Thermal Design Simulation Software “HIEROH”
by Naoki Kimura * and Akira Hideno *Heatsinks are frequently designed specifically for the equipment to which they
are to be applied. In addition to a trend toward shorter development periods
for equipment that incorporates heatsinks, it has become more common to consider cooling
capacity from the earliest stage of overall equipment design. This has led to a need for fasterthermal design.
In the past reliance has been placed on very simple manual calculations or computational fluid
dynamics (CFD) programs such as the finite volume method, but all of these present problems
in terms of low accuracy, difficulty of use and length of calculation time.
As part of our “total thermal solution” extending from the design to the product, Furukawa-Sky
has developed the HIEROH thermal design software that is quicker to use.
We also propose spread efficiency as a design guideline for optimization.

ABSTRACT1.

INTRODUCTION

The use of cooling systems ordinarily involves the heatbeing ultimately dissipated to the air, and in such cases
heatsinks play an important role. Heatsinks are frequently
designed specifically for the equipment to which they are
to be applied, and designing them requires a balanced
optimization of thermal performance, size, weight, and soon.
1.1 Conventional Thermal Design
In conventional thermal design two methods of thermal
analysis have been used:a) 1-dimensional analog analysis
Based on a certain set of assumptions, 1-dimension
calculations (fin calculations 1) and so on) are carried out,achieving an approximate understanding of performance.
For example, if we solve an equation for thermal balance under conditions in which the end is adiabatic, we
obtainT = C1 exp(kx) + C2 exp (- kx)
Since analog solutions do not depend on the finenessof the mesh, they have the advantage of providing infinite
fineness, but the information that can be obtained is only
1-dimensional; it cannot yield 2-dimensional information
like heat diffusion in the base plate.
b) 3-dimensional digital solution
For 3-dimensional analysis of the heatsink as a whole,reliance has been placed on programs for the finiteelement method or the finite volume method 2)~4) but theprocedure is complex (pre-processor, solver, post-processor), and the setting of conditions —headed by mesh
* Processed Products Dept., Furukawa-Sky Aluminum Corp.

generation— requires a certain level of skill.
Also the computer must have a great deal of memoryand calculation times are long, typically several hours forlarge-scale jobs.
Some problems associated with conventional design
methods are as follows:• 2-dimensional information could not be obtained (inthe case of fin calculation);• In the digital solution, calculations could be performed for any desired configuration but the calculations required skill and were normally performed by aspecific operator.
• Only the expected temperature was displayed; there
was no display of design guideline to improve performance.
• For weight, etc. calculations were done separatelyfrom the simulation software, resulting in inconvenience.
1.2 Effectiveness of Quickly Usable Software
With the quicker pace of development for equipment
incorporating heatsinks, it has become desirable tospeed up thermal design as well.
In the past it was sufficient to deal with the heatsink at
the later stages of design, but recent increases in heating values have created a need for thermal design to betaken up from the earliest stages of development. In suchcases it is no longer effective to devote a lengthy period
simply to carefully estimating thermal performance using
a specific model; it is now necessary to arrive at an estimate very quickly —in a few minutes. It is also desirable
that estimating thermal performance be easily accomplished, even by persons without expertise in thermal
analysis.

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Thermal Design Simulation Software “HIEROH”

We have therefore developed software that can bemastered quickly by anyone. And to facilitate arriving at acomprehensive decision, we made provision that information other than thermal performance —such as pressure
loss, weight and the like— can be obtained with the same
software, and that it would also be possible to display
spread efficiency as a design guideline.

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Figure 3

2.3.1

2.

Base plate converted to circle.

Calculations for Inside of Exothermic Device

GUIDELINES, MODELING AND
PRINCIPLES OF CALCULATION

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2.1 Guidelines• Insofar as possible to use analog solutions;• To apply digital solution only where unavoidable; and• To enable the obtaining of 3-dimensional information
by means of ingenious modeling.
2.2 Modeling
We created a model based on a grasp of characteristics(here we set forth the case of a square configuration):
Setting forth the sequence of thermal flow we have:1) Heat spreads over the base plate surface;2) Heat spreads in the fins; and
3) Heat is dissipated from the fins to the air.
Fo r p u r p o s e s of simplification, (reversin g t h esequence):• We calculated the fin-to-air dissipation of heat in 3)
using the theoretical formula for heat transfer;• We then calculated the spread of heat in the fins 2) interms of fin efficiency (Figure 1).

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Figure 4

Calculation model for inside of exothermic device.

If we apply the equation focusing our attention on theminute part forming the innermost concentric circle
shown in Figure 4, and if we let x be the distance fromthe center, λ the thermal conductivity, d the thickness,and q0 the exothermic heat per unit volume, we find thattemperature θ satisfies
d×[{-λ (2π x)dθ /dx}x-{-λ (2π x)dθ /dx}x+dx]=-a' (2π x) θ dx (heat transfer term)+π {(x+dx)2-x2}d q0 (exothermic term)
Since the first term on the right side is dramaticallysmaller than the second term, it may be omitted giving���� ���������������� � ����������� �������� �����������

Figure 1

Heat transfer coefficient of heatsink.

Assuming that the fins have been placed with sufficientdensity with respect to the base plate surface, and thatthe hypothetical heat transfer coefficient of the base platesurface is a' , a' may then by calculated as a (heat transfer
coefficient of the fins) × (the surface multiple seen fromthe base plate surface) × (fin efficiency) (Figure 2).

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Figure 2

Hypothetical heat transfer coefficient of base plate.

Again the base plate surface is converted to a circle
having the same area as the base plate surface (Figure
3).
2.3 Principles of Calculation
First let us explain on the basis of a 2-dimensional
calculation.

Here if we take q0 / λ = 4k we may write- xdθ /dx = 2kx2 + c'
and since, when x = 0, the slope of temperature
becomes 0, we get c' = 0. And thus we find that θ = -kx2 + c
At the boundary, analytical continuation is applied
between this thermal field and the one described by theequations in Section 2.3.2 so that agreement is achievedin the value and gradient, and the two fields may be represented by C1 and C2.
2.3.2 Calculations for Outside of Exothermic Device
In Figure 5, if we construct a differential equationconsidering the thermal balance between x and x+dx inthe x direction, we obtaind×[{-λ (2π x)dθ /dx}x-{-λ (2π x)dθ /d x}x+dx]=a' (2π x)θ dx
which may be ordered as�������������� ���������� � �����������
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and if we substitute �� �������� using a progression solu���tion with �������� �� ������ �����

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Thermal Design Simulation Software “HIEROH”

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���� ���������� (where C is the circumference and S is the��cross-sectional area), we may write the solution asθ (t)=C3 exp (mt)+C4 exp (-mt)
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Figure 5

Calculation model for outside of exothermic device.

we can find��������������� �������If we take A0 = 1, it agrees with type 1 modified Bessel
function I(x). Similarly, finding another solution, if we takean arbitrary constant γ , it agrees with type 2 modified
Bessel function K(x).
Thus it can be seen that θ = C1 Bessel I(mx) + C2 Bessel
K(mx).
The conditions for finding C1 and C2 are:the adiabatic end condition dθ /dx=0 at x=xo andenergy balance����� � �� �� =exothermic heatand given that the number of variables is 2 and thenumber of equations is 2, we can determine C1 and C2.
Solutions can be obtained for the Bessel functions
using Allen’s approximation equation 5) and for integral
calculus using Simpson’s formula 5).
2.3.3 Pseudo 3-dimensional Calculations
By positing a hypothetical component directly beneaththe exothermic device (Figure 6) and calculating the flowof heat into and out of that body, we determined the temperature difference in the thickness direction.
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Figure 6

Calculation model for temperature difference in thickness direction (through base plate).

If we construct an equation under conditions of constant heat transfer coefficient, we would hypotheticallydispose here a cylindrical component directly beneaththe exothermic device, as shown in Figure 6. Taking thevariable for the thickness direction as t, an input heat Qi
equivalent to the exothermic heat enters this component
from above, and the exiting heat Qe can be found by thecalculations shown in Sections 2.3.1 and 2.3.2 above.
From the lateral direction heat is lost, but, using the
hypothetical heat transfer coefficient a" , if we let

Here the way we treat m (and by extension a" ) presents
a problem, but nominally we shall treat it as a variable. (Itis, however, the case that m is assumed to have a fixedvalue irrespective of its position.)
There are three boundary conditions:-λ d θ /dt(t=0)×S=Qi(1)-λ d θ /dt(t=t)×S=Qe(2)
θ (t)=θ t (known)(3)
so they may in principle be solved for the three variables.
In (1) if we take β =1/{S(-λ )m}, C4=C3-β Qi
If we substitute
C3=[{Qt-Qi exp (-mt)}/{exp (mt)-exp (-mt)}]×β
from (2) into (3),
we obtainβ [{exp (mt)+exp (-mt)}/{exp (mt)-exp (-mt)}]Qt-2Qi/{exp (mt)-exp (-mt)}=θ t
Provided mt is sufficiently large, the left term will show
the behavior of β , i.e., 1/m (which will be a graph in
the form of y=1/x), so that with respect to the NewtonRaphson Law it will constitute a “function of good quality”
and will be easily solved.
Here, by treating m as a function without forcibly inserting an inferred value, we can obtain a solution that is selfconsistent (not self-contradictory).
With respect to accuracy, we compared similar calculations for thermal conductivity with the finite-element
method. (Since modeling for HIEROH was carried out
with reference to a practicable heatsink, it does not bearreference to non-practicable heatsinks, such as one having only a single fin.)Comparing for a number of practicable heatsink configurations using values of the finite-element method as abasis, error fell within a range of -2% to +8%.
2.4 Extension to More Complex Configurations
In cases in which the exothermic device is not in thecenter, it may be approximated by means of circles
having different radii, and calculations relating to up-windand down-wind temperature differences can be doneusing thermal conductivity and heat transfer coefficient.
And if there is more than one exothermic device, thecalculations are carried out on the principle of superposition 6).
2.5 Comparison of the Amount of Calculations
Conventionally, taking that the heatsink as a wholeconsists of several million joints of mesh, and that
there are seven variables —air flow in three directions,pressure, temperature, turbulence energy, turbulence
decay rate, etc., tens of millions of equations were solved,
requiring several hours of calculation time.
Modeling in the way described means that these could

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Thermal Design Simulation Software “HIEROH”

Figure 7

Man-machine interface of HIEROH software.

be summarized by a few hundred equations, and operation was greatly speeded up.

3.

REPRESENTATIONAL PROGRAMMING

When we were successful in establishing the calculation
method described in Section 2, we set aboutprogramming in an easy-to-use representational form thatwill run on a Windows† platform.
3.1 Screen and Features of How It Is Used
• Everything is displayed on a single screen making
it easy to see, and reducing setting mistakes during
data entry.
• Operation does not require awareness of preprocessor, solver, or post-processor or meshgeneration, so that there is no need to carry out
operations that were conventionally unnecessary inderiving estimated values from conditions.
• How it is used
On a PC running Windows, double-click on the
HIEROH logo and simply enter numerical values
for size, exothermic heat, etc. into the table. Thenclick on the CALCULATE button at the top right andthe calculated results, including a contour drawing,

will be displayed. There is also a function, to be
described below, that will optimize the configuration.
Simply click on the CALCULATE button for optimizing
to be performed automatically.
3.2 Features of Operation• Easy on/off switching; just double-click on the logo
and the final results from the previous time loads
automatically. When terminating, the calculation thatwas last performed is saved automatically. It has theoperational feel of opening and closing a notebook.
• Quick display: calculations are completed in lessthan 1 second, even on older computers with CPU
clock speeds in the 500-MHz order (Furukawa-Sky
test values). Since a single set of calculations can bedone so quickly, the automated calculations to be
described below can also be completed quickly.
• Includes information on weight, pressure loss, etc. in
the display, so there is no need to perform separate
calculations.
• Displays a contour diagram, allowing checking ofdata entry.
• Rise in air temperature can be calculated: In cases inwhich air speed is slow, even an increase in fin area
has only a limited effect in reducing thermal resistance.

† Windows is a registered trademark of Microsoft Corp.

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Thermal Design Simulation Software “HIEROH”

�������� ����3.3 Automation• Fitting to the PQ curve diagram
If a PQ curve diagram for the fins is given, theworking point can be found as shown in Figure 8.

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Figure 8

Relationship between pressure loss and air flow rate.

• Optimizing the number of finsWith the fins kept at constant conditions, the numberof fins is changed (Figure 9). As the number of finsincreases, the surface area increases but the air flow
tends to decrease and the point of minimum thermal
resistance can be found.
By suppressing the amount of calculation it becomes
possible to increase speed. The process of optimization,
which conventionally took half a day or even a whole day,can be done in about 30 seconds (Furukawa-Sky test values).

4.

SPREAD EFFICIENCY AND
OPTIMIZATION

4.1 Spread Efficiency
Dispersion of heat at the spreader (base plate) is
extremely important for the heatsink. If heat is not spreadat the base plate, even the best fins will not be able toreach their true performance.
In the same way that the thermal performance of fins is������� ����������
������� ������� ������� �� ����Figure 9

Relationship between number of fins and thermal
resistance.

shown by fin efficiency, we have defined spread efficiency.
Comparing with fin efficiency:• Fin efficiency is average temperature for the fins as awhole (one dimension) / temperature at fin base• Spread efficiency is average temperature for the baseplate (2 dimensions) / temperature of exothermic
device thereby effecting an extension from 1
dimension to 2.
Like fin efficiency, spread efficiency takes a numericalvalue from 0 to 1, the larger the number the more the heatspreads to the edges.
Extending to 3 dimensions, heatsink efficiency (3dimensions) is represented by heatsink efficiency =spread efficiency × fin efficiency, and for the thermal performance of the heatsink as a whole we have: fin area ×heatsink efficiency = fin area × spread efficiency × fin
efficiency.
What is needed is to raise fin area × spread efficiency
× fin efficiency in proper balance, while paying attention
to pressure loss. With HIEROH it becomes possible toachieve this with a single software package.
4.2 Optimization at Constant Height
In cases in which the dimension in the direction of finheight is limited, the question of how thick the base plateshould be is a difficult one.
Figure 10 shows a typical representation. Care must be
exercised, since if the base plate is too thin the spread in
the base plate is poorer and thermal resistance tends toincrease precipitously.
With HIEROH, under conditions where total height (finheight + base plate height) is constant, it is possible to������� ����������• Displays time constant: The time required for a rise intemperature to approximately 63 % of steady state isdisplayed.
• Displays temperature difference at the thermal compound part: Temperature difference resulting fromtypical contact resistance is displayed.
• Displays wattage for power loss: Power loss is normally displayed in a unit of pressure such as Pa, but
to be consistent with fin power, etc., there is also a
display converted into Watts.
• Maximum radiation: When the radiative coefficient
has been increased, the estimated value for the maximum wattage that can be dissipated by radiation isdisplayed.
• Comparison with previous time: There is a functionthat retains the value calculated the previous timeand displays the difference from it. When conditions
have been modified, this makes it easy to see whatkind of change has occurred.
• Physical properties can be set independently for thefins and base plate, so that it is equally applicable tocrimped fins.

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Figure 10 Relationship between base plate thickness and thermal resistance.

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Thermal Design Simulation Software “HIEROH”

determine the minimum value for thermal resistance bychanging base plate height (and obviously fin height aswell) as parameters. Here also higher speed is achieved.
4.3 Heatpipe EffectA heatpipe may be used in common on the base plate
when adequate heat dissipation cannot be obtained
without a heatpipe, and in such cases it is possible to
make a rough estimate of the effect.
An algorithm is applied that includes empirical parameters from experimental values. This would be:
HP (ordinary): Number of heatpipes : 1 or 2
HP (superior):: several
HP (best):: as many as canbe inserted

5.

CONCLUSION• Adopting a hybrid analog + digital structure has
made it possible to achieve simple operation as well
as speedy analysis.
• Using spread efficiency as a design guideline for
optimization has made optimization easier.
• It is also possible to calculate estimated values incases when heatpipes are inserted into the baseplate.
• As part of a total thermal solution, extending fromdesign to finished product, we hope to use HIEROH
to speed up the design process as well as to assist in
upgrading customer service.

ACKNOWLEDGMENTS
We would like to express our appreciation to those
persons from Furukawa Electric, Furukawa-Sky andFurukawa Information Technology who have been
involved in this work.

REFERENCES
1) Fin Efficiency, Heat Transfer Data Book, The Japan Society of
Mechanical Engineers, Third Revision, (1974), Fourth Revision,(1986). (in Japanese)
2) Non-Compressive Fluid Dynamics, Publishing Department ofTokyo University, (1995). (in Japanese)
3) Performance of CFD Software and Hardware, Automotive
Engineers, (2001). (in Japanese)
4) Frontier of CFD Design Application, Automotive Engineers, (2003).
(in Japanese)5) Numerical Calculation Handbook (Revised Edition), Ohm
Publishing, (1990), p145, p845. (in Japanese)
6) Japanese Patent Application, TOKUGAN 2002-109702

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