,

the impact on the heat transfer coefficient hwithradiationis much more modest (see Figure 7). This is

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due to the compensation effect of the frontal fins, just before the entrance of the fin row. The

results for h frontal are reported in Table A.2. As can be seen the absolute values for the heat

transfer coefficient h frontal are much larger than the values for hbottom and hsides, although the frontal

surface area is also small.

As can be seen from Figures 5 to 8, the difference in heat transfer coefficient of adding 5

mm, 10 mm or 20 mm to the sides of the fin row has a negligible effect, taking into account the

relative uncertainty of 2% on the numerical results. In other words, there is no need to add a

larger amount of fluid domain, in fact for this case 5 mm of fluid domain was enough. However,

this is only the case when the effect of adding fluid domain to the top of the fin row is neglected.

As will be shown in the next paragraph where both a fluid domain at the sides and at the top will

be added, the flow pattern will completely change (again) and there will be a combined effect of

adding fluid domain at the sides and flow domain at the top.

Effects of adding fluid at top and sides of fin row

Adding fluid domain to the top and both sides of the fin row will change the heat transfer

coefficients further. Figure 6 shows the effect of adding fluid domain to top and sides on hbottom.

Although the impact on the heat transfer coefficient is still large and negative compared to the

reference geometry (Mesh-ref), the relative impact is smaller compared to the case where only

fluid domain was added to the sides. For the rectangular and interrupted fin design the relative

impact on hbottomdecreases from -28% to -24%. For the inverted triangular fin the impact on hbottom

decreases from -27% to -20% (see Figure 6). The fact that hbottom decreases by adding fluid

domain at the sides is explained in the previous paragraph. The fact that the impact of adding

fluid domain on the top and side is more severe for the inverted triangular shape can be

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explained by comparing Figure 11 (for the inverted triangular fin) with Figure 9 and 12 (for the

rectangular fin). Figure 12 shows that e.g. the streamlines in the middle of the fin are oriented

more to the bottom side of the fin row compared to the streamlines at the same location for

inverted triangular case (Figure 11). This explains the differences in impact for hbottom between

the rectangular and inverted triangular fins when adding fluid to top and sides. In case of the

rectangular fins more fluid is passing by the bottom plate.

Comparing the streamlines in both Figure 11 (inverted triangular case) and Figure 12

(rectangular case), it is clear that the impact of adding extra fluid domain on top and sides of the

fin row on hsides is larger for the rectangular and interrupted fin row than for the inverted

triangular fin. For the first two fin rows, the impact on hsides is reduced from -18% to -11%, while

for the inverted triangular fin the impact on hsides is decreased (not significantly) from -20% to -

19% (see Figure 6). This can be explained by the fact that the heated area in case of the inverted

triangular fin row is varied in the direction from the frontal surface area deeper into the heat sink.

This is in contrast with the rectangular fin case, where the heated area remains constant. This

means that a higher outflow is induced compared to the rectangular case, causing the lasting

negative impact for the inverted triangular case.

The influence on the lumped heat transfer coefficient hwithradiation is significant and the

relative influence strongly depends on the fin shape. For the rectangular fin, the impact is large:

going from a decreased heat transfer coefficient (-5%) when only fluid domain at the sides is

added to an increase in the heat transfer coefficient for the case with fluid domain added to sides

and top (+12%). This is completely different from the case of the inverted triangular fin where

the impact goes from -5% (approximately the same as for the rectangular and interrupted fin

row) to -2%, when the largest fluid domain at the top is added. This difference in impact can be

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explained by the frontal facing heat transfer coefficient of the fins: h frontal. Although the absolute

number of h frontal is approximately the same for all the fin rows studied (see Table A.2), the

relative importance of the frontal facing part of the fin row in case of the inverted triangular fin is

larger compared to the rectangular case (relative difference in the frontal fin area is almost

doubled in the case of an inverted triangular case).

Next, the influence of adding extra fluid domain: going from 10 mm to 50 mm is

significant in case of hsides and hwithradiation as can be seen in Figure 5 and 7. The impact on h frontal of

adding extra fluid domain on the top is limited as can be observed in Table A.2. When at least

50mm of fluid domain is added above the heat sink, the results no longer depend on the height of

the computational domain. This corresponds to a height of 130% of the equivalent fin height (in

case of a triangular fin shape the equivalent fin height is half of the total fin height). This

required fluid domain does not depend on the fin shape that is studied.

To check if the equivalent fin height could be taken as a parameter to guess the volume of

the fluid domain to be placed on top of the fin structure, the rectangular fin structure from Figure

2 (a) is taken, however, the fin height is divided by 2. In this case, the fin height is 19 mm

instead of 38 mm. The results for this shorter fin are presented in Table 4. It is clear that for the

fluid domain adjacent to the heat sink, only the existence of some fluid domain is enough. The

same can be observed for the results as presented in the paper. If our statement holds, the fluid

domain that is put on top of the heat sink should be at least 130% times 19 mm = +/- 25 mm.

From the results, it can be clearly seen that for a shorter fin design, the heat transfer coefficient

convergences much faster to a steady value. This indicates that the equivalent fin height could be

taken as an indicator to guess the amount of fluid height above the fin structure, independently of

the fin height itself.

20

CONCLUSIONS

In this work the influence of adding a fluid domain on the heat transfer coefficient is

studied numerically for three different fin shapes. A rectangular fin row, an interrupted

rectangular fin row and an inverted triangular fin row are studied. The rectangular and inverted

triangular fin row are based on the work of Dogan et al. [11]. All studied fin rows have a length

of 127 mm and the height is either 38 mm for the rectangular fins or 76 mm for the inverted

triangular fins. The impact on the heat transfer coefficient for the rectangular and interrupted

rectangular fin row are approximately the same. This is caused by the single chimney pattern.

The influence of only adding fluid domain to the sides is studied separately. A large

decrease is observed on the heat transfer coefficients on both the sides and the bottom of the fin

row. This is caused by the fact that by adding fluid domain, the velocity vector is not

perpendicular to the inlet of the fin row, inducing a different heat transfer coefficient.

Furthermore, the lumped heat transfer coefficient is also studied. This is the average heat transfer

coefficient over all the fin surfaces. Adding fluid domain to the sides, also causes a decrease in

this lumped heat transfer coefficient. However, the decrease in this lumped value is much more

modest, because of the importance of the heat transfer coefficient at the frontal sides of the fin

row. Only adding a small part (some millimetres) of fluid domain to the sides is enough to

capture the flow behaviour there, which is not perpendicular to the plane anymore.

Finally, the influence of adding fluid domain on both the top and the sides is studied.

Here the effect of the fluid domain on the heat transfer coefficient is only significant for the

rectangular and interrupted rectangular fin shape. Adding fluid domain to all sides of the fin row

induces a relative increase in lumped heat transfer coefficient of +12%. However, for the

21

inverted triangular shape, the lumped heat transfer coefficient doesn’t change significantly. This

means that the comparison between two fin types depends on the size of the computational

domain, if it is not chosen large enough. To summarise, the length of the fluid domain of the fin

shapes studied here has to be larger than 130% of the equivalent fin height, increasing the height

of the fluid domain on top of the heat sink doesn’t show any influence on the resulting heat

transfer coefficient. If no fluid domain is added the error that is made over the difference heat

transfer coefficients varies between -1.7% to -23.5%. When only 50% of the equivalent fin

height is added, the made error varies between -1% and only +6%.

In this work, it is also illustrated that the suggestion of using 130% of the equivalent fin

height to be put above the fin structure is independent of the fin height (or fin structure) that is

used.

APPENDIX A

For Figures 5 to 8 the absolute values are reported in Table A.1. Also the percentage

difference with the base case (Mesh-ref) is mentioned in this table.

Table A.1. Calculated values for the heat transfer coefficient as reported in Figures 5 to 8.

Mesh name Rectangular fin

%-diff Interrupted rectangular fin

%-diff Inverted triangular fin

%-diff

hsides

Mesh-ref 4.80 - 5.11 - 7.70 -Mesh-5mmside 3.93 -18.21 4.17 -18.35 6.15 -20.10Mesh-10mmside 3.90 -18.74 4.14 -18.95 6.14 -20.24Mesh-20mmside 3.94 -18.00 4.18 -18.22 6.11 -20.68Mesh-top10mm 4.20 -12.42 4.51 -11.73 6.23 -19.08Mesh-top20mm 4.05 -15.56 4.22 -17.55 6.24 -18.87Mesh-top30mm 4.25 -11.48 4.30 -15.88 6.46 -16.08Mesh-top50mm 4.28 -10.89 4.49 -12.22 6.32 -17.91Mesh-top100mm 4.29 -10.57 4.50 -11.98 6.22 -19.19

22

Mesh-top200mm 4.27 -11 4.49 -12.05 6.20 -19.55hbottom

Mesh-ref 3.46 3.69 - 3.42 -Mesh-5mmside 2.48 -28.31 2.65 -28.21 2.49 -27.04Mesh-10mmside 2.43 -29.63 2.60 -29.37 2.44 -28.73Mesh-20mmside 2.34 -32.30 2.57 -30.36 2.44 -28.73Mesh-top10mm 2.58 -25.47 2.74 -25.67 2.65 -22.53Mesh-top20mm 2.58 -25.23 2.75 -25.43 2.63 -23.01Mesh-top30mm 2.64 -23.51 2.72 -26.31 2.56 -25.24Mesh-top50mm 2.64 -23.57 2.79 -24.36 2.69 -21.27Mesh-top100mm 2.64 -23.51 2.78 -24.55 2.72 -20.46Mesh-top200mm 2.64 -23.75 2.78 -24.76 2.72 -20.55

hwithradiation

Mesh-ref 5.14 - 5.46 - 8.78 -Mesh-5mmside 4.89 -4.87 5.20 -4.89 8.36 -4.79Mesh-10mmside 4.84 -5.79 5.15 -5.79 8.32 -5.19Mesh-20mmside 4.87 -5.22 5.17 -5.32 8.27 -5.77Mesh-top10mm 5.58 8.63 5.90 7.97 8.56 -2.54Mesh-top20mm 5.52 7.46 5.64 3.20 8.63 -1.69Mesh-top30mm 5.66 10.21 5.87 7.43 8.87 0.99Mesh-top50mm 5.82 13.22 5.93 8.60 8.77 -0.09Mesh-top100mm 5.78 12.40 5.93 8.50 8.63 -1.70Mesh-top200mm 5.79 12.55 5.94 8.75 8.64 -1.55

hwithradiation , lim ¿¿

Mesh-ref 4.70 4.98 7.10Mesh-5mmside 3.81 -18.79 4.04 -18.82 5.67 -20.10Mesh-10mmside 3.79 -19.37 4.01 -19.48 5.66 -20.24Mesh-20mmside 3.81 -18.82 4.05 -18.78 5.63 -20.68Mesh-top10mm 4.08 -13.17 4.37 -12.31 5.75 -19.08Mesh-top20mm 3.94 -16.11 4.09 -17.84 5.76 -18.87Mesh-top30mm 4.13 -12.16 4.17 -16.38 5.96 -16.08Mesh-top50mm 4.15 -11.61 4.35 -12.70 5.83 -17.91Mesh-top100mm 4.17 -11.31 4.36 -12.40 5.75 -19.01Mesh-top200mm 4.16 -11.55 4.36 -12.55 5.73 -19.35

In Table A.2 the area-averaged heat transfer coefficient on the frontal side at the inlet of

the fin row h frontal is reported.

Table A.2. Calculated values for the heat transfer coefficient at the frontal side of the fin row.

23

Mesh name Rectangular fin

%-diff Interrupted rectangular fin

%-diff Inverted triangular fin

%-diff

h frontal

Mesh-5mmside 20.99 21.02 19.19Mesh-10mmside 19.35 -7.82 19.36 -7.90 17.59 -8.35Mesh-20mmside 18.24 -13.10 18.24 -13.23 16.36 -14.73Mesh-top10mm 18.94 -9.77 18.91 -10.02 17.50 -8.82Mesh-top20mm 17.58 -16.26 17.79 -15.37 16.18 -15.65Mesh-top30mm 17.37 -17.26 17.24 -17.97 15.89 -17.18Mesh-top50mm 17.72 -15.59 17.77 -15.44 16.19 -15.64Mesh-top100mm 17.69 -15.74 17.75 -15.56 16.16 -15.78Mesh-top200mm 17.66 -15.88 17.71 -15.75 16.15 -15.85

NOMENCLATURE

A area, m²

g gravitational constant, m/s²

GCI Grid Convergence Index, -

h heat transfer coefficient, W/m²K

H fin height, m

H f fin height, m

L (characteristic) length, length of fin, m

M w molecular weight of the gas, kg/mol

P pressure, N/m²

Pr Prandtl number, -

Q electric power supply, W

R universal gas constant, J/K/mol

Ra Rayleigh number, -

S fin spacing, m

24

T temperature, K

v velocity, m/s

W width of the heat sink, m

Greek symbols

α thermal diffusivity, m²/s

β thermal expansion coefficient, 1/K

μ dynamic viscosity, Pa.s

ρ density, kg/m³

ν kinematic viscosity, kg/(s.m)

Subscripts

ave average

env environment

o operating

op operational

b base plate

tot total

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29

Table 1. Overview of numerical work done on heat sinks in buoyancy-driven heat transfer.

Author Year Geometry Simulation

(dimensionality –

fluid – turbulence –

radiation)

Mesh

Ahmadi

et al. [23]

2014 Rectangular

interrupted fins

2D – Boussinesq

approximation – no

turbulence model -

included

One fin row is simulated. Fluid added

before the fin, but not after the fin (2D).

Length of this fluid domain: not reported.

Dialameh

et al. [14]

2008 Rectangular

fins

3D – constant

properties, except

ρ(T ) – no turbulence

model – not included

A half fin row is simulated. Fluid added at

sides and on top of the fin. Dimension of

this fluid domain: ¾ times the fin length to

the sides, 9 times the fin height on top of

the heat sink. Only one fin is taken in to

account, other side is open.

Shen et

al. [26]

2014 Rectangular

fins

3D – incompressible

ideal gas – no

turbulence model –

included

Several half fin rows are simulated. Fluid

domain added before, after and underneath

the fin. Length of this fluid domain: 1 time

the fin length to the sides, 5 times the fin

height on top of the heat sink and 1 time

the fin height underneath the fin.

Mehrtash

and Tari

[27] and

Tari and

Mehrtash

[28]

2013 Rectangular

fins

3D – air is ideal gas -

zero order turbulence

model – included

Complete heat sink is simulated together

with insulating material. Square 3m x 3m

box. Length of fluid domain: ± 6 times the

fin length to the sides, min. ± 116 times the

fin height on top of the heat sink.

Dogan et 2014 Thin-fin arrays 3D – constant One fin row is simulated. The fins at either

30

al. [11] with various

shapes

properties except

ρ(T ) – no turbulence

model – included

side are taken into account. No calming

section before or after.

Table 2. Overview of all geometries used in this study per fin shape

Mesh-ref

Mesh-5mmside

Mesh-10mmside

Mesh-20mmside

Mesh-top10mm

Mesh-top20mm

Mesh-top30mm

Mesh-top50mm

Mesh-top100mm

Side 5 mm

x

Side 10 mm

x x

Side 20 mm

x x x x

Side 30 mm

x

Top 10 mm

x

Top 20 mm

x

Top 30 mm

x

Top 50 mm

x

Top 100mm

x

31

Table 3. Uncertainty analysis for grid discretization

Coarse

grid

Fine grid

hsides[W /m ² K ] 3.394 3.455

hbottom [W /m ² K ] 4.724 4.801

have[W /m ² K ] 4.621 4.697

GCI hsides [%] 7.07 1.77

32

Table 4. Calculated values for the heat transfer coefficients based on a fin structure

presented in Figure 2 (a) but with a fin height of 19 mm instead of 38 mm.

Mesh name Rectangular fin height divided by 2

%-diff

hsides

Mesh-ref 4.87 -Mesh-5mmside 4.48 -8.7%Mesh-20mmside 4.50 -8.2%Mesh-top20mm 4.69 -3.8%Mesh-top50mm 4.72 -3.2%Mesh-top100mm 4.72 -3.1%

hbottom

Mesh-ref 3.78 -Mesh-5mmside 3.63 -4.1%Mesh-20mmside 3.65 -3.6%Mesh-top20mm 3.45 -9.6%Mesh-top50mm 3.47 -9.6%Mesh-top100mm 3.45 -9.6%

hwithradiation

Mesh-ref 5.36 -Mesh-5mmside 5.47 +2.01%Mesh-20mmside 5.50 +2.5%Mesh-top20mm 6.37 +15.9%Mesh-top50mm 6.46 +17.0%Mesh-top100mm 6.49 +17.4%

hwithradiation , lim ¿¿

Mesh-ref 4.71 -Mesh-5mmside 4.36 -8%Mesh-20mmside 4.38 -7.5%Mesh-top20mm 4.51 -4.4%

33

Mesh-top50mm 4.54 -3.7%Mesh-top100mm 4.54 -3.7%

List of Figure Captions

Figure 1 Illustration of the test case from Dogan et al. [11] for a rectangular fin

Figure 2 Illustration of the used fin shapes in this work. (a) rectangular fin, (b) interrupted

rectangular fin, (c) inverted triangular shape. Dashed line indicates the symmetry plane.

Figure 3 Illustration of the boundary conditions in case of a rectangular fin row (Mesh-top20mm)

Figure 4 Illustration of the discussed fin surfaces.

Figure 5 Dependency of the heat transfer coefficient on the fluid domain at the sides of the fin

row (hsides)

Figure 6 Dependency of the heat transfer coefficient on the fluid domain on the bottom side of

the fin row (hbottom)

Figure 7 Dependency of the heat transfer coefficient on the fluid domain, taken all the available

surfaces into account (hwithradiation)

Figure 8 Dependency of the heat transfer coefficient on the fluid domain, taking an area-average

of hsides and hbottom into account (hwithradiation , lim ¿¿)

34

Figure 9 Velocity contour plot and streamlines in the middle of the rectangular fin row without

any fluid domain added (Mesh-ref, right side: symmetry plane)

Figure 10 Velocity contour plot and streamlines in the middle of the rectangular fin row with 20

mm of fluid domain added at the sides (Mesh-20mmside, right side: symmetry plane)

Figure 11 Velocity contour plot and streamlines in the middle of the inverted triangular fin row,

(a) without added fluid domain (Mesh-ref), (b) with 20 mm of fluid domain added at the sides

and top (Mesh-top20mm, right side: symmetry plane).

Figure 12 Velocity contour plot and streamlines in the middle of the rectangular fin row with 20

mm of fluid domain added at the sides and top (Mesh-top20mm, right side: symmetry plane)

35

Figure 1 Illustration of the test case from Dogan et al. [11] for a rectangular fin

36

Figure 2 Illustration of the used fin shapes in this work. (a) rectangular fin, (b) interrupted

rectangular fin, (c) inverted triangular shape. Dashed line indicates the symmetry plane.

37

Figure 3 Illustration of the boundary conditions in case of a rectangular fin row (Mesh-

top20mm)

38

Frontal fin surface

Top fin surface

Side fin surface

Bottom fin surface

Figure 4 Illustration of the discussed fin surfaces.

39

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

h sides[ W

=m2 K

]

ref

5mmsid

e

10mmsid

e

20mmsid

e

top10

mm

top20

mm

top30

mm

top50

mm

top10

0mm

top20

0mm

rectangular interrupted inverted triangular

Figure 5 Dependency of the heat transfer coefficient on the fluid domain at the sides of the

fin row (hsides)

40

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

h bottom[ W

=m2 K

]

ref

5mmsid

e

10mmsid

e

20mmsid

e

top10

mm

top20

mm

top30

mm

top50

mm

top10

0mm

top20

0mm

rectangular interrupted inverted triangular

Figure 6 Dependency of the heat transfer coefficient on the fluid domain on the bottom side

of the fin row (hbottom).

41

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9h w

ithradiation

[ W=m

2 K]

ref

5mmsid

e

10mmsid

e

20mmsid

e

top10

mm

top20

mm

top30

mm

top50

mm

top10

0mm

top20

0mm

rectangular interrupted inverted triangular

Figure 7 Dependency of the heat transfer coefficient on the fluid domain, taken all the

available surfaces into account (hwithradiation).

42

3.5

4

4.5

5

5.5

6

6.5

7

7.5h w

ithradiation

;lim[ W

=m2 K

]

ref

5mmsid

e

10mmsid

e

20mmsid

e

top10

mm

top20

mm

top30

mm

top50

mm

top10

0mm

top20

0mm

rectangular interrupted inverted triangular

Figure 8 Dependency of the heat transfer coefficient on the fluid domain, taking an area-

average of hsides and hbottom into account (hwithradiation , lim ¿¿).

43

Figure 9 Velocity contour plot and streamlines in the middle of the rectangular fin row

without any fluid domain added (Mesh-ref, right side: symmetry plane)

44

Figure 10. Velocity contour plot and streamlines in the middle of the rectangular fin row

with 20 mm of fluid domain added at the sides (Mesh-20mmside, right side: symmetry

plane)

45

Figure 11. Velocity contour plot and streamlines in the middle of the inverted triangular fin

row, (a) without added fluid domain (Mesh-ref), (b) with 20 mm of fluid domain added at

the sides and top (Mesh-top20mm, right side: symmetry plane).

46

Figure 12. Velocity contour plot and streamlines in the middle of the rectangular fin row

with 20 mm of fluid domain added at the sides and top (Mesh-top20mm, right side:

symmetry plane)

47

BIBLIOGRAPIC INFORMATION

Sven De Schampheleire is a Ph.D. student at the Department of Flow,

Heat and Combustion Mechanics, Ghent University, Belgium. He

received his master's degree in electromechanical engineering from Ghent

University in 2011. His master's dissertation investigates the thermo-

hydraulic aspects of metal foam as alternative for louvered fins in heat

exchangers and was honored with the ‘Atlas Copco Airpower’ prize

2011. Current research interest is the numerical modeling of natural

convection (in porous materials) using CFD.

Kathleen De Kerpel is a postdoctoral researcher at the Department of

Flow, Heat and Combustion Mechanics, Ghent University, Belgium. She

received her master's degree in electromechanical engineering from

Ghent University in 2010. Her main research interest is two-phase

refrigerant flow in compact fin and tube heat exchangers.

Bernd Ameel is a postdoctoral researcher at the Department of Flow,

Heat and Combustion Mechanics, Ghent University, Belgium. He

received his Ph.D. from Ghent University in 2014 and his master's degree

in electromechanical engineering from the same university in 2010. His

main research interest is the optimization of compact fin and tube heat

exchangers.

Özer Bağcı is a postdoctoral researcher at the Department of Flow, Heat

and Combustion Mechanics, Ghent University, Belgium. He received his

bachelor’s, master’s and Ph.D. degree from Istanbul Technical

University. His main research interest is fluid flow and heat transfer in

metal foams and compact heat exchangers.

48

Michel De Paepe is professor of Thermodynamics in the Faculty of

Engineering at Ghent University. He graduated as Master of Science in

Electro-Mechanical Engineering at the Ghent University in 1995. In 1999

he obtained the Ph.D. in Electro-Mechanical Engineering at the Gent

University, graduating on ‘Steam Injected Gas Turbines with water

Recovery’. He is currently heading the Research Group Applied

Thermodynamics and Heat Transfer at the Faculty of Engineering at the

Ghent University. Research focuses on: thermodynamics of new energy

systems, performance of HVAC systems and energy in buildings and

complex heat transfer phenomena in industrial applications, as in compact

heat exchangers, refrigerant two-phase flow and electronics cooling. He is

(co-)author of more than 100 papers published in international peer

reviewed journals and more than 250 conference papers. For his master

thesis and his Ph.D. he received the WEL Energy prize.

49