Jamie Fogarty CFD FINAL DRAFT PRINT

ME6062 - Advanced Computational Fluid Dynamics – Spring 2015-2016

Abstract- This paper presents the comparison of 3 Computational Fluid Dynamics (CFD) models in the computation of a conduction-convection conjugate heat transfer problem. As no CFD model is universal to all problems, their comparison can highlight the pros and cons of each model and aid in the determination of the most suitable for the specific problem. The functionalities of turbulence models is introduced and discussed to obtain an appreciation for the computed results.CFD Software, Star CCM+, is used to conduct an analysis on a plate-fin heat sink with 6 fins. A heat load of 160,000W/m2 is applied to the base plate of the heat sink, and the heat sink is subjected to forced convection corresponding to a channel Reynold’s number of 900, at room temperature. The heat sink is modelled using PTC Creo Parametric and imported into Star CMM+ where a polyhedral mesh is applied and refined. 5 simulations are conducted using the laminar model, Standard Low-Reynolds Number k-ε model (low y+ & high y+ wall treatment) and V2F Low-Reynolds Number k-ε model (low y+ & high y+ wall treatment). Convergence was gauged from the residual plots, maximum heat sink base plate temperature monitor and pressure drop monitor. The results were then benchmarked with theoretical calculations for heat sink thermal resistance and pressure drop to ensure their validity.The results for all simulations conducted correlate with theory to within 10% for the heat sink thermal resistance and 20% for the pressure drop over the heat sink, and also correlate extremely close one another. The most efficient model for the problem was deemed to be the laminar solver, as it yielded the same accuracy as the turbulence models in the least amount of time. It was also found that in the cause of a fine, refined mesh, the low y+ wall treatment enables the turbulence models to iterate faster.

22/04/16 [email protected]

I. INTRODUCTIONfluid flow and it’s physical aspects are governed by three fundamental principles,

that is the conservation of mass, the conservation of energy and Newton’s second law (F=ma). These principles can be expressed in terms of, usually, partial differential equations. In basics terms, Computational fluid dynamics (CFD) is, in part, the art of replacing the governing partial differential equations of fluid flow with numbers, and advancing these numbers in space and/or time to obtain a final numerical description of the complete flow field of interest (Wendt and Anderson 2009). In saying so, this is not the inclusive limit of CFD capabilities, as there are some problems that allow for an immediate solution without advancing in space or time, and also contain integrals as opposed to differentials.

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The CFD software used to conduct the analysis in this paper is called Star CCM+. This software provides a comprehensive engineering physics simulation, inclusive of an entire engineering process for solving problems involving flow (of fluids or solids), heat transfer, and stress (CD-Adapco, 2016). This software will be used to obtain a physical description of a conductive-convective conjugate-heat transfer problem. Conjugate heat transfer is the interaction of at least two mediums or subjects (Dorfman 2010). This problem will be the interaction of a cooling fluid, air, flowing over a heat sink with applied heat flux at the base. A heat sink is a device to effectively absorb thermal energy from one location and dissipate it to the surroundings (air) through use of extended surfaces, such as fins. Heat sinks are used in a wide range of applications where efficient heat dissipation is required; major examples included refrigeration, heat engine, and cooling electronic devise (Lee 2010). The most common heat sink design will be considered, a heat sink with longitudinal rectangular fin array as the extended surface. This type of heat sink has the benefit of simple

Comparison of Turbulence Models in Predicting Heat Transfer Parameters of a

Finned Heat Sink under Forced Convection

Jamie FogartyDepartment of Mechanical, Aeronautical & Biomedical Engineering, University of

Limerick

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design, low fabrication costs and high thermal performance achievable.This paper will outline the procedures for modelling such a problem using that CFD software, and will aim to compare three different turbulence models used by the software, in order to arrive at a numerical solution of the problem at hand. Turbulence models, their functionalities and differences will be introduced, discussed and compared. The abilities of each turbulence model will be benchmarked by theoretical calculations and solutions, to determine the most suitable model for this conjugate heat transfer problem. The models will be compared to theoretical calculations for heat sink thermal resistance and pressure drop.

II.OBJECTIVESThe main objective of this paper is to provide a platform of understanding on the functionalities of CFD software and turbulence modelling. This will be achieved through the analysis and benchmarking of a conjugate heat transfer problem of a heat sink under forced convection. The objectives are as follows;

Introduce and use correlations to determine the theoretical values for the heat sink:

o Thermal Resistanceo Pressure Drop

Model the heat sink in CAD software Import software into Star CCM+ develop

an adequate mesh Introduce the concept of turbulence

modelling and develop its concept Determine the most suitable turbulence

model to use in conduction of a mesh sensitivity study

Introduce the turbulence models to be considered, highlighting the reason for choice and the differences and similarities between the models, in the context of equations used by each

Conduct a mesh sensitivity study to determine the most sufficient mesh, in terms of cell count, size, etc.

o Mesh sensitivity will be bench marked from the theoretical calculations of thermal resistance and pressure drop

Once an adequate mesh has been generated and correlates with theoretical calculations, the model used will be compared to other slected models to gauge how they compare.

o The comparison will be drawn from theoretical calculations, Resistance and Pressure drop, and also to determine which turbulence model best captures the behavior of the forming boundary layers between the plates.

III. GEOMETRYIn order for the reader to gain an appreciation for the theory and theoretical calculations, it is effective to firstly define visually the heat sink under consideration. The heat sink under analysis is a plate-fin heat sink with 6 fins. The modelled heat sink has two fins, Figure 1, and advantage was taken of the symmetry plane options when defining the boundary conditions of Star CCM+. The heat sink was modelled using PTC Creo Parametric and imported into Star CCM+.

Figure 1: Front, side and angled view of plate-fin heat sink modelled.

The dimensions of the total heat sink, with the symmetry planes implemented, are presented in Table 1. Note, that the heat sink will have half a channels width at the fins closest to the either side of the width (when looking from a front view).

Dimension (mm)Base Height 6Fin Height 30Channel Thickness 1.5Fin Thickness 1Base Width 15Length 60

Table 1: Plate-Fin Heat sink dimensions, with the symmetry planes in mind.

The heat sink was enclosed in a bounding box of width 15mm, height 31mm and length 360mm. The bounding box was placed on the top of the base plate, and just encloses the fins and the exposed area of the base plate top surface. This can be seen in Figure 2. The heat sink was

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positioned 120mm from the defined inlet, and 180mm from the defined outlet. The heat sink is not equidistant from either let, as the effect of the heat sink wake would like to be visualised.

Figure 2: Heat-sink including bounding box.

IV. FLUID & SOLID MATERIAL PROPERTIESThe material properties of both the fluid and solid region (air and aluminium heat sink) are presented in Table 2. These values were used for both the theoretical calculations and for the CFD analysis.

Property

Air Aluminum

Units

ρ 1.18415 2702 Kg/m3k 1.85508 237 W/m-KCp 0.0257 - J/Kg-Kμ 0.9 - Pa-sα 1003.62 - /KPr 0.0333 - -

Table 2: Fluid and solid material properties used in both theoretical calculations and CFD analysis

V.THEORYThe theoretical calculations conducted were aimed to obtain the; heat sink thermal resistance and the pressure drop through the heat sink. This section outlines the theoretical equations used.

The heat sink thermal resistance is given by;

Rtot=Rhs+H −H f

k base . w . L(Eqn.

1)(Simons,

2003)

The second term is the thermal resistance of the base, where H is the total height of the heat sink, Hf is the height of the fins, kbase is the thermal conductivity of the base, w is the width and L is the length. The first term, Rhs, is the thermal resistance of the heat sink fins and is given by;

Rhs=1

h .¿¿(Eqn. 2)

(Simons, 2003)Where Abase is the exposed area of the base (between the fins), Nfin is the number of fins, ηfin is the fin efficiency, and Afin is the surface area per fin taking into account both sides of the fin.

The pressure drop, ΔP (Pa), across the heat sink is given by:

ΔP=(Kc+4. f app . LDh

+K e) . ρ . V 2

2 (Eqn.

3)(Culham & Muzychka,

2001)

Where Kc and Ke are coefficients that represent the pressure losses due to sudden contraction channels and expansion of the flow entering and leaving the heat sink respectively. fapp is the apparent friction factor, that takes into account the developing and developed regions of the flow in the heat sink channels.

Appendix A presents a full representation of the equations and correlations used to calculate both the thermal resistance and pressure drop.

VI. FLOW CONDITIONSFirstly, it is necessary to state that two flow conditions will be present. That is, the flow condition of the fluid in the bounding box, and the flow condition of the fluid through the heat sink channels. The air will be introduced into the inlet at a temperature of 293K and a velocity of 2.9m/s, corresponding to a Volume Flow Rate V̇ of 1.35x10-3 m3/s. Dividing the Volume Flow Rate by the open cross sectional area of the heat sink (i.e. between the fins and the bounding box) gives a velocity of 5m/s through the heat sink channels. The Reynold’s number corresponding to the bounding box is Re=3740, which is in the turbulent regime, and the channel Reynolds number is Rech=900, which is in the laminar regime.The problem under consideration is one of a conductive -convective heat transfer. A heat flux Q̇of 16,000W/m2 will be applied to the bottom of the heat sink base plate, corresponding to a heat load q of 14.4W. This will conduct through the heat sink base and fins, where it will be subjected to forced convection by the working fluid.

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VII. TURBULENCE MODELLINGThis section aims to briefly describe the concept of turbulence modelling and present some of the fundamental equations used in aid to develop a basis for which turbulence models under consideration may be compared. Firstly, consider a laminar flow. In the computation of a laminar flow, the Navier-stokes equations are solved directly. For the computation of a turbulent flow, the starting point for any CFD software is the conservation of mass and continuity. Making the assumption that the fluid is incompressible, with constant-property following the continuum hypothesis, no thermal interaction and no other body-forces, the conservation of mass and momentum can be denoted in its conservation form, after the addition of a time-averaging. The time-averaging process takes the Navier-Stokes equations for the instantaneous velocity and pressure fields are decomposed into a mean value and a fluctuating component (Wilcox, 2010). This equation is otherwise known as the Reynolds-Averaged Navier Stokes (RANS) equation and is;

ρ∂U i

∂ t+ρ U j

∂U i

∂ x j=−∂ P

∂ x i+ ∂

∂ x j(2 μ Sij− ρ μ j

' μ i')

(Eqn. 4)

Where μ j' μi

' is a time-averaged rate of momentum transfer due to turbulence, and ρ μ j

' μi' is known as

the Reynolds-stress tensor. The time-averaged Navier stokes equations are identical to the instantaneous equations aside from the replacement of instantaneous variables with mean values and the addition of a time-averaged rate of momentum transfer due to turbulence.The Reynolds stresses arise from the velocity fluctuations associated with the turbulence of the flow. The purpose of a turbulence model is to compute the Reynold’s stress tensor in terms of the mean flow quantities, and provide closure for the governing equations (Griffin, 2016). Various methodologies exist to try achieving this end goal. The one of interest in this project is the eddy viscosity method.Every turbulence model begins with the Boussinesq eddy-viscosity approximation to compute the Reynolds stress tensor as the product of an eddy viscosity and the mean strain-rate tensor. For computational simplicity, the eddy viscosity, in turn, is often computed in terms of a mixing length that is analogous to the mean free path in a gas. Because of this, the eddy viscosity and mixing length must be specified in advance, most simply, by an algebraic relation between eddy viscosity and

length scales of the mean flow. Thus, each turbulence model computes a different algebraic relation for eddy viscosity (Wilcox, 2010). Boussinesq’s eddy-viscosity in mixing length form is given as:

μt=Cμ ρυ t lt (Eqn. 5)

Where Cµ is a constant, νt is the averaged eddy transport velocity around the flow field (velocity scale) and lt is the distance an eddy travels before it exchanges it original mean momentum (length) scale) (Griffin, 2016).Turbulence models are then categorised by the number of equations used to compute the eddy viscosity into mean flow parameters. Two-equation models are the most popular choice loaning to their effective balance between computational cost and accuracy. The turbulence models used in the analysis will be two-equation models. Two-equation models are reliant on the Boussinesq hypothesis to evaluate the Reynolds stresses, and are based around the transport for turbulent kinetic energy defined k, and in the case of the turbulence models of interest, the turbulence dissipation rate per unit mass ε.

The combination of k- ε brings us to the k- ε turbulence models and its variants. Recalling Eqn. 27, we can infer that the velocity of the large eddies is proportional to √k , defining the turbulence velocity scale as;

υt=√k (Eqn. 6)

The time scale associated with the turbulence is given by the length scale divided by the velocity scale (eqn. 6), and the dissipation rate per unit mass can be expressed as (eqn. 7):

t t=lt

√k (Eqn. 7) ε= k1.5

lt(Eqn.

8)

Using this, the turbulence length scale can be expressed in terms of k and ε;

lt=k1.5

ε (Eqn.

9)

Now substituting into Prandtl’s turbulence viscosity expression gives;

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μt=ρ Cμk2

ε(Eqn.

10)

This is used to compute the turbulent viscosity in the k- ε model, and then the turbulence viscosity is used in conjunction with the Boussinesq approximation to calculate the Reynolds stresses. In order to arrive at a numerical solution, it is necessary for the model to also use a modified transport equation for k and ε consisting of semi-empirical correlations and approximations for the unknown/immeasurable terms, along with closure coefficients. The model describe is called the Standard k- ε model. Although this model is reasonably accurate for a wide range of flows, it performs poorly for flows with large swirls, pressure gradients and separation.Let’s consider two other variations of this turbulence model. Standard Low-Reynolds Number k-ε (Lien et al. 1996) and the V2F Low-Reynolds Number k-ε (Durbin 1991, Lien at al. 1998).The first, Standard Low-Reynolds number k-ε, is identical to the standard model except for the additional turbulence production term in the modelled ε equation. Also, this model computes the eddy viscosity the same way, with addition of a damping function, fμ:

μt=ρ f μC μk2

ε(Eqn.

11)

This damping function is present to ensure that the turbulent viscosity attenuates accordingly in the near-wall region (Griffin, 2016). It should be noted that both the standard model and low Re model cannot be implemented directly to the wall, as the ε- equation contains a term which cannot be computed at the wall. As a result, it is necessary for the models to use wall functions. The law of the wall is one of the most famous empirically-determined relationships in turbulent flows near solid boundaries. Measurements show that, for both internal and external flows, the stream wise velocity in the flow near the wall varies logarithmically with distance from the surface. This behaviour is known as the law of the wall (Wilcox, 2010). This behaviour can be categorised into three sections:

The viscous or linear sublayer – viscous stresses dominate in this region

The buffer layer –both viscous and turbulent shear stresses are of equal magnitude.

The log-law layer– influence of turbulent shear (Reynolds) stresses is strongest and viscous stresses are small.

In order to compute the near wall region, Star CCM+ offers three alternative wall treatments. These are Low y+, High y+ and all y+ wall treatment. y+ is a dimensionless parameter that defines the central distance of a wall bounded cell. Low y+ is used when the centre of the near-wall cell is as a y+ value of under 1 (viscous sublayer), high y+ is between 5-30 (buffer layer) and the all y+ is a hybrid model that attempts to provide a more realistic modelling than either low or high Re wall treatments if the wall adjacent cell lies in the buffer layer.The V2F model is also a low Reynold number k- ε model variant that can model the anisotropy of near-wall turbulence. The model is similar to the standard model but rather than use the turbulent kinetic energy to calculate the eddy viscosity it used a velocity scale v '2. The V2F model is able to provide the correct scaling for the representation of the damping of turbulence in the near-wall region without actually using exponential damping or the wall functions (Griffin, 2016). Anisotropic turbulent behaviour close to the walls is modelled through an elliptic relaxation function f. The model solves this, along with v' 2, k and ε making it essentially a four equation model. The V2F model was originally developed for attached or mildly separated boundary layers but can simulate with reasonable accuracy flows that are dominated by separation (Griffin, 2016). The eddy viscosity is given by:

μt=ρ Cμ 2 v '2T (Eqn. 12)

Where T is a turbulent timescale. The V2F model can accurately model anisotropic flow and heat transfer effects in wall-bounded, channel and jet flows, and has been chosen as one of the models to compare.

VIII.PROCEDURE A. Mesh Generation

This section outlines the meshing procedure, in order to enable the conjugate heat transfer solution. The mesh generation procedure was as follows:

Once imported, the heat sink model was sectioned into two parts; Heat Sink and Fluid Region. The Heat sink was then broken further into two subgroups; fins and base plate.

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The base plate of the heat sink was split (by patch) so that the base plate of the heat sink could be considered an individual part among the heat sink sub-group. This was conducted in order to create a boundary where a heat flux could be applied. This will simulate a heat flux from a heater matt/component/etc.

Next, the fluid region was extracted by the subtract function. This function would separate the region where the fluid will flow from the heat sink fins.

Next an imprint function was executed. This function allows for the option of a conformal mesh i.e. the mesh faces match one-to-one at interfaces to ensure heat transfer occurs smoothly.

Next it was necessary to apply parts to regions. The subtract was assigned fluid region, and the heat sink was assigned solid region, with each part surface having a boundary.

Next it was necessary to create interfaces between certain faces within the model. The boundaries were created were:

o Heat sink Base plate – Heat sink Base

o Heat sink Base – Heat sink Finso Fluid Region: Heat sink fin faces –

Heat sink: Finso Fluid Region: Heat Sink Base Faces

– Heat sink: Base This was done so that the mesh of the

solid and fluid region could communicate at these interfaces and exchange values, in order for the conjugative heat transfer (conduction-convection) to solve correctly.

Two automated mesh operations were used for the fluid region and the solid region. This was done to have the option of turning the prism layers on or off for the fluid region.

A coarse polyhedral mesh was applied initially, and then further refined as necessary, ensuring correlation with theory and ensuring consistency through visual inspection.

A mesh sensitivity study was conducted and is in Section IX. The resulting refined mesh is shown in Figure 3. This figure presents a top view of plane section taken across the heat sink, displaying the mesh concentration at the fin’s leading edge and in between. The image below it shows a plane section taken to display the mesh through the

middle of the entire section. This displays the concentration of the mesh approaching, and leaving the heat sink region. The final image at the bottom illustrates the mesh concentration at the trailing edge.

Figure 3: Top (plane section cut through the heat sink fins and fluid domain), middle (plane section taken to present a side view of mesh), bottom (cross

sectional plane to show mesh concentration at trailing edge)

B. Physics models and ContinuaIn order to arrive at a full solution of the flow field, it is necessary to enable various models, in addition to the k- ε- models, to aid the solution. The following models were used:

Fluid Continua Solid ContinuaBoussinesq Model -Constant Density Constant Density

Gas Multi-part SolidGradients GradientsGravity -

Segregated flow -Segregated fluid

temperatureSegregated solid

energySteady Steady

Three Dimensional Three DimensionalTurbulent -

Table 3: Physics continua models used in addition to the k- ε models in order to arrive at a full description of the flow field

Boussineq modelo This model provides a buoyancy

source term that only applies when there are small variations of density due to temperature variations.

Segregated Fluid/Solid Temperatureo This model solves the total energy

equation with temperature as the solved variable.

Gradientso This model accounts for:

Secondary gradients for diffusion terms

Pressure gradients for pressure-velocity coupling

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in the segregated flow model

Strain-rate and rotation-rate calculations for turbulence models

Segregated Flow Modelo This model solves the flow

equations (one for each component of velocity, and one for pressure) in a segregated, or uncoupled, manner.

Gravityo The Gravity model accounts for the

action of gravitational acceleration in STAR-CCM+ simulations.

o For fluids, it provides two effects: The working pressure

becomes the piezometric pressure.

The body force due to gravity can be included in the momentum equations.

Note the descriptions of each models purpose are sourced from Cd-Adapco’s Star CCM+ User Manual.

C. BenchmarkingIn order to ensure the validity of the model, the computed results will be benchmarked from the theory presented earlier. This will be conducted in conjunction with a mesh sensitivity study. Initially, the mesh sensitivity study will only factor in the heat sink thermal resistance and pressure drop. This will ensure the validity of the model. After hand, as the boundary layer is also of interest, a visual sensitivity study will be conducted in which the essential mesh to model the development of the boundary layer will be conducted.The Theoretical calculations are as follows:

Channel Reynold’s

number (Rech)

Thermal Resistance

(Rhs)

Pressure Drop (ΔP)

900 1.03 K/W 53.32 PaTable 4: Theoretical calculations corresponding to an inlet velocity of

2.9 m/s, a channel average velocity of 5 m/s and a channel Reynolds number of 900.

D. Numerical Methods and ConvergenceTo monitor convergence, the software automatically creates a residual monitor plot. This plot computes the error in the solution of the

solved equations. The software takes an initial guess and solves discrete equations for each cell. From this, the error is computed and processed into a residual to then be normalized so that it varies between 0 and 1. Although it can be said that the solution is converged when the residuals tend towards a small number, it is more effective to ensure convergence by monitoring quantities of interest. To help assess the stability and extent of convergence of the solution, a maximum temperature monitor (of the base plate) and pressure drop monitor (between the inlet and the outlet) were set up.The convergence of the solution was achieved when the residuals stabilised under a residual value of 1x10-3 and the maximum temperature monitor’s, and pressure drop monitor’s slope tended towards 0 and preceded in a stable manor. Figure 4 and 5 present a residual plot and a maximum temperature monitor respectively for the laminar simulation run. The residuals consist of 5 components, 4 residuals resulting from the solved transport equations and another residual from the solved energy equation with temperature as the variable. Convergence can be identified from the maximum temperature plot as the slope tends to 0, and the temperature remains constant.

Figure 4: Residual plot example, corresponding to the laminar solution whose results are presented in the next section

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Figure 5: Maximum temperature monitor plot, corresponding to the laminar solution previous mentioned

IX. MESH SENSITIVITY STUDYIn the conduction of the mesh sensitivity study, the Standard Low-Reynolds Number k-ε Model (Lien et al. 1996) was used, as it is effective in conjugate heat transfer and also computationally efficient in comparison to the V2F model. This model will give a good representation of the validity of the simulation.In the study, the initial wall treatment was set to low y+. Low y+ wall treatment was used as the y+ values corresponding to each base size used were in the range of 1-2. The wall y+ values were obtained from a surface average monitor on the fins alone. Appendix A contains the settings used in the mesh sensitivity study. In general, within the heat sink volume a polyhedral mesh was used. In the fluid region the same mesher was used, with addition of the surface repair, thin mesher and prism layer option. The thin mesher maximum thickness was set to the heat sink channel thickness. This allowed for a consistent mesh through the channels. The prism layer mesher was used in the fluid region and between the heat sink fins. In the fluid region only one prism layer was used, and between the fins 5 were originally used. The prism layer gap fill percentage was increased to the maximum in order to enable the prism layers to fill the gap between the fins. Custom controls enabled the mesh to be concentrated at the leading and trailing edge of the heat sink fins. At the leading edge this allows for a smooth transition of the mesh into the restricted flow area and at the trailing edge to facilitate the wake left by the fins. The results obtained are as follows:

With the applied settings, the base size was varied from 30-5mm in increments of 5mm. The results are presented in Table 5.

Base Size (mm)

Rhs Error

(%)

ΔP Error (%)

y+ Cell Count

30 1.1736

14.08 43.59

18.25 1.4 24187

25 1.1240

9.26 43.35

18.70 1.4 31084

20 1.1295

9.80 43.42

18.57 1.41

46727

15 1.1356

10.38 42.98

19.40 1.41

75764

10 1.1396

10.77 39.81

25.34 1.39

156072

5 1.14 10.81 38.15

28.45 1.38

617042

Table 5: Mesh sensitivity study results for a base size varied from 30mm-5mm in 5mm increments

Studying Table 5, it is evident that the initial mesh correlated quite close with theory. As the mesh base size decreases the thermal resistance sits around 10% error from theory, which is sufficiently close. On the other hand, as the base size decrease the pressure drop digresses from the theoretical value. This may be due to the fact that the pressure drop correlation used does not facilitate the flow conduit that the heat sink is in, and is an idealisation. With the given results it was decided to take the base size of 15mm and add some additional refinements. This was selected in consideration of the iteration time and accuracy. The number of prim layers between the fins was increased to 10, while their stretching was changed to 1.15. The results are as follows:

Base Size (mm)

Rhs Error (%)

ΔP Error (%)

y+

Cell Coun

t15 1.13

8710.6

941.5

722.03 0.

5247623

Table 6: Refinement to the 15mm base size mesh used in the initial mesh sensitivity study

With further refinements conducted, Table 6 displays that the percentage error increases slightly. It should be noted that the validity of either the computational model or theory is undermined by experimentation. However, no experimental results are available for this heat sink under the given conditions. Regardless, the benchmarking conducted indicates agreeance in particular with the heat sink thermal resistance, and some agreeance in regards to pressure drop.

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X.RESULTSThe refined mesh was used the compare the results of 3 different models; V2F Low-Reynolds Number k-ε model (Low Re), V2F Low-Reynolds Number k-ε model (V2F) and laminar model. The results are presented in Table 7.

Model (wall function)

Rhs Error (%)

ΔP Error (%)

Iterations to

converge

Low Re (low y+)

1.13875

10.69%

41.57

22.03%

120

Low Re (all y+)

1.13889

10.70%

41.56

22.06%

120

Laminar (none)

1.13986

10.80%

41.54

22.10%

140

V2F (low y+) 1.13875

10.69%

41.60

21.98%

200

V2F ( all y+) 1.13375

10.20%

41.71

21.78%

200

Table 7: Results obtained from three different models; Low Re, V2F and Laminar. For the Low Re and V2F the wall treatment was varied between

low y+ and all y+.

Note in Table 7, Convergence was determined as the number of iterations for all plots to stabilise i.e. Thermal Resistance, Pressure Drop and Residuals.Each model, and associated wall function option, was run for 100 iterations and a solver iteration time elapsed monitor was generated. This was then exported to excel, where an average time was obtained. This average time was then normalised about the laminar model, as it was the shortest, to gauge the comparison of the models in terms of iteration time. Table 8 presents the results.

Low Re (Low y+)

Low Re(all y+)

Laminar

V2F(Low y+)

V2f(All y+)

1.19 1.35 1.00 1.31 1.46Table 8: Iteration time elapsed (seconds) normalised by the laminar iteration

time elapsed (1.375s) for each simulation conducted.

Multiply the normalised iteration time by the number of iterations to convergence, yields a time scale for comparison between the simulations. Table 9 presents the results:

Low Re (Low y+)

Low Re(all y+)

Laminar

V2F(Low y+)

V2f(All y+)

142.8sec 162sec 140sec 262sec 292secTable 9: Number of iterations multiplied by the normalised iteration time

(sec).

The boundary layer was visualised between the fins through use of a section plane. Figure 5 and 6 present the scalar planes for the Low Re model for low y+ and high y+. These plots were generated for each simulation, and the rest are available in Appendix B.

Figure 5: Standard Low Reynold’s Number, all y+. Scalar plot of a plane section cut through the flow domain. Y direction coming out of page and Z

direction is from bottom to top.

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Figure 6: Standard Low Reynold’s Number, low y+. Scalar plot of a plane section cut through the flow domain. Y direction coming out of page

and Z direction is from bottom to top.

XI. DISCUSSIONFirstly, considering the heat sink thermal resistance computed from each simulation in Table 7 and rounding up to two decimal places, each model computes the same value of 1.14 K/W. This is around 11% error from theory. Next, considering the pressure drop through the heat sink, each model is computing a value of 41.6 Pa, plus or minus 0.1 Pa, corresponding to around 22% error with theory. The percentage errors may arise from the fact that the theory used is an idealisation and does not consider as much variables in the flow field as the computational model. Regardless, the correlation is adequate and the validity of the model is sound.Considering the time per iteration, it would be expected that the quickest would be laminar (as it solves 5 equations), then the standard low Reynold’s number model (as it solves the same 5 equations as the laminar plus 2 more), and finally the V2F model (as it solves the same 7 equations as the low re model plus an additional 2). The Laminar simulation solves the 4 transport equations, being continuity and x, y, & z momentum. The standard Low Reynold’s model solves these in addition to the turbulence dissipation rate and the turbulent kinetic energy.

Finally the V2F model solves the stated in addition to an elliptical function f, which is a redistributed term used to solve the last variable required, v' 2. Analysing Table 7, it is clear that this trend is satisfied, except however that the V2F (low y+) simulation had a quicker iteration time that the Low Re (all y+). Considering this, and the fact that the low y+ was quicker for both simulations than the associated all y+ for the same model, displays the additional computational requirement needed to use the ‘hybrid’ wall function. The all y+ attempts to merge the low y+ and high y+ wall treatment. It should be noted that the all y+ is designed for a more coarse mesh, with a y+ value ranging from 5-30. If used in this range, the computation time may have been reduced to lower than the corresponding low y+ values, at the potential expense of accuracy.Table 9 presents the normalised iteration time as a product of the number of iterations to convergence. These figures enable the comparison of the computation time for each simulation. The shortest convergence time being laminar, following to Low Re (low y+ first and all y+ after), and finally the V2F model (low y+ first and all y+ after). Recalling the proximity of the variables of interest (thermal resistance and pressure drop), the most efficient model is the laminar. The accuracy of the laminar model may loan itself to the fact that the flow is laminar between the heat sink channels, which is the area of interest. Therefore, the model is capable of competing with the more complex turbulence models. For the purpose of this heat transfer problem, the laminar model or Low Reynold’s models provide a sufficient solution in the shortest time.Analysing Figure 5 and 6, the scalar plot of a plane section cut through the flow domain, it can be seen that the development of the hydrodynamic and thermal boundary layer of both the all y+ and low y+ are in agreeance. Studying the fins leading edge in both plots (Z direction runs from bottom to top), the all y+ simulation displays a higher reduced velocity than the low y+, while at the trailing edge the low y+ model illustrates a larger wake than the all y+. Considering the V2F scalar plots, Appendix B, the same trend is evident for the V2F simulations. In comparison to these simulations, the laminar scalar plot, Appendix B, experiences a leading edge reduced velocity analogous to the low y+ simulations and has the smallest wake at the trailing edge. All in all, the scalar plots of all the simulations are near identical.

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XII. CONCLUSIONThe most significant conclusion that can be drawn from the analysis is that each model used correlates, with nearly the same proximity, with theory. Further from this;

The laminar model provided the quickest solution, in terms of time per iteration and iterations to convergence, when compared to the Low Re and V2F models.

For both the Standard Low Reynold’s model and the V2F model, the low y+ option reached convergence quicker than the corresponding all y+. This was determined to be an effect of the all y+ wall treatment attempting to emulate both the low y+ and high y+ wall functions.

Despite the fact that the V2F model solves 9 equations and the Low Reynold’s number 7, the low y+ V2F simulation had a quicker time per iteration than the high y+ Low Re model. However, the high y+ simulation reached convergence quicker.

Analysing the scalar plot of a plane section cut through the flow domain, it was seen that

o All simulations agreed upon the development of the hydrodynamic and thermal boundary layer.

o The all y+ simulations generated a higher reduced velocity at the leading edge than the low y+ model, while the low y+ model generated a larger wake at the trailing edge than the all y+ model.

o The laminar plot displayed a leading edge velocity analogous to the low y+ simulations and a lower wake than all simulations.

It is evident from the conduction and analysing of the simulations, that for a conduction-convection conjugate heat transfer problem of a plate-fin heat sink under the given flow conditions, the laminar model is just as adequate at achieving a thermal resistance and pressure drop value than the other turbulence models.

APPENDIX

A. Appendix AThe Reynold’s number is a ratio of inertial forces to viscous forces used to categorise a flow into three regimes. The Reynolds number is given by:

ℜDh=

ρu Dh

μ(Eqn.

13)

Where ρ= density of fluid (kg/m3), μ=dynamic viscosity (Pa-s), u=fluid velocity (m/s), and finally Dh=hydraulic diameter, which is a diameter measure defined to correlate the flow in a non-circular duct to that of a circular duct, and is given by:

Dh=4 AP

(Eqn.

14)

Where A=Cross sectional area (m2) and P=wetted perimeter (m).

The Nusselt number is defined as the ratio of convection heat transfer to fluid conduction heat transfer. This dimensionless parameter is dependent on the flow regime. It should be noted that when the flow constricts to the heat sink channels, the fluid will have to redevelop and entrance length effects may not be omitted. Therefore it is appropriate to use a Nusselt number correlation that accounts for both developing and developed flow. The correlation proposed by Teertstra et. al. (1999) factors both criteria and is given by:

Nui=¿¿ (Eqn. 15)

Where Nui=Ideal Nusselt Number (η=1), Pr=Prandtl Number and Reb* is defined as a modified Reynolds number and is aimed to combine the channel width, length and Reynolds number, otherwise known as Elenbass Rayleigh number for natural convection:

ℜb¿=ℜb . b

L(Eqn.

16)The convective heat transfer coefficient can be related to the ideal Nusselt Number by:

h=Nui .k f

b(Eqn.

17)

where kf is the thermal conductivity of the fluid, and b is the channel width.

The heat sink thermal resistance, Rhs (K/W) i.e. the resistance of the heat sink to, under the given flow conditions, the flow of heat and is given by:

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Rhs=1

h .¿¿(Eqn. 18)

Abase is the exposed area of the base (between the fins), Nfin is the number of fins, ηfin is the fin efficiency, and Afin is the surface area per fin taking into account both sides of the fin.

The fin efficiency, ηfin, is given by:

η=tanh (mH )

mH(Eqn.

19)

Where H is the height of the fins, and m is defined as:

m=√ hPk Ac

(Eqn.

20)

Where P is the perimeter (P=2t+2L), h is the heat transfer coefficient, k is the thermal conductivity of the fins, and Ac is the cross sectional channel area of the fins (Ac=tL).It may be noted that the relationship for Nusselt number (Eqn. 3) includes the effect of the temperature rise in the air as it flows through the fin passages. To obtain the total thermal resistance, Rtot, to the base of the heat sink it is necessary to add in the thermal conduction resistance across the base of the heat sink. For uniform heat flow into the base Rtot is given by:

Rtot=Rhs+H −H f

k base . w . L(Eqn.

21)The pressure drop, ΔP (Pa), across the heat sink is given by:

Δ P=(K c+4. f app . LDh

+K e) . ρ . V 2

2 (Eqn.

22)Where Kc and Ke are coefficients that represent the pressure losses due to sudden contraction channels and expansion of the flow entering and leaving the heat sink respectively. These coefficients are given by:

K c=0.42 (1−σ2 ) (Eqn. 23)

K e=(1−σ 2)2 (Eqn. 24)

Where σ is the ratio of the area of the flow channels to that of the flow approaching the heat sink.

fapp is the apparent friction factor, that takes into account the developing and developed regions of the flow in the heat sink channels. It is defined as:

f app=[( 3.44

√ L¿ )2

+( f . ℜDh )2]

1 /2

ℜDh

(Eqn.

25)

Where L* is a dimensionless length defined as:

L¿=L/ Dh

ℜDh

(Eqn.

26)The Poiseuille number is given by:

f . ℜDh=24−32.527 ε+46.721 ε2−40.829 ε 3+22.954 ε4−6.089 ε5

(Eqn. 27)

Where ε is defined as the fin aspect ratio:

ε=bh

(Eqn. 28)

Where b is the channel width, and h is the channel height (fin height).

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B. Appendix B

Figure B1: Laminar. Scalar plot of a plane section cut through the flow domain. Y direction coming out of page and Z direction is from bottom to top.

Figure B2: V2F Low Reynold’s Number, all y+. Scalar plot of a plane section cut through the flow domain. Y direction coming out of page and Z


Figure B3: V2F Low Reynold’s Number, low y+. Scalar plot of a plane section cut through the flow domain. Y direction coming out of page and Z


NOMENCLATURE

ρ Density Kg/m3

Pressure drop Pa-sVolume Flow rate m3/s

heat flux W/m2

A AreaCp Specific heat

capacityJ/Kg-K

D Diameter mh heat transfer coefficientH Height mk Thermal

conductivityW/m-K

K contraction/expansion coefficnet

-

N number -Pr Prandtl number -

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q heat load WR Thermal Resistance K/W

Re Reynold's number -

v velocity m/sα Thermal expansion

coefficient/K

η fin efficiency -μ Dynamic viscosity Pa-s

Subscriptshs heat sink

ch channel

tot totalf finsbase basec contractio

napp apparente expansionh hydraulic

XIII. REFERENCES

[1] Culham, J.R., and Muzychka, Y.S. “Optimization of Plate Fin Heat Sinks Using Entropy Generation Minimization,” IEEE Trans. Components and Packaging Technologies, Vol. 24, No. 2, pp. 159-165, 2001.

[2] Dorfman, A. (2010) Conjugate Problems In Convective Heat Transfer, CRC Press: Boca Raton.

[3] Lee, H. (2010) Thermal Design, Wiley: Hoboken, N.J.[4] Simons, R.E., “Estimating Parallel Plate-Fin Heat Sink

Thermal Resistance,” ElectronicsCooling, Vol. 9, No. 1, pp. 8-9, 2003.

[5] Simons, R.E., and Schmidt, R.R., “A Simple Method to Estimate Heat Sink Air Flow Bypass,” ElectronicsCooling, Vol. 3, No. 2, pp. 36-37, 1997.

[6] Teertstra, P., Yovanovich, M.M., and Culham, J.R., “Analytical Forced Convection Modeling of Plate Fin Heat Sinks,” Proceedings of 15th IEEE Semi-Therm Symposium, pp. 34-41, 1999.

[7] Wendt, J., Anderson, J. (2009) Computational Fluid Dynamics, Springer: Berlin.

[8] Griffin, P. (2016) Advanced Computational Fluid Dynamics.

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