TUT 5: Natural Convection

Heat Transfer in FluentCFD 814

Adhikar Hariram (18121004)

2013

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ContentsTable of figures......................................................................................................................................3

Introduction...........................................................................................................................................4

Technical Section...................................................................................................................................4

Problem Description..........................................................................................................................4

Problem Setup...................................................................................................................................5

Results...............................................................................................................................................6

Dimensionless Numbers..................................................................................................................19

Conclusion...........................................................................................................................................20

References...........................................................................................................................................21

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Table of figures

Figure 1: Geometrical Description of Problem......................................................................................4Figure 2: Mesh Generated.....................................................................................................................5Figure 3: Surface Temperature Integral Convergence SIMPLE..............................................................7Figure 4: Temperature Contour Plot SIMPLE Scheme............................................................................8Figure 5: Surface Temperature Integral Convergence Coupled.............................................................9Figure 6: Temperature Contour Plot Coupled; 2nd Order Upwind........................................................9Figure 7: Temperature Contour Plot Coupled; 1st Order Upwind.......................................................10Figure 8: Temperature Contour Plot Coupled; QUICK.........................................................................11Figure 9: Larger Domain Size...............................................................................................................11Figure 10: Surface Temperature Integral Convergence; Large Domain...............................................12Figure 11: Temperature Contour Plot Large Domain...........................................................................13Figure 12: Surface Temperature Integral Convergence; Rough Mesh.................................................14Figure 13: Residuals for Rough Mesh Simulation................................................................................14Figure 14: Temperature Contour Plot Rough Mesh.............................................................................15Figure 15: Thermal Boundary Conditions Issues..................................................................................16Figure 16: Temperature Contour Plot Inviscid.....................................................................................16Figure 17: Velocity Vectors for Inviscid Flow.......................................................................................17Figure 18: Temperature Contour Plot Boussinesq...............................................................................18

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Introduction

A problem that needs analysis when looking at components such as heat sinks or even ram modules on a motherboard is that of natural cooling. These components will transfer heat to the surrounding fluid which increases its temperature, resulting in a change in density of the fluid. This change in density causes the heated fluid to rise, creating a plume of heated fluid above the component. This effect has been investigated in FLUENT with a custom defined fluid around an aluminium cooling fin that has a heat flux into its base, as it would in reality from the component it is cooling. It has also been investigated using various discretisation schemes, solvers and density approximations.

Technical Section

Problem DescriptionThe problem that was modelled consisted of an aluminium fin, heated at its base, surrounded by a fluid of custom properties. The domain also had a symmetry plane across the centre of the fin and was open to the atmosphere on the side and top of the domain. The base of the fluid domain was also specified as being insulated. It was modelled as a 2-D geometry as only the plume for a section of the fin was of concern. A geometrical description of the problem can be found in figure 1.

Figure 1: Geometrical Description of Problem

The values for H, W as well as the other variables used in the simulation can be found in table 1. All of the values used were derived from the student number 18121004. It should also be noted that the size of the domain was not pre-defined as its effect on the solution was also of interest. Thus its size was varied for different simulations in order to investigate the effect that the domain size had on the results.

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Table 1: Values used in Simulation

Property/Dimension ValueCp (specifc heat) 1000 J/KgKR (gas constant) 4000 J/kgK

k (thermal conductivity) 0.001 W/mKt 2 mmH 10 mmW 20 mm

q'' (heat transfer rate) 4000 W/m2 (changed by lecturer during tutorial session)

T (operating temperature) 313.15 K

Problem SetupThe first step of the problem setup was to create the geometry for the domain, which was done using the ANSYS Design modeller. The first step was to create a sketch for the solid, which was the aluminium fin. Since the problem was being modelled as 2-D; the option to create surface from sketches was used to create a planar surface. This surface was then frozen so as not to merge with the fluid region, which was sketched and created the same way as the solid region; with the bottom left corner of the fluid domain coinciding with the bottom left corner of the solid region. The next step was then to create a Boolean operation to subtract the solid surface from the fluid region. The option to keep the tool body was enabled in order to apply the q'' in at the base of the solid in FLUENT. Finally both regions were selected and one part was created in order for the interaction region to automatically be recognised by FLUENT. Once the geometry was created the next step was to create the mesh for the problem. The mesh was created in ANSYS meshing and in order to create a perfectly square mesh on the region the edge sizing function was used in conjunction with the face sizing function in order to enforce the cell size everywhere in the domain concerned. This was done by selecting every edge, both solid and fluid in the domain, and then applying the edge sizing. The surface sizing however was used only in the fluid domain as the edge sizing function sufficed to enforce the cell size in the solid region. One of the resultant meshes can be seen in figure 2.

Figure 2: Mesh Generated

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During the meshing stage the named selections were also created in order to assign the appropriate boundary conditions in FLUENT. The selections created were: Pressure_Inlet, Pressure_Outlet, Symmetry, Q_in, Insulated and Interaction_region which was the wall surface of the fin in contact with the fluid region. The next step was to setup the problem in FLUENT, however this step varied in order to investigate the effect of the different discretisation schemes, different solvers, different density approximations and the effect of inviscid flow on the solution. Apart from these differences, the common setup steps in FLUENT involved first turning on gravity and defining its magnitude and direction. Next the energy equation was turned on so as to model the heat transfer between the solid and the fluid. Once this was done the fluid properties were changed to those given in table 1 as they would be constant for every simulation. Next the operating conditions were set with the operating pressure being set to 100 kPa, as this was specified by the problem, and the operating temperature was set to that in table 1. After this step the boundary conditions had to be set. The pressure inlet and outlet were both defined to have 0 Pa gauge pressure and temperatures of 313.15 K, as per table 1. The Q_in value was set to 4000 W/m2 as per the values in table 1 and all the other conditions were set as default due to FLUENT recognising that a boundary named Symmetry is a symmetry plane and any unidentified boundary names are walls. Walls are also automatically set to have zero heat flux across them, as with an insulated wall, thus there was no need to change the boundary conditions at the insulated wall. Once the boundary conditions were set, the discretisation schemes as well as pressure-velocity coupling was chosen before setting up surface monitors to better judge the convergence of the solution as compared to looking purely at the residual plots. The monitors that were setup included a mass flow rate across the outlet as well as a surface temperature integral over the Interaction_region wall. When the plots of both these monitors became approximately constant it was an indication of convergence of the solution as the values being monitored were longer changing. The solution was then initialised using Hybrid Initialisation which solves the Laplace equation to produce a velocity field and smooth pressure distribution in the domain (1). Once initialised the solution was calculated and results extracted.

ResultsWhen investigating the effects of the previously mentioned factors in the Finite volume method, the following order of simulations was decided on in order to compare the effects that they would have on the results as well as convergence times:

1. SIMPLE versus Coupled pressure-velocity coupling2. Different discretisation schemes3. Domain size4. Mesh independence5. Viscid versus Inviscid fluid6. Incompressible ideal gas versus Boussinesq density approximation

The first simulation was run using the SIMPLE solver with the discretisation schemes being left as their default values, namely, 2nd order upwind discretisation for both momentum and energy. With the upwind differencing scheme, cell face values upwind of the centroid is taken to be the same as the value of the cell centroid upwind of the cell concerned (2). This is what is done when the first order upwind scheme is used. With the second order upwind scheme, the cell face values are instead computed using a Taylor series expansion of the cell-centred solution about the cell centroid (1). See (1) for the equation governing the calculation of the cell face values. The SIMPLE velocity-

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pressure coupling scheme uses a relationship between the velocity and pressure corrections in order to satisfy mass conservation and consequently solve for the pressure field before moving onto the next iteration (1). The density model that was also used for this simulation, as well as all the following simulations, excluding the final one, was the incompressible ideal gas model. The incompressible ideal gas model ignores the effect that pressure has on the density and approximates the pressure based solely on the temperature according to the following equation:

ρ=PoperatingRuniversalM

T−(1)

This equation has been obtained from (1). When using this approximation the density was calculated based on the operating pressure and temperature and was specified as the operating pressure in the operating conditions in FLUENT. The molecular weight M was also specified for the fluid material after specifying the density as obeying the incompressible ideal gas law. After then following the steps described in the Problem Setup section, the solution was found to have converged after approximately 400 iterations when investigating the monitor plots that were set up. This can be seen from figure 3 which shows the convergence history of the temperature integral over the surface of the fin in contact with the flow.

Figure 3: Surface Temperature Integral Convergence SIMPLE

Whilst the surface monitors had converged by 400 iterations, the residuals were set to converge only at a point where the continuity and energy reached a value below 1x10 -6. Thus the simulation was run until the point where the residuals reached this value which occurred after approximately 660 iterations. The resultant temperature contours can be seen in figure 4 which clearly illustrates the plume above the fin.

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Figure 4: Temperature Contour Plot SIMPLE Scheme

The second simulation was run using the same settings as the first, however the pressure-velocity coupling was changed to the Coupled solver. This algorithm solves both the momentum and pressure based continuity equation simultaneously (1). With this single change it was found that the solution converged in 137 iterations. It was however found that the residuals converged before the surface temperature integral plot did, and hence the solution needed to be run further. This still resulted in a solution still converging twice as fast as with the SIMPLE scheme in approximately 250 iterations. This was again judged by the surface temperature integral plot as seen in figure 5.

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Figure 5: Surface Temperature Integral Convergence Coupled

The temperature contour plot for the second simulation can be seen in figure 6.

Figure 6: Temperature Contour Plot Coupled; 2nd Order Upwind

From figure 6 it can be seen that the solution has been largely affected by the change from a SIMPLE to Coupled algorithm. This can be seen by the greater width of the plume at the top of the domain. This solution however would appear to be more accurate as the plume would be wider as a result of the width of the base of the fin also heating the surrounding air, causing it to rise a result of the change in density as a result of its rise in temperature. Due to this width of heated air at the fin base; the width of the heated air at the top of the plume would also be larger than simply the thickness of the fin.

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Due to the faster convergence as well as producing an apparently more accurate result, all subsequent simulations were run using the coupled algorithm. Once this decision was made, the following two simulations were run in order to compare the effect of the different discretisation schemes on the result as well as the convergence rate. Seeing as the previous simulation was done using the second order upwind discretisation scheme; only the effect of the first order upwind and QUICK discretisation schemes needed to be investigated. The first comparison was made with the First order upwind discretisation scheme. With this scheme it was found that the convergence rate was almost exactly the same as that for the second order scheme with convergence occurring at approximately 255 iterations. The resultant temperature contour plot from as a result of this scheme can be seen in figure 7.

Figure 7: Temperature Contour Plot Coupled; 1st Order Upwind

From figures 6 and 7 it can be seen that there is no noticeable difference in the results from using a second and first order upwind differencing scheme. The final discretisation scheme that could be compared to was the QUICK scheme.

The QUICK scheme is a quadratic upwind differencing scheme which uses a quadratic function passing through the cell centroid and two upwind cell centroid's in order to find the face value of the cell concerned (3). When the simulation was run using the QUICK discretisation scheme it was found that the solution converged within approximately 240 iterations. This shows that it was fractionally faster than the first and second order upwind schemes (only 4% faster). The temperature contour plot resultant from the simulation run using the QUICK scheme can be seen in figure 8.

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Figure 8: Temperature Contour Plot Coupled; QUICK

From the comparison between figures 8,7 and 6, no apparent difference in the results can be seen thus showing that the three different discretisation schemes would lead to the same result if all other factors affecting the solution remained constant.

After investigating the results from the coupled scheme, the effect of the domain size on the solution needed to be investigated. This is especially true after noticing that the domain used in the previous simulations did not capture the full height of the plume above the fin. The larger domain size that was used can be seen in figure 9.

Figure 9: Larger Domain Size

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From figure 9 it can be seen that the new domain size that was used was greater than twice the domain size used in the previous simulations. When running the simulations with the new, larger domain, the Coupled pressure-velocity coupling was used along with the second order discretisation schemes. This was done as the coupled solver converged the fastest as well as providing seemingly more accurate results and the second order upwind schemes were the default in FLUENT. It was found that even with the domain being more than twice as large, the solution still converged within approximately 270 iterations. This was found, as before, by monitoring the surface temperature integral plot due to the residuals converging before the plot became constant. This plot can be seen in figure 10.

Figure 10: Surface Temperature Integral Convergence; Large Domain

The resultant temperature contour plot from the simulation run using the larger domain can be seen in figure 11. It should be noted that with this larger domain; the size of each cell remained the same as for the simulations run with the smaller domain, that is, the mesh density was kept constant.

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Figure 11: Temperature Contour Plot Large Domain

From figure 11 it can be seen that the larger domain captures a much larger portion of the plume than as with the previous domain sizes. The general shape and temperature contours of the plume however appear to be consistent as with those found from the simulations run with the smaller domain size. Thus the only observable effect of running the simulation with a larger domain size is the fact that greater detail of the solution can be viewed at the cost of a fractional increase in the solving time. For this reason, the subsequent simulations were run with this larger domain size in order to gather greater detail of the results.

Beyond investigating the effect of domain size on the result of the simulations, the mesh independence of the solution was also of concern. Due to the mesh that had been used in the previous simulations being a relatively fine mesh, as well as the FLUENT academic licence having a limit on the number of cells that can be used; in order to check the mesh independence of the solution the domain was re-meshed using cells that were twice as large. This resulted in a mesh that was half as dense as previously used and would thus provide a result that could be compared to the previous simulation in order to check the mesh independence of the solution. The newly used mesh resulted in the convergence of the solution occurring at approximately 140 iterations, as judged by the plot in figure 12. However after investigation of the residuals after 140 iterations it was decided that greater accuracy was desired and hence the simulation was run to 200 iterations, as can be seen in figure 13.

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Figure 12: Surface Temperature Integral Convergence; Rough Mesh

Figure 13: Residuals for Rough Mesh Simulation

When the simulation was run using the rough mesh, the temperature contour plot generated can be seen in figure 14. It should be noted that the simulation was run with the Coupled solver as well as second order upwind differencing schemes as this would allow for a comparison to previous results.

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Figure 14: Temperature Contour Plot Rough Mesh

From the comparison between figures 14 and 11, it can be seen that the results produced are unaffected by the change in mesh density. It should be noted however that whilst the solution converged in less iterations, the actual runtime of the solution was still not significantly less than that for the fine mesh due to the fairly simple geometry. Thus, whilst the solution was seemingly unaffected by the mesh, the following simulations were still run with the fine mesh due to the fast solving time as a result of the simple geometry.

The next comparison that could be made was the effect of inviscid flow on the results. When the model was switched from viscous-laminar flow to inviscid flow however, it was found that thermal boundary conditions on the solid were not available, which is shown in figure 15. In order to overcome this peculiarity, the flow was set to viscous laminar but all the walls in the domain were set have a specified shear stress of 0 Pa as this would simulate the effects of inviscid flow at the walls. Using this approximation along with the coupled scheme and second order upwind discretisation it was found that the solution converged within approximately 260 iterations which was not much different from the solution using the viscous-laminar model. The resultant temperature contour plot can be seen in figure 16.

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Figure 15: Thermal Boundary Conditions Issues

Figure 16: Temperature Contour Plot Inviscid

From figure 16 it can be seen that the results are drastically different than when compared to the viscous laminar results. This can be attributed to the fact that for the zero shear stress at the wall, the fluid velocity at the wall is non-zero and hence the heat transfer rate to the fluid is not nearly as high as that for the viscous-laminar flow. The inviscid solver also does not contain terms for molecular diffusion which would largely affect the solution as the flow is at such a low speed, the

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diffusion term would be a large contribution to the fluid temperature within the domain (1). This is especially at the start of the simulation when no flow is present within the domain. The velocity vectors of the flow were also investigated due to the unexpected results, and this revealed that the flow at the fin was in the opposite direction than the expected flow which would result in a plume above the fin. Instead of the air rising as would be expected, the velocity vectors of the air are as shown in figure 17.

Figure 17: Velocity Vectors for Inviscid Flow

From figure 16 it would appear as of the plume is now in the horizontal plane, and this can be explained by figure 17 as the flow across the fin is now from top to bottom and right to left. This flow gets heated and carries the heated air to the left of the fin, creating the plume seen in figure 16.

This illustrates that the results obtained for an inviscid fluid, approximated by applying zero shear at the walls, are highly inaccurate as it is known for this scenario that a plume would be created in the opposite direction of the gravity vector. Once it was established that the solution found using an inviscid fluid was inaccurate, a comparison between the Boussinesq density approximation and the incompressible ideal gas law density approximation needed to be investigated.

With the Boussinesq approximation, the density is set as constant in all the solved equations with the exception of the buoyancy term in the momentum equation (4). The main purpose for this approximation is to reduce the convergence time of the solution and thus the affects of the Boussinesq approximation on both the convergence time as well as final solution has been investigated. When setting up the problem using the Boussinesq approximation; the density of the fluid material had to be changed from incompressible-idea-gas to boussinesq, and its constant density had to be specified. This was entered simply as the operating density specified in the

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previous simulations. The thermal expansion co-efficient, beta, had to also be defined, which is given by the equation:

β= 1T

−(2)

In equation (2), the temperature T was taken as the operating temperature of 313.15 K. In the operating conditions specification the operating density check box had to also be disabled in order to implement the Boussinesq approximation, whereas the operating pressure and temperature were left as 100kPa and 313.15 K respectively. The Coupled solver as well as second order upwind discretisation schemes were again used for this simulation in order to compare the results to previous simulations.

When using the Boussinesq approximation it was found that the solution converged within approximately 150 iterations, which was in the region of 40% faster than using the incompressible-ideal-gas law. The resultant temperature contour plot found using the Boussinesq approximation can be seen in figure 18.

Figure 18: Temperature Contour Plot Boussinesq

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From the results obtained it can be seen that the results are indeed affected by the use of the Boussinesq approximation as compared to the incompressible ideal gas law. This approximation shows a much thinner plume, which was established as being an incorrect result. This may be due to the fact that the Boussinesq approximation is only valid for (4):

β (T−T0 )=(T−T0 )T

≪1−(3)

whereas the value for the left hand side of the above equation is close to 1 in regions near the fin. This would result in the Boussinesq approximation providing inaccurate results in the region near the fin which is exactly where the plume base is. This would propagate through the domain resulting in an inaccurate representation of the plume of hot air. It should be noted however that the results from the Boussinesq approximation are fairly close to those achieved using the incompressible ideal gas law as seen in figure 11. From this it can be said that; whilst the results from the Boussinesq approximation appear to be correct, the relationship given by equation (3) needs to be evaluated in order to check the validity of the solution generated. The likely reason that this could not be used is the fact that the heat into the base of the fin was fairly large at a value of 4000W/m2.

Dimensionless NumbersWhen analysing flows where buoyancy can play a dominant role, the ratio of the Grashof to Reynolds number squared is of concern in order to evaluate the strength of the buoyancy contributions to the flow (1). This ratio is as follows:

Gr

ℜ2= gβ∆TL

ν2−(4)

For the case of this ratio being 1 or greater then there would be large buoyancy contributions to the flow. In this scenario, L is given as L+t/2 for the fin (5), and this ratio was calculated to be in the region of 8, which indicates that indeed there are strong buoyancy effects in the flow. The other number of concern with regards to the flow is the Raleigh number, given by (1):

Ra=gβ ΔT 3 ρC p

νk−(5)

For this scenario the Raleigh number was calculated to be in the region of 7.45x1015 which indicates a very strong buoyancy effect as anything above 108 would indicate turbulent buoyancy induced flow (1). This is a result of the large temperature difference in the domain which dominates the numerator in equation (5) and hence results in a large Raleigh number. When looking at the heat transfer we can evaluate the Peclet number and the Prandtl number. For a given flow geometry the Peclet number is given as (6):

Pe=VLρC p

k−(6)

This number provides an indication as to whether the transport terms or diffusion terms are stronger and for a high Peclet number the transport terms dominate. For this flow scenario, based on the area weighted averaged velocity, the Peclet number is approximately 25, which indicates that

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majority of the heat transfer is via transport as compared to diffusion, as is expected with a natural convection problem with buoyancy terms. The Prandtl number is given by:

Pr¿ν ρCpk

−(7)

and indicates the ratio of momentum to thermal diffusivity (6). For this flow this is in the region of 147, which indicates a large amount of fluid movement with relatively little thermal diffusion in the fluid.

Conclusion

From the simulations conducted it can be concluded that when analysing a natural convection cooling problem, the different discretisation schemes are able to handle a simple geometry all equally. It can also be concluded that the solution is mesh independent for a simple geometry such as that which has been simulated. A conclusion can also be drawn about the size of the domain for the specific problem that has been modelled in that; the larger the domain size, the more information can be extracted from the results at a relatively small increase in the computational cost of the domain. It can also be concluded that the larger domain does not affect the solution itself whilst allowing for more information to be extracted from the results. When looking at the pressure-velocity coupling of the solution it can be concluded that the SIMPLE algorithm results in greater solution times and possibly less accurate results than the Coupled method. The inviscid solution for the problem modelled can also be concluded as providing inaccurate results due to not accounting for diffusion terms which largely affect the solution at the start of the simulation. Finally it can be concluded that for this specific problem the Boussinesq approximation is invalid but does however provide results that appear to be correct except when compared to the solution obtained with the incompressible ideal gas law. Due to the invalidity of the Boussinesq approximation it can also be concluded that the incompressible ideal gas law resulted in the most accurate solution to this problem.

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References

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2. HK, Versteeg and W, Malasekera. 5.6 The upwind differencing scheme. An Introduction to Computational Fluid Dynamics The Finite Volume Method. Essex : Pearson Education Limited, 2007.

3. Versteeg, HK and Malasekera, W. 5.9 Higher-Order differencing schemes for convection diffusion problems. An Introduction to Computational Fluid Dynamics The Finite Volume Method. Essex : Pearson Education Limited, 2007.

4. Ansys-help. Ansys FLUENT User Guide. [book auth.] Ansys-Help. Ansys FLUENT User Guide. s.l. : ANSYS, 2011.

5. Incropera, F P and De-Witt, D P. Overall Surface Efficiency. Fundamentals of Heat and Mass Transfer. Massachusetts : John Wiley & Sons, 2002.

6. —. Physical Significance of the Dimensionless Parameters. Fundamentals of Heat and Mass Transfer. Massachusetts : John Wiley & Sons, 2002.

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