978-1-61284-736-8/11/$26.00 ©2011 IEEE 76 27th IEEE SEMI-THERM Symposium

Heat Sink Design Optimization Using the Thermal Bottleneck Concept

Robin Bornoff, Byron Blackmore & John Parry Mentor Graphics Mechanical Analysis Division

81 Bridge Road, Hampton Court, Surrey UK robin_bornoff@mentor.com

Abstract

Calculation and display of a thermal bottleneck scalar field as an integrated part of a CFD simulation enables a practitioner to interact with and understand the physical mechanisms by which heat is removed from an electronics system. By applying the characteristics of this thermal bottleneck scalar to heat sink design aspects, one can identify near optimal solutions with a minimal number of simulations. This work will detail the principles of using thermal bottleneck information to optimize fin thickness distribution and copper slug design and compare the results to that obtained by more traditional Design of Experiments and numerical optimization techniques.

Keywords Thermal, Bottleneck

Nomenclature BN – BottleNeck Number Dh – hydraulic diameter = 2 x fin gap DoE – Design of (Numerical) Experiment Lhy – Hydrodynamic Entrance Length Re – Reynolds Number RSM – Response Surface Modeling SC – ShortCut Number SO – Sequential Optimization

1. Introduction Electronics thermal management involves the design of an

electronics system to facilitate the effective removal of heat from the active surface of an integrated circuit (the heat source) out to a colder ambient surrounding. As the heat travels from the source it passes through various objects and length scales; from the die through the package to the board, into a chassis and out to an operating environment.

How ‘easily’ the heat passes from the source(s) to the ambient will determine the temperature rise at the source and all points in-between. The often complex 3D heat flow paths carry proportions of the heat with varying degrees of ease. Those paths that carry a lot of heat and which offer large resistances to that heat flow are considered ‘thermal bottlenecks’. Identifying and relieving these bottlenecks through a redesign will allow the heat to pass to the ambient more easily, thus reducing temperature rises along the heat flow path, all the way back to the heat source.

The addition of new heat flow paths that allow the heat to bypass these bottlenecks and pass directly to colder areas and on to the ambient more easily will also result in a decrease in temperature rises. Identification and implementation of such thermal ‘shortcut opportunities’ also allow targeted design changes to be made with maximum effect.

2. BottleNeck and ShortCut Numbers and Their Characteristics

2.1. BottleNeck Number (BN) The dimensionalized BN number is the dot product of the

heat flux and temperature gradient vectors (Fig. 1).

Figure 1: Misaligned Heat Flux and Temperature Gradient vectors

In vector notation: BN = Heat Flux • Temperature Gradient. In scalar notation: BN = |Heat Flux x Temperature Gradient x cos(�) |.

If the angle between the two vectors is zero, i.e. the heat flux is aligned with the temperature gradient as it would be for conductive heat flow in a homogenous thermally isotropic material, then BN is the product of the magnitudes of the vectors, since cos(0°) =1 .

Large values of this BN scalar, computed as a part of the thermal simulation, pinpoint areas of high heat flow experiencing a large local thermal resistance (characterized by a large, aligned temperature gradient), and thus identify the thermal bottlenecks in a design. Normalizing this scalar by the maximum value in a model will provide an indication of the relative levels of thermal bottleneck in a single simulation model.

2.2. ShortCut Number (SC) The dimensionalized SC number is also calculated from

the heat flux and temperature gradient vector fields. The SC scalar value at any point is calculated as the magnitude of the cross product of the two vector quantities. In vector notation: SC = Heat Flux x Temperature Gradient. In scalar notation: SC = | Heat Flux x Temperature Gradient x sin(�) |.

If the temperature gradient is orthogonal to the heat flux, then SC is simply the product of the vector magnitudes, since sin(90°) =1.

Large values of the SC field pinpoint areas where large heat flux vectors are misaligned with large temperature gradient vectors (i.e., the heat is not moving directly toward a

Bornoff et al, Heat Sink Design Optimization Using the… 27th IEEE SEMI-THERM Symposium

significantly cooler area), and thus identify locations where the benefit in establishing a new heat transfer path to shortcut the heat to colder areas of the design is highest. Normalizing this scalar by the maximum value in a model will provide an indication of the relative levels of shortcut opportunities in a single simulation model.

3. Literature Review Examples of the application of the BN and SC Numbers to

thermal management applications have been previously published [1, 2]. In [1], BN and SC are introduced, and distributions of both are shown for a single TO 263 device on a board to illustrate where the optimal locations in the package to implement thermal design changes are to be found. In [2], BN and SC distributions are used to derive a sequence of thermal design changes for a board in a forced convection situation. BN distributions suggested the shape of the optimal copper pad for two overheating packages, while SC distributions suggested the use of a heat sink and the optimal locations for an array of thermal vias.

4. Optimization vs. Insight for Heat Sink Design The BN and SC Numbers can be used individually, or

together with any electronics thermal design. In this paper we will focus on the application of the BN Number to heat sink design. Heat sink design is a generic challenge that now forms part of the thermal design of many electronics systems, from consumer electronics to military avionics. The traditional approach to heat sink design optimization is automated parametric design optimization, using numerical Design-of-Experiment (DoE) techniques coupled with Response Surface Modeling (RSM) or Sequential Optimization (SO) in conjunction with CFD [3]. While these approaches have been shown to be very effective, they can also consume prohibitive amounts of engineering time and computing resource. The basic methodology is as follows. A base model is defined in the analysis software following standard modeling practices for the tool in use. Then, the parameters to be varied as part of the DoE have to be selected and their ranges defined. Next a sufficiently large set of design points needs to be generated within the design space, usually requiring a minimum of 5-10 design points per design parameter. Each of these designs then has to be solved to a converged state. Once all the designs have been solved a Response Surface can be created by fitting the data points for a given objective (or cost) function, comprised of quantities that should be optimized, such as weight and temperature rise. The minimum value of the Response Surface can then be found within the design space, and this design created and solved to check the performance of the design. An alternative to using RSM is use SO, which entails starting from the best design found from the DoE set of simulations, and stepping sequentially towards the optimum using analysis of previously solved models to guide the step size and direction within the design space.

In contrast to the traditional approach, using the BN Number field and the physical insights it offers allows designers to create near-optimal design configurations directly from a single simulation.

The following sections will discuss the implementation of both techniques for two selected subsets of heat sink design, and draw conclusions about the results and calculation time.

5. Heat Sink Fin Thickness Optimization The first heat sink design task considered is the

optimization of individual fin thicknesses for an extruded plate fin heat sink while minimizing the heatsink mass but maximizing thermal performance. The test case for this design task is described in Table 1 and Figure 2. This is a simplified case which consists of a deliberately oversized heat sink with a small heat source at the center of a thick base. We have also fixed the device velocity, i.e. the velocity of the flow entering the finned region constant and uniform across the fin gap. Note that the deliberate over sizing of this heat sink was necessary to effectively isolate the effect of the fin thickness on heat sink performance. Otherwise, the conclusions of the study would be affected by heat spreading in the base and varying channel flow rates.

5.1. Case Description Due to symmetry, only half of the width of the heat sink

was modeled. The data shown in Table 1 is for the full heat sink.

Material Aluminum, k=137 Wm-1K-1 Base Thickness 15 mm Base Length 400 mm Base Width 400 mm Fin Thickness 0.5mm* – 6.0 mm Fin Height 360 mm Number of Fins 28 Device Air Speed 1 ms-1 Heat Source Size 5 x 5 mm Heat Source 100 W Cost Function Total Mass (g) + 2xThermal

Resistance (KW-1) Table 1: Heat Sink and Optimization Parameters (*=Base Case)

Figure 2: Heat Sink Model Configuration. Heat Source is Marked in Red

Bornoff et al, Heat Sink Design Optimization Using the… 27th IEEE SEMI-THERM Symposium

The length of the heat sink fins has been chosen to ensure that the thermal boundary layers formed on the fins do not touch before the flow has reached the end of the heat sink (again, to allow study of fin thickness effects in isolation). Using the following equation for a flat duct [4]:

���

��

� ���� ��� �

� ������

The hydrodynamic entrance length is 603mm for the

narrowest fin channel (as derived from the maximum fin thickness in Table 1).

5.2. DoE Description To demonstrate the power of the BN approach we first

adopt a classical DoE approach, where the thickness of every fin in the heat sink is independently allowed to vary within it’s design limits, followed by SO from the best design. A DoE of 300 designs (representing approximately 21 designs per design parameter) was constructed so that the fin thickness of each of the 14 fins was varied independently over the range shown in Table 1. The cost function includes a contribution from both the heat sink mass and the base-to-ambient thermal resistance. The cost function had an observed range of 150.8 to 160.9 over the 300 DoE simulations. The cost function for the optimum design arrived at by SO was 145.4. The results of the DoE are plotted in Figure 3 as well as a depiction of the optimal heat sink fin thickness distribution. The four central fins in this optimal design were at the maximum fin thickness of 6 mm. Moving outwards from these central fins, the next two fin thicknesses drop down to ~1mm and ~0.75 mm and all remaining fins were determined to be optimal with the minimum thickness of 0.5 mm.

Figure 3: DoE and SO Results for Cost Function. Initial

and Optimized Heat Sink Geometry are shown.

5.3. BN-based Optimization For a general thermal design application, use of the BN

Number for design optimization proceeds as follows. The region of highest BN is found and the design is changed to reduce the values. This usually involves decreasing the local thermal resistance in the vicinity of the highest BN by either increasing the thermal conductivity of the material, or by increasing the local cross-sectional area of the high BN flow

path in question. With the design change in place, the new model is then solved to identify the region of highest BN for the new design. This process is repeated, until a thermally satisfactory design is arrived at. If this process is continued until its logical conclusion the result is a uniform BN field, characterized by all possible heat flow paths being equally conductive. We therefore hypothesize that the optimal design will have the same BN value for each fin and this hypothesis is used as the basis for manual optimization of the heat sink design.

The BN field for the base case was plotted and the BN maximum value for each fin, which occurs at the root of each fin, was noted. The design was then modified as follows.

The thickness of each fin was scaled between the minimum value of 0.5 mm and the maximum value of 6 mm linearly with BN number. The fin with the smallest BN number retained a thickness of 0.5 mm, and the fin with the largest BN number was set to a width of 6 mm. The resulting design is shown in Figure 4. The optimal heat sink design from the DoE and SO approach is overlaid for comparison.

Figure 4: Thickness Distribution Based on Maximum BN number

Solving this new BN-derived design resulted in a cost

function of 145.5, within 0.1% of the results achieved through the use of DoE and SO. Further, the shape of the fin thickness distribution is similar for both techniques, with the thickest fins located centrally, and a sharp drop off to the minimum allowable thickness as we move away from the center.

With the BN approach we are not directly attempting to limit the mass of the heat sink. However, the BN approach increases the fin width from its minimum value only where it is needed to improve the thermal performance, thus having a very similar effect to including the heat sink mass in the cost function for the SO calculations.

6. Introducing a Slug in the Heat Sink Base As a further demonstration of the use of the BN number

we investigated the BN field in the heat sink base around the heat source in the base case reported above, but with the base thickness increased to 35 mm to allow for a larger variation of BN within the base (see Figure 5). Again, this step was taken to allow the investigation to isolate, as much as possible, one

Bornoff et al, Heat Sink Design Optimization Using the… 27th IEEE SEMI-THERM Symposium

aspect of the heat sink design, namely the heat spreading in the base near the heat source.

Figure 5: BN Distribution in the Heat Sink Base

The BN field goes from a maximum of 1 just above the

heat source and decreases as we move away from the center. In this case, as the base is relatively thick, the BN isosurfaces (for larger values of BN) are approximately hemispherical a short distance from the heat source as the heat flux lines are not significantly influenced by the rest of the heat sink geometry until we move closer to the fins. To address the thermal bottlenecks indicated by this BN distribution, and hence improve the thermal design, we can introduce a slug of higher conductivity material where the largest BN values are observed. It follows then, that the most optimal shape for a slug of a given volume will closely resemble a BN isosurface, as the integral of BN over the slug’s volume will be maximized.

To investigate this, copper inserts were designed using the BN distribution in Figure 5 to guide the insert geometry creation. Eight BN cutoff values were selected, and hemispheres of copper (in some cases, truncated hemispheres were required, as the BN isosurface extended upwards far enough to extend past the top of the heat sink base) were created to approximate the isosurface corresponding to that cutoff value. Each copper slug was placed in the heat sink base immediately above the heat source in turn. The model for each of these inserts was solved and the results are presented in Figure 6.

Figure 6: Effect of BN Cutoff value on Temperature Rise.

In Figure 6, ‘All Aluminum’ records the temperature rise when the entire heat sink is all aluminum (no copper insert). ‘All Copper’ shows the temperature rise when the heat sink is made entirely of copper. Figure 6 clearly shows that as the heat slug is made bigger (the smaller BN Number Cutoffs), the incremental improvement reduces. This is aligned with expectations, as areas of the heat sink base experiencing low levels of BN are less important to address with a design change than high BN level areas. The corollary to this observation is that the smaller copper inserts represent the most efficient use of copper, as the gain in thermal performance per unit volume of copper is highest for the largest set of BN cutoff values.

The BN field gives a clear indication of where the thermal bottlenecks are in the design and therefore where to include a bottleneck relieving insert. Based on Figure 6, the thermal designer can select a BN cutoff as the best balance between cost, performance, and manufacturability, recognizing that diminishing returns is a concern for smaller BN cutoff values.

To verify the assumption that the BN isosurface shape should guide the design of the copper insert shape, we compared the performance of the hemispherical insert (for a selected BN cutoff value of 7.5x10-5, the second point from the left in Figure 6, with a diameter of 40.2 mm) with cuboidal inserts having the same volume of copper, but different shapes. These shapes ranged from covering the entire bottom surface of the heat sink base to spanning the entire thickness of the base, with thicknesses or footprints chosen to maintain an equivalent volume of copper. The resulting temperature rises for the nine tested insert shapes are shown in Figure 7.

Figure 7: Effect of Conserved Volume Copper Slug Dimensions on Temperature Rise

The optimum shape found for the cuboidal insert was a block that had dimensions of 32.4 x 32.4 x 16.2 mm high. This shape is one half of a cube, and when comparing this to the 40.2 mm hemispherical insert evaluated earlier, represents an identical bounding box aspect ratio (i.e., the height of the insert is ½ the dimensions of the bounding box footprint). This matches our supposition that the BN isosurface chosen as the cutoff defines the shape of the region within which the thermal conductivity should be increased. In this case the

Bornoff et al, Heat Sink Design Optimization Using the… 27th IEEE SEMI-THERM Symposium

temperature rise for the hemisphere and the optimum equi-volume cuboidal shape are within 0.2%. This suggests that approximating the BN isosurface for the selected cutoff value with an equivalent volume, equivalent bounding box aspect ratio representation produces satisfactory results.

7. Conclusions Identification and visualization of the BN scalar field

offers insight into the physical mechanisms of how heat moves from source to ambient. This work has demonstrated how the BN number can be applied to two aspects of heat sink design. Deriving optimal heat sink fin thickness via inspection of the BN distribution in each fin was demonstrated to produce similar results to that of a best practice DoE and SO simulation approach, but using a small fraction of the computing resources. The optimal location and geometry for a copper insert of a fixed volume was found to be governed by the shape of a BN isosurface at an assumed cutoff value.

Acknowledgments CFD simulations and BN and SC post-processing were

carried out with Mentor Graphics’ FloTHERM V9.1 software.

References 1. John Parry, Robin Bornoff, Byron Blackmore, “Thermal

BottleNecks and ShortCut opportunities; innovations in electronics thermal design simulation” Electronics Cooling Magazine, Vol. 16, No. 3, Fall 2010, pp. 24-25.

2. Byron Blackmore, John Parry and Robin Bornoff, “New 3D thermal quantities help designers address thermal problems as they arise” Cover Story in Printed Circuit Board Design & Fabrication Magazine, Vol. 29, No. 11, pp. 30-32

3. J. Parry, Robin Bornoff, P. Stehouwer, Lonneke Driessen and Erwin Stinstra, “Simulation-Based Design Optimisation Methodologies Applied to CFD”, Proceedings of 19th SEMI-THERM Symposium, San Jose CA, March 2003, pp. 8-13

4. Kakaç, Sadik, Ramesh K. Shah & Win Aung, Handbook of Single-Phase Convective Heat Transfer, John Wiley & Sons (New York 1987) Ch. 3 pp. 35