robin bornoff bottleneck y shortcut.pdf

May 13, 2010 Copyright © 2010 Mentor Graphics Corporation Page 1 of 78

AppNote 10811

Thermal Bottlenecks and Shortcut Opportunities; Innovations in Electronics Thermal Design by Simulation By: Byron Blackmore, Robin Bornoff, PhD, Mechanical Analysis Division Last Modified: May 13, 2010

A P P N O T E S SM

Thermal Bottlenecks and Shortcut Opportunities; Innovations in Electronics Thermal Design by Simulation

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Table of Contents 1. Introduction ....................................................................................................................................................... 3 2. Description of Parameters ................................................................................................................................ 6

2.1. Bottleneck Number (Bn) ......................................................................................................................... 6 2.2. Bottleneck Example ............................................................................................................................... 8 2.3. ShortCut Number (Sc) .......................................................................................................................... 10 2.4. ShortCut Example ................................................................................................................................ 11 2.5. Bn and Sc Fields for a PCB Mounted Component ............................................................................... 16

3. Validation Cases ............................................................................................................................................. 18 3.1. Conduction Bottleneck Example .......................................................................................................... 18 3.2. Conduction Shortcut Example .............................................................................................................. 23 3.3. Conduction Labyrinth ........................................................................................................................... 29 3.4. Developing Boundary Layer in Channel with Fixed Temperature Walls .............................................. 36 3.5. Developing Boundary Layer in Channel with Local Fixed Temperature Walls .................................... 40

4. Real World Examples ...................................................................................................................................... 43 4.1. Package Level Modeling ...................................................................................................................... 43 4.2. Heatsink Design Optimization Worked Example ................................................................................. 58 4.3. System Level Refinement Worked Example ........................................................................................ 65

5. Discussion ....................................................................................................................................................... 75 6. Summary – Usage Advice .............................................................................................................................. 77


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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 scales; from the die

through the package to the board, into a chassis and out to an operating environment.

Figure 1 – Heat flow from die to ambient

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’.

Figure 2 – The complex 3D heat flows drawn as a thermal resistance network. Two of the thermal

resistances are identified as being the largest 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 pass to colder areas and on to the ambient more

easily will also result in a decrease in temperature rises. Identification of such thermal ‘shortcut opportunities’

would allow targeted design changes to be made with maximum effect.


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Figure 3 – Introducing a new heat flow path to improve thermal performance.

Electronics thermal simulations have always provided an indication of component junction and case

temperature, information that can be used to judge thermal compliance by comparing the simulated

temperatures to maximum rated operating temperatures. Useful information of course, but provides very little

insight about why the temperature field is the way it is.

Figure 4 – Typical thermal simulation results showing temperatures and air flows

Thermal simulations can allow further insights into the heat removal paths by, for example, examining heat

flux vectors. However, indications of the direction and magnitude of the heat flux vectors do not provide a

measure of the ease by which the heat is leaving the system, nor does it provide insight into where and how

the heat flux distribution could be better balanced or re-configured to improve performance.


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Figure 5 – Typical thermal simulation results showing heat flux vectors

One of the innovations introduced here is the calculation of a new, 3D scalar field that represents the strength

of the local thermal bottleneck. This scalar is computed as a part of the thermal simulation and will pin point

areas of the design where there is high heat flow and that heat flow is finding it difficult to pass. Such a

property is called the ‘Bn’ number; the BottleNeck number.

Figure 6 – Thermal bottlenecks in a generic TO 220 Package

The second innovation presented here is a new 3D field that indicates where heat is travelling ‘parallel’ to a

locally colder area, i.e. it is not moving towards a colder region of the design. This new field will identify

where there is an opportunity to bypass that heat to the colder area by the insertion of a new heat flow path.

Such a property is called the ‘Sc’ number, the ShortCut number.


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Figure 7 – Potential thermal Shortcuts in a generic TO 263 package

Together, the Bn and Sc fields offer tremendous insight into why the thermal performance of a design is

what it is, and better, provide graphical information as to where the best opportunities for design improvement

can be found. The Bn field provides clear identification of thermal bottlenecks and the Sc field indicates

opportunities for thermal shortcuts. As we’ll see in this document, application and interpretation of the Bn and

Sc distributions within typical electronics design constraints offers a systematic method to determining what

thermal design modifications are most promising, and provides evidence beyond engineering intuition and

experience that such modifications should be pursued.

2. Description of Parameters

The mathematical description of the Bn and Sc post-processing parameters are described in this section.

The following description and its application to the post-processing of simulation data has been developed by

Mentor Graphics and a patent is pending at the time of publication.

2.1. Bottleneck Number (Bn)

Heat flow can be defined in terms of a heat flow through a given cross section area. This measure is known

as a heat flux.

The presence of a heat flux vector will always result in a temperature gradient vector. The temperature

gradient field is taken to be an indicator of conductive thermal resistance as, for a given heat flux, the greater

the temperature gradient is the larger the thermal resistance will be.

The dimensionalized Bn number is the dot product of the above 2 vector quantities. At each point where there

exists a heat flux vector and temperature gradient vector:


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Figure 8 – Misaligned Heat Flux and Temperature Gradient vectors

The Bn scalar at that point is calculated as:

Heat Flux magnitude (i.e. length of the heat flux vector arrow) x Temperature Gradient magnitude (i.e.

length of the temperature gradient arrow) x |Cos(θ)|

Note that |Cos(θ)| is always positive

If the angle between the two properties 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 the Bn number is the

product of the vector magnitudes (i.e., |Cos(θ)| =1) . If the temperature gradient is orthogonal to the heat flux

then the Bn number is zero.

SI units for such a dot product are WdegC/m3, akin to a volumetric power density multiplied by a

temperature rise. In FloTHERM, the Bn number is normalized by dividing through by the maximum value of

that variable in a model, thus producing a range that always has a maximum value of 1 to facilitate results

inspection and interpretation (i.e., Bn = 1 will always be the ‘worst’ local bottleneck in the design). The Bn

number is therefore dimensionless.

Regions of the analysis that exhibit large values of Bn, will have three qualities:

• A large value for heat flux, i.e., it will be on an existing heat transfer path of significance.

• A large value of temperature gradient, i.e., there will be significant thermal resistance at that point.

• The heat flux and temperature gradient vectors will be aligned

Therefore, areas that are thermal bottlenecks, i.e., are on an existing heat transfer path and experience

significant resistance to heat transfer, will result in large values of the Bn parameter. Once these bottleneck

areas have been ‘designed out’, heat will leave the model more ‘easily’ thus reducing all ‘upstream’

temperature rises, all the way back to the heat source.


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As implied by the m3 term in the original unit denominator, Bn is a ‘per volume’ parameter. It is important

to recognize this and consider the integral value of Bn over a volume of interest when using Bn to analyze a

system. For example, a small pocket of large Bn may easily be less important that a vast area of moderate

Bn.

2.2. Bottleneck Example

To further illustrate what one can expect from the Bn number, and how it correlates with temperature, heat

flux and temperature gradient, we’ll investigate the case of a point source of heat.

Consider a small radius, infinitely long cylindrical source of heat inset into a solid material with a constant

isotropic thermal conductivity. The source of heat is set with a fixed temperature of 100 °C. The far periphery

of the solid is set to 0 °C with a heat transfer coefficient of 110 W/m2K. The temperature drops off from the

maximum at the heat source as the heat spreads radially out toward the 0 °C boundary condition:

Figure 9 – Radial Temperature Distribution

The magnitude of the heat flux (W/m2) drops off even faster with radial distance from the heat source, visible

in both the scalar magnitude plot and the heat flux vector plot.

100degC


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Figure 10 – Heat Flux Distribution

The temperature gradient is what drives the heat flux, the gradient will be positive in the direction that points

towards the heat source (heat flows ‘downhill’, from hot peak to cold trough). Again, the gradient decreases

with radial distance away from the heat source:

Figure 11 – Temperature Gradient Distribution

The Bn number is the dot product of both the temperature gradient and heat flux vectors, normalized by the

maximum Bn value in a simulation model. For a conduction situation within a homogenous material both

vectors will always be aligned so the Bn number is simply the product of both heat flux and temperature

gradient scalars (Figure 10 and Figure 11). Being the product of the two, the drop off in Bn radially away from

the source is much more pronounced than either the heat flux or temperature gradient. This Bn region is

indicative of the increase in thermal bottleneck close to the source only.


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Figure 12 – Bn Distribution

The radial Bn field implies that there is no preferred direction for the heat to flow. It is the perfect

conduction situation in a way, as there is no thermal resistance dependency on conduction path. Heat will

find it equally easy/difficult to leave the system regardless of the direction.

2.3. ShortCut Number (Sc)

The Sc number is also calculated from the heat flux and temperature gradient vector values. The Sc scalar

value at any point is calculated as the magnitude of the cross product of the two vector quantities:

Heat Flux magnitude (i.e. length of the heat flux vector arrow) x Temperature Gradient magnitude (i.e.

length of the temperature gradient arrow) x |Sin(θ)|

Note that |Sin(θ)| is always positive

If the angle between the two properties is zero, i.e. the heat flux is aligned with the temperature gradient, then

the Sc number is zero. If the temperature gradient is orthogonal to the heat flux then the Sc number is simply

the product of the vector magnitudes (i.e., |Sin(θ)| =1).

Like the Bn number, SI units of such a cross product are WdegC/m3. In FloTHERM, the number is

normalized by dividing through by the maximum value of that variable in a model, thus producing a range that

always has a maximum value of 1, again for reasons of facilitating results interpretation. The Sc number is

therefore dimensionless.

Regions of the analysis that exhibit large values of Sc will have three qualities:

• A large value for heat flux, i.e., it will be on an existing heat transfer path of significance.

• A large value of temperature gradient orthogonal to the heat flux direction, i.e., there will be areas of

lower temperature adjacent to the heat flux path


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• The heat flux and temperature gradient vectors will be misaligned, i.e., existing heat transfer path is

not moving heat toward the largest temperature gradient.

So areas that exhibit high levels of heat flux, but have it moving in the ‘wrong’ direction will produce large

values of Sc. It’s the potential benefit gained by establishing a new heat transfer path in the direction of the

colder area of the design (thus aligning the heat flux and temperature gradients) that is important here.

As implied by the m3 term in the denominator, Sc is also a ‘per volume’ parameter. It is important to

recognize this and consider the integral value of Sc over a volume of interest when using Sc to analyze a

system. For example, a small pocket of large Sc may easily be less important that a vast area of moderate

Sc.

Considering the definition of Sc in the realm of conduction, one will note that within a homogeneous,

isotropic material, Sc will always be zero as the heat flux and temperature gradient vectors will always be

aligned. However, at the interface of two or more dissimilar isotropic materials, or within an orthotropic

material, this will not be the case. Sc will thus typically manifest itself with non-zero values at the interfaces

between materials, highlighting the areas where conduction shortcuts should begin or end.

Evaluating Sc in a moving fluid will have a very different interpretation. Efficient convective heat transfer is

characterized by a large temperature gradient, usually normal to the solid surface and a large heat flux

convected with the fluid, usually parallel to the solid surface. As the vectors are misaligned in this

circumstance, we can use Sc near a convection surface to gauge where convective heat transfer is most

effective.

2.4. ShortCut Example

To illustrate what one can expect when working with the Sc field in a conduction situation, we’ll investigate a

model which is similar geometrically to traces on a PCB. The geometry is shown below in Figure 13. A

constant width, unnecessarily twisty, copper path moves 10 W from one location to another that’s held at 0 °C.

For the purposes of this example, the only design consideration that is important is the reduction of

temperature at the 10 W injection point.


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Figure 13 – Copper conduction path within a dielectric material.

The temperature results are shown in Figure 14 for this simulation.

Figure 14 – Temperature Field


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The heat flux field in Figure 15 is nearly uniform within the copper. This indicates that virtually all of the 10 W

travels the long copper path in preference to traversing the short segment of dielectric material.

Figure 15 – Heat Flux Magnitude

The temperature gradient field shows a very large gradient in the region between the 10 W source and the

fixed temperature sink as implied by the temperature plot in Figure 14.

Figure 16 -Temperature Gradient


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The Sc distribution for this case is shown in Figure 17.

Figure 17 – Sc Distribution. Largest areas of Sc = 1 are highlighted. Thermally connecting the two

red areas would create a shortcut for heat flow.

The largest values of Sc are found on the faces of the vertical copper strips nearest the 10 W source and sink.

This is indicating the area between the copper strips would be an excellent place to insert a new conduction

path. This observation certainly agrees with intuition, as building a copper bridge in this location would

eliminate nearly all of the copper track length and thus would serve as a great thermal shortcut.

Other shortcuts are revealed when the Sc scale is reduced. Each one is an area where a new conduction

path would eliminate a large fraction of the copper track length.


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Figure 18 – Sc field with a smaller range. Additional thermal shortcuts are highlighted here.

It’s important to note that the temperature gradient vector field is an excellent partner to the Sc results. The

Sc field can be used to identify promising areas for the start or end of a new thermal shortcut, the temperature

gradient vectors will point out the direction of the shortcut if it’s not obvious. Figure 19 superimposes

temperature gradient vectors on top of the Sc distribution in the vicinity of the topmost shortcut highlighted in

Figure 18.

Figure 19 – Sc field with Temperature Gradient Vectors


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2.5. Bn and Sc Fields for a PCB Mounted Component

Consider a 100 mm x 100 mm x 2 mm PCB with a centrally mounted cuboidal representation of an active IC

package dissipating 10 W. Both component and PCB have an isotropic thermal conductivity of 10 W/mK. The

PCB is placed in a computational wind tunnel with an approach air speed of 3 m/s and temperature of 0 °C.

Figure 20 – Velocity vectors over the component and PCB

Temperature contours show the heating up of the component and the board around the component as well as

the heating up of the air above and below the PCB.

Figure 21 – Temperature Distribution

Examining the temperature gradient vectors show the largest gradients to be up into the underside of the PCB

beneath the component and from the air down into the top of the component.


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Figure 22 – Temperature Gradient Vector Distribution

80% of the heat dissipated in the component passes down into the PCB whereupon it spreads outwards to be

convected away from both top and bottom PCB surfaces. 20% of the heat is convected directly away from the

top surface of the component.

Figure 23 – Heat Flux Vector Distribution

The Bn number distribution shows higher values in the PCB at the edge of the component/PCB interface,

bottlenecks as the heat spreads outwards, areas where there is an opportunity to relieve the bottlenecks thus

reduce source temperatures. There is also a local ‘jet’ of higher Bn number in the air as it leaves the top of the

component in the flow direction. This convective Bn area is due to the air flow (thus heat flux) direction , as

such it is not considered an area of potential bottleneck improvement, just a by product of the flow field.


Spreading resistance bottleneck

Convective Bn ‘jet’


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The Sc number is highest in the thermal boundary layers on the top of the component and below the PCB

under the component. For solid/fluid interfaces, where there is heat transfer occurring across those interfaces,

the Sc number in the air adjacent to those surfaces is actually indicative of the effectiveness of that heat

transfer. This is opposed to the interpretation of Sc number in conductive situations which will be more

indicative of the opportunity to add new heat flow paths. For this application where all solids have the same

thermal conductivity there are low Sc numbers within those solids as the heat flux vectors will all be aligned

with the temperature gradients.

Figure 25 – Sc Distribution

3. Validation Cases

In this section, several simplistic examples will be investigated. The Bn and Sc fields will be inspected and

used to determine the best locations to apply thermal corrections so as to improve overall thermal

performance. Such an approach will be validated by either correlating such a location with one of many

locations that a design modification is made at or alternatively validated against empirical correlations for

Nusselt number.

3.1. Conduction Bottleneck Example

Motivation

This example is focused on validating usage of the bottleneck number. The model discussed in this

section will involve a clear and obvious thermal restriction with the intention of demonstrating that the Bn

number can capture and display the bottleneck. Further to this, proof will be provided that the locations of the

largest values of Bn are indeed the optimal locations to investigate when selecting a thermal design solution

to an existing heat transfer path.

Effective solid/fluid convective heat transfer


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Description

The geometry for the first example considered is shown in Figure 26.

Figure 26 - Model setup for Conduction Bottleneck Validation Example

There are two areas defined with a fixed temperature, connected by a 30 mm long, variable cross section

conduction path (k = 100 W/mK). The path is 1.5 mm wide for the right and left sections and tapers down to

0.5 mm wide in the central region. The conduction path is completely surrounded by an insulating material

with k = 0.01 W/mK. All external boundaries are defined as adiabatic. The primary result of interest in this

example is the heat conducted from the 100 °C area to the 0 °C area.

Expectation

The expected result is that most of the heat will flow along the 100 W/mK conduction path. The largest

values of dT/dx and heat flux will be observed along the portion of the path with the smallest cross-sectional

area. As Bn is the product of those figures, and within a isotropic material, one can expect the heat flux and

temperature gradient to be aligned, we expect Bn to have large values along this narrow portion of the path.

The expectation for the validation phase of this example is that the best locations for a design modification

will be proven to be in the vicinity of the narrow section of the path as well.

Results

Figures 27-30 show temperature, heat flux, temperature gradient, and Bn distributions for this model. The

heat flow through the system is 315 mW.


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Figure 28 – Heat Flux Magnitude Distribution


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Figure 29 – Temperature Gradient Magnitude Distribution

Figure 30– Bn Distribution

The plot of Bn indicates two items of interest. 1) The largest value of Bn is located at the bottom corners

of the tapers as the cross sectional area reaches its minimum and 2) The entirety of the reduced cross

section area of the bar shows a Bn value of ~ 0.5. The moderate value of Bn in the middle section, combined


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with its large area, makes it the optimal location to pursue thermal design improvement by bottleneck

reduction.

Validation

The validating design modification for this case will be a copper block with dimensions of 1.5 mm x 0.5

mm. The block will always be centered on the conduction path. The 1.5 mm dimension is selected so as to

enable the block to completely replace the original conducting material at any given cross section. The block

will start at the 100 °C boundary and parametrically be moved at 0.5 mm intervals over the length of the

model as shown in Figure 31. The heat flow from the 100 °C boundary is measured for each case.

Figure 31 – Copper block size and range

Figure 32 shows the normalized increase in heat flow from the original design versus the lateral position

of the copper block. The optimal location for this copper block is seen to be between 10 mm and 20 mm.

Figure 32 also plots normalized Bn along the bar centerline for the original simulation (without the copper

block). There is an excellent level of correlation between the observed values of Bn and the amount of heat

flow increase for each block location.


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Figure 32 – Plot of additional normalized heat flow and normalized Bn number distribution versus

lateral location. The increase in heat flow is the value recorded when the center of the copper block was simulated at that lateral location. The Bn values are taken from along the bar centerline in the

original simulation.

It’s important to note that the absolute maximum values of Bn were found in the corners of the taper and

did not cover a large area of the model. When considering Bn as an integral property, this would be

considerably less important than the rather large coverage of moderate Bn values predicted along the center

section of the bar. This integral nature of the Bn parameter is captured well by the validation exercise in this

section.

3.2. Conduction Shortcut Example

Motivation

This example is focused on validating usage of the Shortcut number. The model discussed in this section

will involve a clear and obvious thermal shortcut that is not utilized in the original design with the intention of

demonstrating that the Sc number can capture and display the shortcut. Further to this, proof will be provided

that the regions with the largest integral values of Sc are indeed the optimal locations to consider when

selecting a thermal design solution that addresses a potential heat transfer path.


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Description

The geometry for the second example considered is shown in Figure 33.

Figure 33 – Model Setup

In this example, there are three fixed temperature boundary conditions. A fixed temperature of 100 °C is

thermally connected to a fixed temperature of 30 °C by a 30 mm long, thin Aluminum bar. A short distance

below that is a 13 mm, wider, bar of copper that is connected to a fixed temperature setting of 0 °C. Filling

the rest of the model is dielectric material. This serves as a thermal break between the two metallic bars and

inhibits heat from flowing from the aluminum bar to the copper bar. The key result again is the total amount of

heat that flows from the 100 °C fixed temperature boundary.

Expectation

The expectation for the initial design is that a large percentage of the heat from the 100 °C boundary condition

will flow directly along the aluminum bar to the 30 °C boundary condition. The layer of dielectric material will

effectively block access to the potential thermal shortcut available in the form of the copper bar connected to

the 0 °C boundary condition. The overlap area between the copper bar and thin aluminum bar is expected to

have the largest values of Shortcut number. The validation stage of this section should reveal that creating a

thermal conduction path down to the existing copper bar (which is connected to the 0 °C boundary) in this

overlap area will best increase the amount of total heat flow. The optimal locations for the new thermal

conduction path are expected to be indicated by the Sc distribution.


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Figure 34 – The yellow block arrows indicate the expected heat flow route after the shortcut is in place

Results

Figures 35-39 show temperature, heat flux, temperature gradient, Bn, and Sc distributions for this model.

The heat flow through the system is 0.267 W.



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Figure 36– Heat Flux Distribution



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Figure 39 – Sc Distribution

The plot of Bn indicates that nearly all of the bottleneck in this design is along the aluminum bar. Looking

at this model from only a Bn perspective the best improvement would be to increase the cross sectional area

of the bar.

The plot of Sc for this analysis shows the area on the bottom of the aluminum bar immediately above the

copper bar has Sc values ranging from 0.5 to 1. The largest values of Sc are located near the left most end

of the copper bar. This meets expectations, as the sooner the heat can be utilizing the copper block, the

lesser the distance it must conduct through the more thermally resistive aluminum bar.

Again, the validation approach for this scenario is to introduce a copper block to the analysis. The block in

Figure 40 is 1 mm wide and 3 mm high, which matches exactly the vertical distance between the aluminum

Location of Maximum Sc at ~18 mm from

100 °C edge


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and copper bars. A parametric study moved the copper block horizontally at 1 mm intervals and recorded

the heat flow from the 100 °C surface.

Figure 40 – The range of copper block locations investigated.

Figure 41 is plotting the normalized increase in heat flow and Sc distribution along the bottom of the

aluminum bar versus location. The optimal location for this copper block is seen to be 17 mm which is an

excellent match with the pocket of large Sc observed in Figure 39 (the maximum value of Sc is at roughly 18

mm).

Figure 41 – Normalized increase in heat flow and Sc on the bottom of the Aluminum bar versus

lateral position


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The correlation between Sc and +Q for locations greater than 17 mm is excellent, with the curves nearly

parallel in this section of the plot. The lack of correlation for locations smaller than 17 mm is due to the

chosen shape of the copper block shortcut introduced. The Sc parameter has elevated values along this

range as it is detecting possible shortcuts at an angle as shown in Figure 42. The validating copper block test

was only able to consider pure vertical shortcuts as a consequence of it being an axis aligned rectangle.

Figure 42 – The yellow arrows indicate angled potential shortcuts from the aluminum bar to the

copper bar. The Sc field will detect these shortcuts, but the validating copper block used in this section will not.

Once more, the integral nature of these parameters is shown in this example. The best range of positions

to introduce a thermal shortcut was predicted well by the large pocket of Sc predicted.

3.3. Conduction Labyrinth

Motivation

This example is focused on validating usage of the Bottleneck number and Shortcut number

simultaneously. The model discussed will not have a clear and obvious best answer, but instead present a

scenario that has many different conduction paths of unknown (or at least, unknown via geometric inspection)

thermal resistance. The intent is to demonstrate that the relative importance of thermal problems can be

captured with Bn and Sc and to demonstrate that considering Bn and Sc as integral values can be more

important that determining the absolute maximum value.


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Description

The geometry for this example is shown in Figure 43.

Figure 43 – Conduction labyrinth of variable kx, ky, and kz.

In this example, the geometry consists a 100 mm x 100 mm x 1 mm block consisting of a 100 x 100

regular array of orthotropic materials. Each orthotropic material has different values for kx, ky, and kz. The

colors of each section in Figure 43 give a qualitative measure of percentage copper in that section: yellow

indicates pure copper (k = 385 W/mK), green indicates pure dielectric (k = 0.3 W/mK). The entire left edge of

the model is fixed at 100 °C and the right edge is fixed at 0 °C. Again, the key result is the total amount of

heat observed flowing out of the 100 °C edge.


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Expectation

It is expected that multiple heat transfer paths will be observable in the results where the largest

concentrations of high percentage copper are found, and the narrowest of these heat transfer paths will result

in largest Bn values. Similarly, it’s expected that the pockets of relatively low percentage copper will not

conduct much heat and will result in the largest values of Sc in those regions.

Results

The heat flow in this original design is 18.92 W and plots of the field variables are provided below.



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Figure 45 – Heat Flux Magnitude Distribution



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Figure 47 – Bn Distribution. The solid white ovals indicate where the integral value of Bn within is

high.

Figure 48 – Sc Distribution. The dashed white ovals indicate the areas of the model where the

integral value of Sc is high.


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The plot of heat flux in Figure 45 meets expectations. The highly randomized copper distribution has

resulted in a number of distinct heat transfer paths throughout the conduction maze. The plot of temperature

and temperature speaks to the torturous paths available for heat flow as well.

The Bn plot in Figure 47 shows that are several areas in the model where the integral value of Bn would

be relatively large. The largest such area is found near the upper half of the right side, where there are

several pockets of maximum Bn values. Note that the Bn field is qualitatively much different than that the

heat flux field. This wasn’t the case for the earlier examples. For more complicated scenarios like this

conduction maze, the variation in thermal conductivity along the heat flow path causes the different behavior,

and leads to more valuable information than heat flux alone.

The Sc plot in Figure 48 shows that are several areas in the model where the integral value of Sc would be

relatively large. The largest such area is again found near the upper half of the right side, where there are

several pockets of maximum Sc values. There is significant overlap between the Bn ovals and the Sc ovals

on the right edge. This would be an excellent place to begin when determining the optimal location to

investigate thermal design changes, as it contains both bottlenecks to correct and shortcuts to leverage.

For this case we’ll determine the optimal location for a 10 mm x 10 mm block of copper and compare this

with the maximum Bn and Sc locations. The block of copper will be simulated in 100 different locations by

shifting it in steps of 10 mm so as to cover every section of the model individually.

Figure 49 shows the results of this investigation. For each block position considered, the surface plot is

showing the increase in heat flow due to the inclusion of the block at that position.


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Figure 49 – Map of improvement to heat flow (W) with the inclusion of 10 mm x 10 mm block at

various locations. The solid and dashed white ovals are transferred directly from Figures 47-48

The areas of large Bn and Sc integral values show an excellent correlation with the measured

improvement to heat flow. Every block position that resulted in a heat flow increase of greater than 1 Watt is

accounted for with an oval. Recall, the ovals were originally drawn around those regions of the model

deemed to have a large Bn or Sc integral values in Figures 47-48. The maximum improvement is found

along the right edge showing excellent correlation with the locations of where both the Bn and Sc integral

values are estimated to be highest.

It’s important to reiterate that in complicated conduction scenarios like this, the heat flux and temperature

gradient fields in isolation cannot provide sufficient data to draw the conclusions made here. It’s the

combination of heat flux and temperature gradient, and their relative alignment, in the form of Bn and Sc that

provides the insight.


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3.4. Developing Boundary Layer in Channel with Fixed Temperature Walls

Motivation

This example is intended to show a qualitative correlation between the variation in local Nusselt number and

shortcut number on the surface of a fixed temperature wall that is transferring heat to a cooler developing

boundary layer across it. A condition encountered in flow between say heatsink fins. A local Nusselt number

provides a measure of the effectiveness of heat transfer from a surface to the surrounding fluid. It is

notoriously difficult to fully automate the extraction of a ‘local ambient’ temperature used in the Nusselt

number definition. By showing a correlation of Nusselt number with Sc number, and the fact that the Sc

number is itself much simpler to define, use of the Sc number to identify and appreciate areas of effective

convective heat transfer can be proposed. Note that this appears to be in contradiction to the previous

ascertation that the Sc number is indicative of where there is an opportunity to insert new heat transfer paths

where currently there exists none. However the nature of the Sc number, involving a temperature gradient

orthogonal to a predominant heat flux path, when examined at solid/fluid interfaces where the air flow is

parallel to the surface, should also be indicative of the local effectiveness of the heat transfer.

Description


Figure 50 - Parallel Fixed Temperature Plate Model Geometry

A solid wall at a fixed temperature of 0 °C is defined with a length of 0.2 m. A constant velocity of air is

introduced over the wall at a speed of 1 m/s and at a temperature of -100 °C. Symmetry boundary conditions

are applied on the upper plane of the model at a height of 5 mm above the fixed temperature wall. This

equates to a parallel wall channel type flow with a channel gap of 10 mm. A Reynolds number of 1261

‐100 degC 1 m/s

0 degC


May 13, 2010 Page 37 of 78

(laminar flow) is realized. Symmetry boundary conditions are applied on the two sides of the model

representing an infinitely wide channel.

Expectation

For a fixed wall temperature, the effectiveness of heat transfer will be seen as the ratio of the amount of heat

that is transferred to the air and how close that air temperature is to the wall temperature. More heat moved

for less temperature difference is the result of effective heat transfer! This effectiveness will be high in the

entrance to the channel as the fluid is still relatively cold compared to the wall and the local velocity gradient is

high thus the wall friction will be high and thus, due to the Reynold’s analogy, heat transfer will be more

effective.

A Nusselt number is defined as: ′

Where:

L = characteristic length (m)

k = thermal conductivity of fluid (W/mK)

q’ = heat flux (W/m2)

Twall = local wall temperature

Tambient = local ambient temperature**

**taken as flow weighted average vertically across the channel at a plane adjacent to where the local heat flux

is recorded

The effectiveness of heat transfer is inherent in its definition, i.e. the ratio of q’ and dT.

The Sc number will be large where the predominant heat flow direction is perpendicular to the local

temperature gradient. For this geometry the predominant heat flow direction is in the direction of the air flow:

Figure 51 - Parallel Fixed Temperature Plate Heat Flux and Temperature Gradient Directions

Dominant temperature gradient direction

Dominant heat flux direction


May 13, 2010 Page 38 of 78

As the air heats up as it flows through the channel the perpendicular temperature gradient will reduce thus the

Sc number should also reduce. So, although the Sc number is NOT a direct analogy to the local Nusselt

number, it should be high where there are symptoms that are indicative of high Nusselt number conditions.

Results

Plots of temperature, heat flux, temperature gradient and Sc number are shown in the following figures (for a

few channel heights in the flow direction only for clarity).

Figure 52 - Parallel Fixed Temperature Plate: Temperature Distribution

Figure 53 - Parallel Fixed Temperature Plate: Heat Flux Distribution


May 13, 2010 Page 39 of 78

Figure 54 - Parallel Fixed Temperature Plate: Temperature Gradient Distribution

Figure 55 - Parallel Fixed Temperature Plate: Sc Number Distribution

Validation Validating the defined expectations involves correlating the variation in Nusselt number to Sc number. Local

Nusselt number variation on the surface of the fixed temperature wall is calculated for each grid cell face at

that interface in the model and plotted against the flow direction. Similarly the Sc number is plotted for the row

of air cells immediately adjacent to the wall surface, again in the flow direction. As we are looking for a

qualitative correlation between the two parameters, the Nusselt number is normalized along that profile. The

Sc number is itself already normalized.

Figure 56 shows the variation in normalized Nusselt number and Sc number plotted against the flow direction


May 13, 2010 Page 40 of 78

Figure 56 - Sc and Normalized Nusselt Number vs. Development Length

The main difference is seen in the local spike in Sc very near the entrance to the channel. This is due to the

detachment of the maximum Sc area from the wall surface as seen in Figure 55 where the line of maximum

Sc number is not parallel to the surface whereas the Nusselt number line is by its definition is taken along the

solid/fluid interface. Apart from this entrance effect difference there is a strong enough correlation between

the two parameters, enough to consider Sc variation near walls as a parameter that will indicate where the

local solid to fluid heat transfer is effective

3.5. Developing Boundary Layer in Channel with Local Fixed Temperature Walls

Motivation

This example is almost identical to the previous example apart from only a small section of the wall being held

at a fixed temperature. This configuration being more indicative of heat transfer from a locally heated PCB

surface.

Description



May 13, 2010 Page 41 of 78

Figure 57 - Parallel Plate Fixed Temperature Region Model Geometry

Setting the wall to have a thermal conductivity of 5 W/mK allows for lateral spreading of the heat.

Results

Plots of temperature, heat flux, temperature gradient and Sc number are shown in the following figures.

Figure 58 - Parallel Plate Fixed Temperature Region: Temperature Distribution

‐100 degC 1 m/s

0 degC


May 13, 2010 Page 42 of 78

Figure 59 - Parallel Plate Fixed Temperature Region: Heat Flux Distribution

Figure 60 - Parallel Plate Fixed Temperature Region: Temperature Gradient Distribution

Figure 61 - Parallel Plate Fixed Temperature Region: Sc Number Distribution

Validation As with the previous example, validating the defined expectations involves correlating the variation in Nusselt

number to Sc. The normalized Nusselt number variation at the wall surface and the Sc variation in the first

row of grid cells in the air adjacent to the wall are plotted in Figure 62. The extent of the fixed temperature

region is shown by the two vertical red lines.


May 13, 2010 Page 43 of 78

Figure 62 - Sc and Normalized Nusselt Number vs. Development Length

Overall an excellent correlation is seen, more than enough to give confidence in using the Sc variation in a

model to determine the areas that are being most effective in transferring heat from a solid surface to the air.

4. Real World Examples

4.1. Package Level Modeling

Motivation

The following exercise will explore the application of Bn and Sc results to a typical package scale

application. Using Bn and Sc as a guide, we’ll attempt to reduce junction temperature by reducing the

thermal resistance between the junction and the board.

Description

The application is shown in Figure 63. There is a typical 44 lead SOIC package centrally located on a

PCB and dissipating 2 Watts. The PCB has 6 layers and features a molybdenum core to enhance lateral heat

transfer within the board. The PCB is wedge-locked in place with the chassis temperature being fixed at 12

°C.


May 13, 2010 Page 44 of 78

Figure 63 – Package Scale Application

The SOIC is modeled with much of its geometry explicitly included. There are objects present for the die,

die attach, bond wires, tie bars, lead frames, and the encapsulant. Each of the metallic layers on the PCB is

modeled as a single object with a uniform material property appropriate for its metallic content and

distribution. The junction temperature will be measured at the center of the active surface of the die. The

board temperature is measured in the top signal layer, at the middle of the longest side of the package, and 1

mm away from the leads (towards the chassis wall). In the forthcoming results sections, the thermal

resistances presented will be referring to the temperatures at these locations.


May 13, 2010 Page 45 of 78

Figure 64 – SOIC-44 Geometry

Results

The initial simulation had the following summary results:

Design Tjunction (°C) Tpcb (°C) Rth JB (°C/W)

Original 104.3 42.7 30.8

Figures 65-68 illustrate the temperature field, and interesting locations where there are large volumes of

increased Bn and Sc levels.


May 13, 2010 Page 46 of 78

Figure 65 – Temperature distribution along the three package centerlines.


May 13, 2010 Page 47 of 78

Figure 66 – Maximum Bn values along the tie bar. The horizontal red line indicates the vertical

position


May 13, 2010 Page 48 of 78

Figure 67 – A small area of maximum Sc along the top of the tie bar

Figure 68: A large area of Sc ~ 0.1 on the top of the signal layer


May 13, 2010 Page 49 of 78

The location of maximum Bn in these results is inside the tie bars, at the ends where they meet the die

attach. This is demonstrating that in the initial design most of the heat is bypassing the bond wire-lead frame

route and opting instead to move downwards through the die attach and into the tie bar, dispersing then

through the lead frame to reach the board and then the ambient.

Figure 69: The primary heat flow paths from the die to the lead frame. Most of the heat uses the tie

bar to reach the lead frame.

There are two locations of interest with respect to Sc. The maximum value of Sc in these results is nearly

in the same spot as maximum Bn, but occupying the top side of the tie bars, i.e., the face of the tie bar

nearest the lead frame structure as shown in Figure 67. The identified thermal shortcut in this case is

conduction from the tie bar up to the lead frame and then out to the leads.


May 13, 2010 Page 50 of 78

More noteworthy is the large continuous area of Sc ~ 0.1 on the top layer of the PCB immediately below

the die shown in Figure 68. The integral value of Sc here would be much larger that the small pocket of

maximum Sc on the tie bars. This indicates a potential (and commonly used) thermal shortcut is available

from the bottom of the encapsulant to the PCB. This Sc information forms the basis of the next design

change; the addition of an exposed solder pad and associated copper land to the package design

Design Change One

The first design change is therefore to add a solder pad to the bottom of the package as shown in Figure

70. The pad dimensions match that of the die flag, which in turn is a reasonable approximation to the shape

of the Sc field seen earlier.

Figure 70 – Addition of a solder pad to the bottom of the package to take advantage of the thermal

shortcut highlighted by the Sc distribution


May 13, 2010 Page 51 of 78

The updated results:

Design Tjunction (°C) Tpcb (°C)

Rth JB (°C/W)

[% Decrease from Previous Design]

Original 104.3 42.7 30.8

Added solder pad 88.7 41.6 23.6 [-23.4%]

A reduction of over 20% in Rth JB is achieving by addressing areas of note in the Sc distribution with a

practical engineering solution.

It’s important to recheck the Sc and Bn fields after making a design change, as the heat flow topology will

have changed. The new Sc field looks like this

Figure 71 – Sc from the side. Scale truncated at 0.05 for clarity


May 13, 2010 Page 52 of 78

Figure 72 – Sc from the top. Plane positioned at the top of layer 2 as indicated by the horizontal

red line.


May 13, 2010 Page 53 of 78

With the solder pad and copper land now included in the design, the Sc field is very clearly indicating that

the best opportunity to build a thermal shortcut is downward, into the PCB, from the copper land to the 2nd

layer and molybdenum core beyond. The next design change will create this shortcut with the introduction of

an array of thermal vias.

Design Change Two

The 2nd design change is the addition of 60 filled thermal vias under the solder pad and copper land as

shown in Figure 73. The vias are 0.3 mm in diameter, 0.025 mm plating thickness, and are arranged in a

uniform staggered pattern. The thermal vias are disconnected from the signal layers but do make contact

with the internal plane layers.

Figure 73 – The locations of the 60 thermal vias

With the vias in place, the updated results are:


May 13, 2010 Page 54 of 78


Rth JB (°C/W)


Original 104.3 42.7 30.8

Added solder pad 88.7 41.6 23.6 [-23.4%]

Added thermal vias 72.5 40.7 15.9 [-32.6%]

A further reduction in Rth JB of 32.6% is achieved by again applying valid engineering changes to areas

suggested by the Sc field.

As before, best practice is to re-check the Sc and Bn fields to understand the effect the new heat flow

patterns have on the design. Now the most promising Sc distribution is shown in Figure 74.

Figure 74 – Sc Distribution. The red line indicates the vertical position of the contour plane.


May 13, 2010 Page 55 of 78

With the creation of the shortcut down to the molybdenum layer in place, the largest area of increased Sc

is now along the top surfaces of the tie bars nearest the die. While this may represent a good opportunity to

improve thermal, we’ll have to reject the pursuit of this shortcut due to electrical design constraints.

Now that all of the major Sc opportunities have been considered, it’s an appropriate time to investigate

where the bottlenecks may lie. The most interesting area of the design from a Bn distribution is shown in

Figure 75.

Figure 75 – Bn Distribution in the center of the package

There are three regions of potential interests: 1) The bond wires, 2) the volume immediately under the die

flag, and 3) the spreading of the heat outwards from the thermal via field within the molybdenum core. While

the values of Bn are highest within the bond wires, the volume between the die flag and the solder pad has

the largest integral value of Bn and it is the most promising area to consider for a design change.

Further note that reducing the scale of Bn considered in a plot is an excellent manner in which to spot

areas that have moderate Bn numerical values but have large volumes. These generally are more important

than small pockets of maximum Bn as stated previously.

1

2

3


May 13, 2010 Page 56 of 78

Design Change Three

With this information in mind, we’ll adopt a common design change to address the bottleneck between the

die flag and solder, namely, we’ll drop the entire die structure down so as to have the die flag residing directly

on top of the solder pad. This is an excellent strategy in this case as it completely removes this identified

bottleneck from the heat transfer path. The geometry change is shown in Figure 76.

Figure 76 – Die structure dropped downwards to meet the solder pad. The bond wires (yellow) are

now longer so as to reach up and make contact with the unchanged lead frame structure.

The updated results with the new die structure:


Rth JB (°C/W)


Original 104.3 42.7 30.8

Added solder pad 88.7 41.6 23.6 [-23.4%]

Added thermal vias 72.5 40.7 15.9 [-32.6%]

Dropped die structure 53.5 39.1 7.2 [-54.7%]

The bottleneck identified in Figure 75 has been completely removed and an improvement of 54.7% in Rth

JB is the result. It’s reassuring to note that sequential inspection of the Sc and Bn fields is able to provide

sound reasoning for each of these design changes. The end result of applying engineering judgment to the

courses of action suggested by the Sc and Bn fields is massive improvement in the thermal performance of

the package; a cumulative reduction of 76.6% in Rth JB.

There is no alternative systematic approach based on simulation or experimental results inspection that

could yield the same results in the same timeframe. Performing a Design of Experiments simulation set to

identify shortcuts and restrictive areas of the design is the most efficient alternate approach, and even that

would require hundreds and hundreds of simulations to accurately identify problem areas. As the heat flow


May 13, 2010 Page 57 of 78

topology changes with each design modification, the Design of Experiments simulations would need to

redone to determine the next design change. This is prohibitively expensive when compared with the

methodology used in this paper.

As a recap, the procedure followed in this example was to sequentially address the major areas indicated

by Sc for improvement and then use the Bn parameter to reduce the bottlenecks on the new heat flow

topology. The changes made and associated improvements in Rth JB are given in Figure 77.

Figure 77 – Summary of Package Design Modifications


May 13, 2010 Page 58 of 78

4.2. Heatsink Design Optimization Worked Example

Motivation

Examination of the Sc number variation in and around an oversized heatsink should provide an indication of

regions of the heatsink that are not performing optimally. Knowing this, design modifications can be made so

as to reduce the size of the heatsink without unduly affecting its thermal performance.

Examination of the Bn number should indicate regions of the heatsink that are restricting the flow of heat from

the heat source to the air. Knowing where these areas are will allow design modifications to be made so as to

improve the thermal performance of the heatsink.

Description

An extruded plate fin heatsink with a width of 300 mm, length of 300 mm, base thickness of 5 mm with 52

1mm thick equispaced 295 mm high fins is placed in a computational wind tunnel with no allowance made for

by pass (Figure 78). Air is blown through the heatsink with a mean air speed of 1 m/s at an inlet temperature

of 0 °C through each fin channel. The heatsink has a thermal conductivity of 137 W/mK. A 100 W 5 mm x 5

mm heat source is placed at the geometric center of the base.

Figure 78 Heatsink and Heat Source

Expectation

As described in Section 3.4, taking the near wall Sc variation as being correlated to the local Nusselt number,

identifying regions of low Sc number will determine areas of the heatsink that are not contributing to the

effectiveness of the heatsink and therefore can and should be removed. This will be done by changing the

extruded profile outline.


May 13, 2010 Page 59 of 78

Areas of the large Bn number will be those that are most restrictive and therefore require remedial design

modifications, most commonly changing the geometric cross section of either the base or fin widths. The

intention is to ‘smooth out’ the variation in Bn number. There will always be a biggest Bn number in any

simulation; an optimal design will be one where there is no one heat flow path that carries the ‘lions share’ of

the overall thermal resistance.

Results

The original heatsink design has a maximum temperature of 106degC, equating to an overall thermal

resistance of 1.06 °C/W. The Sc variation through the center of the heatsink perpendicular to the air flow

direction is shown in Figure 79 (the heatsink itself is hidden for clarity).

Figure 79 - Sc Number Distribution through the Heat Source

Note that a log scale is used to plot the Sc variation. Such an approach will better indicate large areas of

medium Sc. The thin vertical lines of higher Sc number in Figure 79 are in the air immediately adjacent to the

fin surfaces.

A similar plot parallel to the flow direction, in the air immediately next to the central fin, is shown in Figure 80.


May 13, 2010 Page 60 of 78

Figure 80 - Sc Number Distribution through the Heat Source (Parallel to the air Flow Direction)

Figures 79 and 80 can be used to qualitatively determine the Sc number that demarks the interface between

effective and non effective heat transfer regions. For this application that interface is taken to be at Sc = 0.002.

Plotting iso-surfaces of the Sc number at that value will produce a 3D geometric description of the volume of

the heatsink that is effective. Various views of such a surface are shown in Figure 81. Note that the surface is

colored according to air speed at that surface though no correlations are being sought, the choice was simply

to provide a color variation to aid inspection of the shape of that surface.

Air flow direction


May 13, 2010 Page 61 of 78

Figure 81 - Iso-Surface of Sc = 0.002

The shape of the iso-surface is then used to redefine the profile of the extrusion. This is shown in Figure 82

where the original heatsink is shown in wireframe for reference.

Figure 82 - Reduced Heatsink Extrusion Profile

Air flow direction

Air flow direction


May 13, 2010 Page 62 of 78

The thermal performance of the reduced profile heatsink is 1.12 °C/W, an increase of 5.5% but with large

decreases in the overall volume (88% lower) and pressure drop (23% lower).

Now that the Sc motivated design modification is made, examination of the Bn number distribution will allow

subsequent changes to be made to reduce thermal bottlenecks and to improve the thermal performance. Bn

variation through the heat source perpendicular to the flow direction is shown in Figure 83.

Figure 83 - Bn Number through the Heat Source

Figure 84 - Zoomed-in View of Bn Number through the Heat Source

A log plot is used again to better show the extents of the high and medium Bn region. When determining what

changes to make based on the Bn variation it is important to speculate as to what the preferred Bn variation

should be. This situation is very similar to that of Section 2.2, where a point source was considered in a 2D

homogeneous material. The ‘ideal’ Bn field was observed there, a radial distribution of Bn that removed any

bottleneck dependency on direction. The 3D analog of that case is a spherical Bn distribution centered on a

heat source. In this heatsink example, a spherical distribution of Bn is not possible (as the heat source is


May 13, 2010 Page 63 of 78

fixed on the bottom of the heat sink base), but creating a hemisphere of Bn values certainly is. Applying that

to this application, modification to the fin thickness and heatsink base thickness so as to achieve a more

hemispherical Bn variation should lead to a more improved, if not fully optimized, thermal effectiveness.

To investigate this, the fins that offer the largest restriction to heat flow are made wider and the base

thickness in the region of the Bn ‘hemisphere’ is thickened to the extent of that hemisphere. The value that

demarks the edge of this hemispherical Bn region is taken to be 0.001, 1/1000th of the maximum.

Figure 85 - Bn Number through the Heat Source after Base and Fin Modifications

The result is an increase in thermal performance. The overall thermal resistance is 0.95 °C/W, a decrease of

10% compared to the original oversized heatsink.

Having refined the heat sink base and fin thickness to achieve a desired Bn semi-sphere in the base there is

now the option of further improving the thermal performance by increasing the thermal conductivity in that

high Bn region. This is investigated by placing a copper slug over the extents of the Bn hemisphere.

Figure 86 - Iso-surface of Bn = 0.001


May 13, 2010 Page 64 of 78

Figure 87 - Copper Slug in the Heatsink Base

Thermal performance is dramatically increased, with the overall thermal resistance dropping to 0.5 °C/W, a

decrease of 53% compared to the original oversized heatsink. This is a good example of how the Bn number

can be used to make much more (cost) effective thermal design modifications, targeting thermal remedial

changes to exactly where they will be most effective. To verify this, a final simulation with the entire heatsink

made of copper is performed. The overall thermal resistance drops to 0.42 °C/W, 40% of the original

oversized heatsink, a further reduction of 13% compared to the use of the slug but likely an increase in costs

that is much higher than the decrease in thermal resistance.

Summary

An initially oversized heatsink with a small heat source on the underside of the base was reduced in its

extrusion profile shape based upon examination of the extents of the Sc number distribution. Values of Sc <

0.002 were taken to represent areas of the heatsink that were not contributing efficiently to its thermal

performance. This resulted in a slight drop in performance with the thermal resistance increasing by of 5.5%

but more than balanced by a drop in volume of 88% and a drop in pressure drop of 23%. Examination of the

Bn number distribution led to design modifications of increasing central fin thicknesses and increasing the

base thickness around the heat source so as to achieve an uninterrupted Bn distribution of values of Bn >

0.001 that was hemispherical in shape. This resulted in a decrease of 10% of the thermal resistance

compared to the original oversized design. The volume of the base that the Bn hemisphere occupied was

then replaced by a copper slug. This resulted in a decrease of 53% of the thermal resistance compared to the

original oversized design. Finally, the entire heatsink was modeled as copper resulting in only a modest

further improvement.


May 13, 2010 Page 65 of 78

Figure 88 – Summary of Design Changes and Results

4.3. System Level Refinement Worked Example

Motivation

System level applications offer the best opportunity to demonstrate the value of Bn and Sc inspired design

modifications. Even the simplest system level model often has a number of different heat sources and

complex intertwined heat removal paths. This example will involve sequential design modifications based on

identification of high value Bn and Sc areas.

Description

0

0.2

0.4

0.6

0.8

1

1.2

Original oversized

heatsink

Redu

ced profile based

on Sc iso‐surface

Mod

ified

base and fin

based

on Bn

he

misph

ere

Copp

er slug placed

over Bn

hem

isph

ere

Entire he

atsink

made of cop

per

Thermal Resistance (degC/W)


May 13, 2010 Page 66 of 78

A wall mounted sealed system contains a motherboard and two canned daughterboards (Figure 89). A plate

fin heatsink forms the rear of the enclosure with a critical component (Comp1) conduction cooled to the

heatsink via a thermal pad and aluminum block on the underside of the motherboard directly beneath Comp1

(Figure 90). Note this is the ‘Wall Unit’ application example installed with FloTHERM.

Figure 89 - ‘Wall Unit’ Application Example

Figure 90 ‘Wall Unit’ Motherboard

Expectation

Examination of the variation of Bn throughout the model will identify those areas that are bigger bottlenecks to

the removal of heat than others. Such a relative scale will enable a prioritization of which design modifications,

and where, should be investigated first.


May 13, 2010 Page 67 of 78

Examination of the variation in Sc number throughout the model will identify two things. Generally it will

identify where in the model there is an opportunity to insert a new heat flow path so as to shortcut the heat

more directly to colder areas. More specifically, at solid/fluid interfaces it will provide an indication of where

the heat transfer across that interface is effective.

Results

The following figures show the variation in temperature and air speed on a plane bisecting Comp1, the edge

of one of the canned PCBs and through the center of a heat sink plate fin. The images have been rotated by

90 degrees so that gravity is acting from right to left.

Figure 91 - Temperature Distribution

Figure 92 - Air Speed Distribution

To determine the effectiveness of any design modifications, comparisons against Basecase temperature rises

above ambient of Comp1 and the air in the two PCB cans. Such temperature rises in °C for the Basecase are

shown in the following table:

Model Temperature Rise Over Ambient (°C)

Comp1 Air in Can1 Air in Can2

Basecase 36.3 87.3 86.6


May 13, 2010 Page 68 of 78

1st Bn Design Modification

Figure 93 has a plane plot showing the variation in Bn, taken at the same location as in Figure 91. The Bn

range has been reduced to 0-0.2 to better identify the relative levels of Bn that would otherwise be swamped

by the locally very large values in the connectors where they join the PCB can to the motherboard.

Figure 93 - Bn Distribution

It is at those locations that Bn number is at or near its maximum value of 1.0. A large proportion of heat

dissipated in the sealed cans conducts down through the connectors into the motherboard. Despite a relative

high conductivity of 180 W/mK, their cross sectional area is quite small and they abut a much lower

conductivity PCB. To break down this bottleneck at the connector/PCB interface the connectors from both

cans are modeled as punching through the motherboard, not abutting it as shown in Figure 94.

Figure 94 - Can Connector Design Modification

The effect on this modification is shown in the following table. The temperature rise in the two cans drops by

10%. The temperature rise in Comp1 increases by a small amount, due to the increased amount of heat that

now passes into the motherboard.


May 13, 2010 Page 69 of 78

Model Temperature Rise Over Ambient (°C) : [% of Basecase]


Basecase 36.3 87.3 86.6

Bn Modification 1 -

Connectors through PCB 37.3 [+2.8%] 78.1 [-10.5%] 78.0 [-10.0%]

2nd Bn Design Modification

Figure 95 has a plane plot showing the variation in Bn, taken at the same location as in Figure 93.

Figure 95 - Bn Distribution

Note that now that the first bottleneck has been relieved, the relative difference between the bottlenecks in the

canned PCB, at the can/motherboard connector interface and under Comp1 have been reduced. The

remaining candidate targets for Bn remedial action are

1. In the canned PCB

2. Where the canned PCB connects to the internal side of the can

3. In the motherboard around the can connectors

4. In the motherboard under Comp1

Choice of where a remedial action can be performed will always be constrained by other design goals, be

they electrical performance, manufacturability, cost etc. In this case an increase in PCB copper content so as

to increase in-plane effective thermal conductivity could be considered as a possible remedial action for 1 and

3. We will however focus on 4 where it should have a direct effect on the temperature of Comp1. To relieve

this bottleneck an array of thermal vias will be placed in the motherboard, increasing the effective through

plane thermal conductivity of the motherboard in that area as shown in Figure 96.


May 13, 2010 Page 70 of 78

Figure 96 - Thermal Vias Under Comp1

The effect of this modification is shown in the following table. A 24% decrease in Comp1 temperature rise

over ambient compared to the Basecase. A small improvement in the Can temperatures is also seen, due to

the fact that some of the heat leaving the cans also passed through that area of the PCB down to the heatsink.



Basecase 36.3 87.3 86.6

Bn Modification 1 -


Bn Modification 2 –

Comp1 Thermal Vias 27.4 [-24.1%] 77.6 [-11.1%] 77.6 [-10.5%]

1st Sc Design Modification

Further Bn motivated design modifications could be investigated, further chasing down remaining bottlenecks.

At this stage however we turn to examination of the Sc field to identify where new heat flow paths could be

added to more effectively shortcut the heat from its source to the ambient.

Figure 97 has a plane plot showing the variation in Sc, taken near the same location as in Figure 95 but with

the plane positioned so as to be in the air gap between heatsink fins. A log scale is used this time to better

highlight the relative variation in Sc over the model. To better understand existing the heat flow paths Figure

98 is positioned back to bisect a heatsink fin, showing the heat flux distribution with the lower heat flux vectors

clipped away for clarity.

Array of thermal vias

Thermal pad

Aluminum connector to heatsink

Heatsink

Comp1


May 13, 2010 Page 71 of 78

Figure 97 - Sc Distribution

Figure 98 - Heat Flux Distribution

The Sc number will be highest where the heat flux vector is at 90 degrees to the local temperature gradient.

For point 2 this occurs where the heat flux in the thin enclosure runs orthogonally to the temperature gradient

between the enclosure and the local air. Heatsinking of the enclosure may capitalize on the opportunity to

pass the heat more effectively to the cooler air although as the enclosure is made of plastic such area

extension methods may not be effective. Point 3 indicates the area of the heatsink that is providing efficient

heat transfer from the fins to the air. Point 1 has the largest value of Sc and it is at this location that the first

Sc design modification is considered.

Further extending the can connectors down onto the heatsink base should shortcut the heat most directly to

the ambient, this is shown in Figure 99.


May 13, 2010 Page 72 of 78

Figure 99 - Further Can Connector Design Modification

The effect of this modification is shown in the following table. Can temperature rises over ambient reduce by

47% of the Basecase values. There is a slight increase in the Comp1 temperature rise, this time due to an

increase in the heatsink temperature with more heat passing directly down to it.



Basecase 36.3 87.3 86.6

Bn Modification 1 -




Sc Modification 1 –

Connectors dropped to

Heatsink

28.9 [-20.2%] 46.9 [-46.2%] 46.7 [-46.1%]


May 13, 2010 Page 73 of 78

2nd Sc Design Modification

Re-plotting the Sc variation shows a much more even distribution of Sc, Figure 100. A completely uniform

distribution of Sc (and Bn) would indicate a perfectly balanced design, where heat finds it equally easy to pass

to the ambient regardless of the path it took and where there is no advantage to shortcut the heat to cooler

areas from any one location to another. Successive design modifications based on Bn and Sc should evolve

the design closer to this perfect condition.

Figure 100 - Sc Distribution

As discussed in Section 3.4, the Sc number in the fluid, adjacent to a heatsink fin, is correlated to local

Nusselt number and thus indicative of regions of effective heat transfer. For the current state of this design

that region is (qualitatively) determined to be bound by the red box. The heatsink outside of this area is not

operating as effectively and so can be removed. This design modification is shown in Figure 101.

Figure 101 - Heatsink Fin Length Design Modification

The effect of this modification is shown in the following table. The results are initially surprising in that

foreshortening the heatsink actually improves the thermal performance slightly. This is due to an increase in

air flow through the heatsink as, being shorter, it offers less flow resistance.


May 13, 2010 Page 74 of 78



Basecase 36.3 87.3 86.6

Bn Modification 1 -





Connectors dropped to

Heatsink

28.9 [-20.2%] 46.9 [-46.2%] 46.7 [-46.1%]


Reduce Heatsink Length 26.6 [-26.7%] 45.2 [-48.1%] 45.0 [-48.1%]

Summary

Successive Bn and Sc prompted design modifications result in a series of thermal improvements of the

original design. Examination of the initial Bn variation, using a reduced range of Bn 0-0.2 to better highlight

the main bottleneck areas, indicated that the largest bottleneck was at the interface between the can

connectors and the motherboard. This bottleneck was relieved by having the connectors bisect, not abut, the

motherboard. Examination of the subsequent Bn variation led to the inclusion of an array of thermal vias

under Comp1. Two Sc inspired modifications were then implemented. Firstly the can connectors were pushed

further down so that they abutted the heatsink base. Finally the heatsink was foreshortened based on

identification where Sc number was biggest in the heatsink fin gaps. This resulted in a further decrease of the

temperature rises due to an increase in air flow rate through the heatsink.


May 13, 2010 Page 75 of 78

Figure 102 - Summary Results

5. Discussion

An Alternative to Parametric Design Optimization

The inspection of thermal bottlenecks and shortcut opportunities provides a thermal engineer with the ability

to identify where in a design specific modifications could be made so as to reduce temperature rises. Such a

methodology will enable a design to be improved until such time as it is judged to be thermally compliant. Bn

and Sc prompted design modifications should be viewed as an alternative to a parametric and optimization

approach. Specifying design parameter bounds, defining and solving a ‘design of experiments’ then

determining an optimum configuration of input parameters by minimizing a cost function that encapsulates a

thermal design goal will also lead to an improved if not optimized design. This however can be

computationally expensive. Also, a design might be overly constrained to the extent where the freedom to


May 13, 2010 Page 76 of 78

arbitrarily change design parameters is severely limited by electrical, manufacturing and cost considerations.

Examination of Bn and Sc distributions simply offers insights into where the design might benefit from being

changed. It is then up to the designer to apply knowledge of whatever constraints exist to select which high

and/or large Bn or Sc area a modification should be made in.

Serial and Parallel Heat Flow Paths

Relieving a thermal bottleneck through a redesign will enable heat to pass more easily through that area. Real

world electronics systems exhibit complex 3D heat removal path topologies that contain combinations of

thermal resistances in parallel and in series. Relieving a thermal bottleneck may not have as large an effect

as expected should there be other resistances further ‘along’ that part of the heat removal path (nearer

ambient) that lie in series. For a system that has one heat source then a collection of thermal resistances all

in parallel connecting the source to the ambient (e.g. a star type network with spoke resistances) then

relieving the largest bottleneck will have the maximum effect. Examination of the heat flux vector field in

conjunction with the Bn field should provide additional insight into the topology of the heat removal paths such

that more informed prioritizations of which bottlenecks to relieve first could be made.

Shortcutting Directly to Ambient

The Sc field will provide an indication of where there is high heat flow ‘adjacent’, and running in parallel to,

cooler areas. Insertion of a new heat flow path will allow heat to flood ‘down’ to that cooler area. If however

that cooler area itself is surrounded by high resistances, blocking off it’s path to the ambient, then the benefit

of the insertion of the heat flow path may be limited. Peripheral short cut opportunities, those that lie on the

edge of the system, where the cooler area is the ambient itself, are the ones where, when acted on, should

provide the biggest benefit.

The concept of insertion of a ‘new heat flow path’ is a slight misnomer in that heat will be passing, to varying

degrees, though all parts of a system. A ‘new heat flow path’ should be considered as the reduction of an

existing extremely high resistance down to a level where a more appreciable amount of heat will flow.

Geometric Resolution

It is important to appreciate that high Bn or high Sc areas also indicate where the temperature prediction is

sensitized to modeling assumptions. For example, if a large Bn feature is present under a component, and

the PCB under that component is modeled as a simple orthotropic block, you may need to enhance the

geometric resolution of the PCB in that area (to include the layer stack, copper distribution, etc) to better

resolve the heat flux, temperature gradient, and Bn predictions locally.

The important Bn and Sc features identified in the results should be considered in the same way as other

important outputs (e.g., maximum temperatures) and should be given appropriate levels of detail and meshing.


May 13, 2010 Page 77 of 78

Interpretations of Sc in Solid and Fluid

It is important to recall that the interpretation of Sc is dependent on location:

1. Within a solid object: High values of Sc indicate the start or end of a new thermal shortcut that

could be created. Low values indicate there is little scope to consider a thermal shortcut at this

location.

2. Within a fluid: Adjacent to a solid-fluid interface: High values of Sc indicate where efficient

convective heat transfer is occurring and where the local Nusselt number will be relatively large.

Low values indicate where the convective heat transfer is not efficient or not occurring.

3. Within a fluid: No thermal design implications. The heat flux and temperature gradient vectors

are dominated by the fluid velocity distribution.

Interpretations of Bn in Solid and Fluid

It is important to recall that the interpretation of Bn is also dependent on location:

1. Within a solid object: High values of Bn indicate that there is a thermal bottleneck present at this

location.

2. Within a fluid: No thermal design implications. The heat flux and temperature gradient vectors

are dominated by the fluid velocity distribution.

6. Summary – Usage Advice This section contains a summary of the usage advice mentioned in previous sections.

Iterative Design

Each time you make a design modification based on either the Bn or Sc fields, it is important to re-analyze the

Bn and Sc fields before making the next thermal design decision. Addressing a Bn issue or Sc opportunity

will always change the heat flow topology of your design, sometimes drastically, and will invalidate previous

Bn and Sc observations. The correct workflow is to:

1. Evaluate the Bn and Sc fields for a design

2. Implement a design modification

3. Re-solve the model

4. Evaluate the new Bn and Sc fields for the design

5. Repeat Steps 2-4 as needed

ShortCuts First, Bottlenecks Second

In general, and especially for single heat source problems, it is best practice to consider Sc opportunities prior

to investigating Bn issues. The reason is that introducing a new heat flow path based on a Sc feature usually

has a stronger disruptive effect on the heat flow topology than relieving a bottleneck on an existing heat flow

path. For example, it’s quite possible to completely bypass a serious bottleneck by creating a thermal


May 13, 2010 Page 78 of 78

shortcut, which of course pushes that existing bottleneck into irrelevance. Any previous design modifications

that addressed this serious bottleneck will have their effectiveness lessened.

Note that this is not a hard and fast rule, as significant changes to heat flow topology can arise from

bottleneck relief measures as well.

Integral Nature of Bn and Sc

When inspecting the Bn and Sc fields, recall that that area/volume of the Bn or Sc feature identified is

important. If there are two Bn features that exhibit equal Bn levels, the most problematic bottleneck will be

the larger feature. Similarly, a 1 mm3 cube of Bn = 1 would be considered less important than a 100 mm3

feature of Bn = 0.2. The same volumetric considerations are valid for Sc as well.

Display Ranges

The normalized display of Bn and Sc in FloTHERM can occasionally mask large areas/volumes of smaller Bn

and Sc. There are two techniques that can be used to identify these areas:

1. Switch to a logarithmic scale for Bn or Sc. This has the effect of reducing the disparity in color

between medium and large values of these parameters and can improve the ‘findability’ of large

feature with small levels of Bn or Sc.

2. Reduce the range displayed. Decreasing the maximum Bn or Sc for display purposes can

effectively screen out all of the large numerical values of Bn and Sc that you are not interested in.

Work with these techniques individually or together to most efficiently locate areas of interest for your design.

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