Proceedings of IPACK03
International Electronic Packaging Technical Conference and Exhibition
July 6–11, 2003, Maui, Hawaii, USA
InterPack2003-35104
THE THERMAL RESISTANCE OF PIN FIN HEAT SINKS IN TRANSVERSE FLOW
Maurice Holahan
IBM Corporation
Rochester MN
mholahan@us.ibm.com
Sukhvinder Kang
Aavid Thermalloy
Concord NH
kang@aavid.com
ABSTRACT
This paper presents a physics based analytical model to predict the
thermal behavior of pin fin heat sinks in transverse forced flow. The
key feature of the model is the recognition that unlike plate fins,
streamwise conduction does not occur in pin fin heat sinks. Thus, the
heat transfer from each fin depends on its local air temperature or adiabatic temperature and the local adiabatic heat transfer coefficient. Both
experimental data and simplified CFD simulations are used to develop
the two building blocks of the model, the thermal wake function and
the adiabatic heat transfer coefficient. These building blocks are then
used to include the effect of the thermal wake from upstream fins on the
adiabatic temperature of downstream fins in determining the fin-by-fin
heat transfer within the pin fin array. This approach captures the essential physics of the flow and heat transport within the fin array and
yields an accurate model for predicting the thermal resistance of pin fin
heat sinks. Model predictions are compared with existing experimental
data and CFD simulations. The model is expected to provide a sound
basis for a consistent performance comparison with plate fin heat sinks.
area
minimum free flow area between pins (or pin fins)
specific heat
D
diameter of pins
f
h
i,j,k
L
M
N
Nu
O
p
P
Pr
q
Q
ρDU o
Reynolds number  ---------------
µ
SL
ST
T
Tb
Ua
longitudinal pitch of pin array
transverse pitch of pin array
temperature
temperature at boundary
mean air velocity in the duct upstream of the pins
Uo
mean air velocity through minimum free flow area between
pins
XL
XT
normalized longitudinal pitch of pin array (SL/D)
transverse pitch of pin array (ST/D)
X
( XT – 1 )
pitch ratio used for non-square arrays = ------------------( XL – 1 )
Greek Symbols
λ
thermal conductivity
µ
viscosity
ρ
density
η
fin efficiency
θ
wake function or normalized adiabatic temperature rise
χ
friction factor ratio for non-square arrays ( S L ≠ S T )
NOMENCLATURE
A
Amin
cp
Re
Subscripts
a
air or approach
ad
adiabatic
f
fin
in
air inlet conditions
m
metal thermophysical property


∆p ⁄ N
per-pin friction factor  ----------------
2
1
 --- ρU o 
2
heat transfer coefficient
index variables
length of pin
number of rows in the pin fin array
number of pins in a row
Nusselt number
number of segments on each fin
pressure
perimeter
Prandtl number
heat transfer rate
heat input at boundary
INTRODUCTION
Air cooled heat sinks are the workhorse device for all sorts of components in the electronics industry. The widespread availability of userfriendly computational fluid dynamics (CFD) software now enables
heat sink designers and to obtain a reasonably accurate prediction of
heat sink performance. Although desktop computers have become
powerful enough to run CFD models, a full parametric design study
using CFD alone remains impractical because solutions times are still
measured in hours and post processing analysis of large CFD data sets
remains time consuming. Simple accurate numerical models for heat
sinks remain highly desirable to expedite design trade-offs and optimization analyses, e.g. [1-3].
1
Given a set of design constraints and an appropriate heat sink model,
a multi-parameter optimization can be carried out using the procedure
illustrated by Culham and Muzychka [3]. The derivatives of the minimization function with respect to each independent parameter in [3] can be
determined from a function call to a heat sink model with any desired
level of complexity. The accuracy of the design optimization thus
achieved will depend on how faithfully the model embodies the physics
governing heat transport within the heatsink structure.
tive of the present work is to describe such a model by extending the
approach developed in Holahan et. al. [7]. While the modeling approach
presented here is applicable to heat sinks with a variety of pin fin shapes
including circular, rectangular, elliptical, interrupted plate etc., appropriate heat transfer and pressure drop models are required in each case. In
this work, we use empirical correlations and CFD simulations with a
commercial CFD code, Icepak [14], to develop a model for heat sinks
with in-line arrays of circular pin fins.
Numerous investigations in the literature have been directed at parallel plate heat sinks in transverse flow e.g. Iwasaki et. al. [4], Teerstra
et. al. [5], Lee [6] using fin efficiency calculations based on a duct
length averaged Nusselt number. We call this a one-dimensional (1-D)
approach since the fin temperature contours are implicitly approximated
as horizontal isotherms. The flowstream based model described by
Holahan et. al. [7] uses the two-dimensional (2-D) temperature field in
the fin to determine heat transfer and enables alternate flow paths such
as top-inlet-side exit to be modeled as in [8].
DEVELOPMENT OF THE ANALYTICAL MODEL
The heat transport within the type of heatsink structures considered
in this work differs from that of plate fin heatsinks because of the
absence of streamwise heat conduction within the fins. Heatsinks in this
category include a variety of pin fin and interrupted plate fin designs like
those illustrated in Figure 1. To correctly represent the fluid and heat
transport in these heatsinks the model needs to include the following:
For pin fin heat sinks, the information available is relatively sparse.
A large experimental data set for cross-cut pin fin heat sinks by Shaukatullah [9] used a test configuration with considerable flow bypass but the
pressure drop across the heat sink was not reported. Jonsson and Palm
[10] tested heat sinks with plate fins, square pin fins, circular pin fins,
and elliptical pin fins and reported both thermal resistance and pressure
drop. However, their setup had large bypass and the pressure taps were
too far upstream and downstream of the heat sink to correctly measure
its driving pressure drop. Recent research undertaken by Ortega’s group
whose initial results are reported by Dogruoz et. al. [11] shows promise
for generating detailed pressure drop data that can be used to develop
accurate design models.
Because of the limitations of the available pin fin heat sink data,
most designers turn to the well documented heat transfer and friction
factor correlations developed by Zukauskas [12] for tube banks in crossflow. Jonsson and Moshfegh [13] provide new experimental data that
was used in the present work to extend the geometry range to sparser pin
fin arrays. A common modeling approach is to use the heat transfer
coefficient from the Zukauskas correlations in an equation of the following form:
θ sa
•
path taken by the fluid streams through the heatsink
•
pressure gradient due to frictional and form drag
•
heat transfer coefficient on the wetted surfaces of the heat sink
•
1-D heat conduction within each fin from the base to the tip
•
adiabatic temperature rise of a fin due to heat added to each fluid
stream by upstream fins
•
heat conduction within the base of the heatsink, and
•
thermal boundary condition applied to the base of the heatsink
Figure 1. Pin fin and Interrupted plate fin heat sinks
Q - + --------φQ
= ---------hAη m· c p
In the present work we consider transverse laminar flow through a
fully ducted heat sink with Reynolds number ranging from 10 to 3000.
We restrict our attention to in-line arrays of circular pin fins since a suitable body of empirical data and correlations are available in the literature. Figure 2 depicts the pin fin array geometry considered.
SL
Domain of single-row
D
CFD model
where φ ≤ 1 is a multiplier whose value depends on the reference temperature difference used as the basis for the heat transfer coefficient h.
For example, φ = 0.5 for a uniform temperature boundary condition at
the base where the driving temperature difference would be the average
bulk temperature of the fluid in the array. We again designate this as a
“1-D” approach since the variation of the fluid bulk temperature, fin
temperature and fin efficiency along the flow direction is not accounted
for.
Ua
Tin
A considerable improvement in the accuracy of heat transfer prediction from the heat sink could be made by accounting for the “2-D”
nature of the fluid and pin fin temperature distribution in the pin fin
array (this still ignores “3-D” flow patterns at the base and tips of the
fins). A further improvement could be made by recognizing and
accounting for the real driving potential for convective heat transfer
between the fin and the fluid: namely the fin surface temperature to fin
adiabatic temperature difference. Knowledge of the thermal wake in the
pin fin array is required to predict fin adiabatic temperatures. The objec-
...........
y
ST
x
Pin Number 1
2
3
.....
N
Figure 2. Circular pin fin array studied
2
The development of the heat sink model is presented in two steps. In
the first step we use CFD models to illustrate the thermal wake within an
array of fins and to show the grid density that is required so that CFD
predictions reasonably match the existing empirical data for pressure
gradient and heat transfer. We use the CFD results and empirical data to
develop correlation equations for the heat transfer coefficient, the pressure drop, and the wake function that are needed as building blocks for
the heat sink model. In the second step we describe the framework of the
pin fin heat sink model that captures the essential physics of the flow
and heat transport within the fin array.
12
θ pin
Dire
ction
6
3
Building Block Correlations
0
1
The first CFD model was for a 3 x 8 array of circular pins situated in
a duct. The longitudinal and transverse pitch to diameter ratio of the
array was 2.0 and the distance between the side walls of the duct and the
centers of the nearest row of pins was half the transverse pitch. A uniform air velocity and temperature were set at the duct entrance while an
outflow boundary condition was specified at the duct exit. Zero velocity
gradient and adiabatic thermal boundary conditions were applied at the
duct walls to simulate a large array in the transverse direction. All the
pins were modeled with a thermal conductivity of 180 W/m-K and zero
heat dissipation rate except for the second pin in the center row which
had a finite heat dissipation rate. Air was modeled with constant thermophysical properties as documented in Table 1.
Thermal Conductivity, λ a
0.026 W/m-K
Viscosity, µ
1.85 x 10-5 kg/m-s
1005 J/kg-K
2
3
4
column
5
6
7
8
Figure 3. Normalized temperature distribution in 3x8 array
with one heated pin
As above finding that in transverse flow the thermal wake is essentially confined to each row justifies the use of a CFD model containing
only a single row of pins for the remainder of the study. The CFD model
extended to the midplane between the adjacent row of pins on either side
of the row modeled. Symmetry boundary conditions were applied on
both these midplanes. The model allows for non-symmetric features like
vortex shedding behind the pins which contribute to the pressure drop
and heat transport characteristics of the fin array. Table 2 lists the range
of parameters included in the CFD simulations.
TABLE 1. Air Properties Used
Density, ρ
1.16 Kg/m3
Specific Heat, cp
F l ow
9
TABLE 2. Configurations Modeled
Figure 3 presents the adiabatic temperature rise of each of the
unheated pins in the array normalized by the bulk air temperature rise
for the row with the heated pin using the “per row heat capacity” as follows:
( T pin – T in )
θ pin = --------------------------∆T bulk
(1)
q
∆T bulk =  --------------------------
 ρU a c p S T L
(2)
Configuration
XL
XT
1
2
3
4
5
6
7
1.5
1.5
2.0
2.0
2.0
2.5
2.5
1.5
2.0
1.5
2.0
2.5
2.0
2.5
A grid sensitivity study was conducted on the geometry with XT=2
and XL=2 at a Reynolds number of about 750. The initial grid used was
representative of the grid density that is commonly used by CFD users in
industry when they model complete heat sinks. For each grid density the
per pin friction factor and Nusselt number for the heated pin were determined. Grid density is calculated as the product of the ratios of pin
diameter to maximum grid size specified in the x and y directions. The
Nusselt number is calculated using the following definition:
Figure 3 shows that only those pins located in the same row as the
heated pin experience a measurable temperature rise, i.e., the thermal
wake is confined to the transverse flow stream containing the heated pin.
The magnitude of the adiabatic temperature rise of the pin immediately
downstream of the heated pin is considerably greater than the bulk temperature rise of the air in the heated row. Further downstream of the
heated pin, the adiabatic temperature rise decays towards the bulk temperature rise. The decay is essentially complete (within 5 percent) by
about five pins downstream of the heated pin. This variation of the adiabatic temperature rise is termed the “thermal wake function”. An interesting finding is the small temperature rise experienced by the unheated
pin upstream of the heated pin. This is caused by flow recirculation in
the wake region of each pin.
q pin
D
Nu =  ---------------------------------------- ----πDL ( T pin – T in ) λ a
(3)
The grid sensitivity results are reported in Figure 4. These results
imply that a very fine grid is necessary to accurately model the behavior
of pin fin heat sinks. Such a fine grid is rarely used in most applications
of CFD within the electronics cooling industry. Therefore, heat sink
models of the kind developed here which embody the results from fine
3
grid CFD simulations will provide a more accurate prediction of fully
ducted heat sink behavior than most practical CFD models of the complete heat sink.
1
Nu
f
0.9
Nu , f Ratio
TABLE 3. Coefficients for the friction factor
correlation
0.8
XL
C1
C2
m
1.5
2.0
2.5
4.3
48
23
12
9
0.30
0.25
0.18
0.10
2.5
2.0
1.7
1.1
0.7
10
Zukauskas (XL=1.5)
0.6
Zukauskas (XL=2.0)
Zukauskas (XL=2.5)
0.5
0
100
200
300
400
500
f/ χ
Grid Density
1
Figure 4. Grid-sensitivity of friction factor and Nusselt
number
A CFD model of the single row of pins with the grid density of 225
from Figure 4 was used to develop the building block correlations for
the analytical model. While not entirely grid independent, the forthcoming discussion will show that the CFD models strike a reasonable balance between providing good agreement with empirical data and
economy of computational effort.
0.1
Results obtained from the CFD simulations include the per-row friction factor, Nusselt number, and the thermal wake function. Figures 5a
and 5b respectively show the per pin friction factor results for square
and non-square arrays using the same format as Zukauskas[12]. For
square arrays the per pin friction factor for each value of X L is com-
100
1000
10000
Re
10.0
Re=2000
Re=200
χ
puted as defined in the nomenclature. The friction factor ratio χ for
non-square arrays is computed by dividing the friction factor result for
the X T ≠ X L simulation by the result for the square array with the same
X L . As shown in Figure 5(a), the CFD results for square arrays are in
Re=20
1.0
0.1
good agreement with the empirical data compiled by Zukauskas over the
range X L = X T = 1.5…2.5 . Results shown in Fig. 5(b) for the non-
0.1
(b)
square arrays agree quite well with the empirical data for X T < X L but
1
10
(XT-1)/(XL-1)
Figure 5. Friction factor results from single-row CFD
model
are low for X T > X L . For calculating the friction factor we use equation
(4) and the coefficients given in Table 3. This correlation is based on
empirical data from Zukauskas [12] except for the last row in the table
which is based on pin fin heatsink data from Jonsson and Moshfegh
[13].
f = X – 0.754 ⋅ ( ( C1Re – 0.8 ) m + C2 m ) 1 / m
10
(a)
Figure 6 compares the Nusselt numbers at the fourth pin in the flow
direction with empirical correlations over the Reynolds number range
from 10 to 3000 and the combinations of XL and XT listed in Table 2.
The predictions agree very well with the empirical correlations of
Zukauskas over the whole range studied except for the intermediate
range 100 < R e < 1000 where the correlation equation appears to be
low probably due to a typographical error in [12]. The CFD results are
well represented by slightly modifying and blending the correlations
provided by Zukauskas for Reynolds number ranges above and below
this intermediate range using the following equation also shown in Figure 6.
10 < Re < 3000
( XT – 1 )
where X = ------------------( XL – 1 )
Nu = ( ( 0.85Re 0.4 Pr 0.36 ) 10 + ( 0.26Re 0.63 Pr 0.36 ) 10 ) 1 / 10
4
(4)
This correlation is used for all the pins in the array based on a pin by
pin Nusselt number study at Re ~ 750 which showed that the fourth pin
data was representative of all the pins in the array within −
+ 2 percent
except for the first pin which was only 5 percent higher. The reasonable
agreement seen between CFD and empirical results for the friction factor and Nusselt number provides support for the thermal wake function
determined from the same CFD simulations.
have higher values of θ 1 . This means that with increasing transverse
pitch the bulk temperature decreases much faster (as 1/ST) than the adiabatic temperature rise in the wake of the heated pin. This is just what
would be expected when fluid convection dominates. Because our CFD
results are not entirely grid independent the reader should note that the
true magnitude of θ 1 is somewhat greater than the values reported here.
3.5
100
XT=2 XL=2
XT=2 XL=2.5
3.0
XT=2.5 XL=2
XT=2.5 XL=2
XT=1.5 XL=2
XT=2.5 XL=2.5
XT=1.5 XL=2
θ1
Nu
XT=2 XL=1.5
10
XT=2, XL=2
2.5
XT=1.5 XL=1.5
Zukauskas
Equation (6)
2.0
Equation (5)
1.5
1.0
1
0
10
100
1000
1500
10000
2000
2500
3000
Figure 7. Wake function at first pin downstream of heated
pin
Figure 6. Nusselt number results from single-row CFD
simulations
The decay of the thermal wake at the downstream pins is well represented by the following correlation function for square and non-square
arrays.
The wake function is modeled in two parts, the first accounts for the
adiabatic temperature rise of the first pin downstream of the heated pin
and the second part accounts for the decay of the wake function downstream from the first pin. In order to arrive at a suitable correlation we
looked for asymptotes for the wake function at low and high Reynolds
numbers and then blended the two together using the method of
Churchill and Usagi [15]. Following Zukauskas, we first addressed
square arrays where X T = X L and then looked at deviations for config-
( θi – 1 )
------------------ = e– n(i – 1) + (1 – X)
( θ1 – 1 )
(7)
The value of the exponent n may be determined for a given longitudinal
pitch by interpolation from the values tabulated in Table 4.
TABLE 4. Coefficients for Wake Decay
urations where X T ≠ X L . The resulting correlations are inspired by the
underlying physics of the data rather than a multi-parameter fit to a large
set of data. The wake function for the first pin is well represented by the
following equation which is compared in Figure 8 with CFD results for
three configurations which represent the upper and lower bound values
of θ 1 in our CFD study.
XL
n
1.5
2.0
2.5
4.3
1.70
1.05
0.90
0.85
(5)
Heat Sink Model Framework
(6)
We consider a heat sink composed of an array of circular pin fins as
shown in Figure 2. The length L of each pin is constant in the z direction. The airflow is assumed to approach the pin fin array in the x-direction with a uniform velocity and temperature and to remain twodimensional within the array (no velocity in the z-direction). Thermal
wake effects are limited to fins situated along the same flow stream, i.e.
within the same y and z segments.
where
ϕ = ( X L – 1 ) ( 1.2X 2 – 0.64X + 0.48 )
1000
Re
Re
θ 1 = 1 + ϕ { 1 – [ 1 + ( 0.015Reϕ 1 / 0.7 ) 2 ] –0.7 / 2 }
500
As shown in Figure 7, the correlation equation provides a good fit
for the data over the whole range studied. The figure shows that the
value of θ 1 starts at 1.0 (i.e. adiabatic temperature rise is equal to the
bulk temperature rise) at low Re where heat diffusion is dominant,
increases sharply up to an Re ~ 200 as convection becomes more and
more dominant and then more slowly at Re > 500 towards a high Re
asymptote of 1 + ϕ . A comparison between the data for the three different geometries shows that configurations with a larger transverse pitch
5
path. The path is constrained only in the sense that it represent a flow
streamtube which must be known apriori.
.
Segment
Number
Equations (10) through (12) can be substituted in equation (9) to
yield the heat balance equation for each segment of each fin
O
dz
Ua
Tf
Tin
∑
Tad
k
1≤l≤k
qk
zk
Pin Number 1
2
T0
...
i
q
...
N
T 0 – T in = ( T 0 – T f ) + ( T f – T ad ) + ( T ad – T in )
∑
q i, k
T f – T ad = -----------hPdz
1
T ad – T in = ---------------------------ρU a c p S T dz
∑
q
(13)
q
boundary condition or 0 for a specified base temperature boundary con-
th pin.
dition for the i
The above equations (13) and (14) comprise a set of N ⋅ ( O + 1 )
equations for a similar number of unknowns (q for O segments on each
of N fins plus T0 for N fins). This set of equations can be written in
matrix form and solved using standard matrix inversion methods to
determine the heat flow and temperature distribution within the row of
fins. These equations were implemented in a commercial mathematics
code MathCad [16]. Correlation equations described earlier for the friction factor, Nusselt number and thermal wake for circular pin fins were
implemented in the model. The solution results for q and T0 can be used
with equations (10) and (11) to determine the fin temperature and adiabatic temperature for each segment of each fin.
(8)
th fin in the flow
zl 
 ---------- λ m A f q i, l
q
where S i is a switch with a value of either 1 for a specified heat input
These separate pieces can be written in terms of the heat transferred
from each segment of the fin. For segment k on the i
direction, these terms become:
q
l
We examine heat transfer from a fin within the array with the aid of
Figure 8. The fin is divided into several segments along its length starting from the base. Heat transfer from each segment of the fin is driven
by the same overall temperature potential; the difference between the
base temperature of the fin and the inlet temperature of the air. This
potential can be separated into three distinct parts comprising the conduction within the fin, convection from the fin surface and upstream
heatup of the air as follows:
1≤l≤k
q l, k θ i – l – T 0 = T in(12)
S i ∑ q i, l + ( 1 – S i )T 0 = S i Q i + ( 1 – S i )Tb i
Figure 8. Heat transfer from an interior fin in the array
T0 – Tf =
∑
1≤l<i
One more equation is needed to specify the heat input or temperature
boundary condition at the base of each fin as follows:
2
1
zl 
q i, k
1
 ----------- q + ------------ + --------------------------- λ m A f i, l hPdz ρU a c p S T dz
Heat Transfer from Fin Segments to Air
Fin temperatures in Array
(9)
(10)
q l, k θ i – l
(11)
1≤l<i
where
q i, k is the heat transfer from segment k on pin i
q
θ i – l is the wake function on pin i due to heat transferred from pin l
T
Figure 9. Sample solution from a one-row pin fin array
model (D=1 mm, ST=SL=3 mm, L=25 mm, Ua=4
m/s, Ta=0 C, Tb=100 C)
z k is the distance from the base to the center of segment k of the fin
dz k is the length of segment k of the fin
Figure 9 presents a typical result from the model for a row with ten
pins each with five segments. This model computes the solution in under
2 seconds on a personal computer with an Intel P3 CPU running at 650
MHz. As seen in Figure 9, this model provides a detailed accounting of
the heat transfer rates and temperature distribution for each fin in the
row. A check of the temperature distribution on the first pin showed
excellent agreement with 1-D models for conduction in a fin with a uniform convective heat transfer coefficient from the fin surface to fluid at
a uniform temperature. On downstream fins, the fluid temperature
2
----------
A f is the cross-sectional area of the fin  = πD

4 
Note that value of Tad can be set equal to the bulk temperature by specifying θ i – l = 1 . The adiabatic temperature rise expressed in equation
(12) contains information on the path of the fluid flow through the pin
fin array and the order in which the various pins are arranged along that
6
becomes non-uniform so temperature predictions depart from the 1-D
fin model.
here is readily extendable to non-orthogonal transverse flows including
impinging flows. This would be done by partitioning the flow field into
streamtubes (see [7]) which delineate the corresponding thermal wake
linkages between segments, representing new effective pin rows, which
may even be curved. Equation (12) provides the means to implement
any “flow stream connectivity” implied by the flow streamtubes that
describe the flow path through the array. If the wake function correlations are not available, a suitable first step is to set the wake function to
unity.
Moving a step further we developed the model to represent a heat
sink having a conducting base and a two dimensional array of fins. Heat
transfer from each row of fins is treated as before but each fin is now
connected to a heat sink base of finite thickness. Heat conduction within
the base is solved using a standard finite volume approach [17] with
boundary conditions supplied on the lower surface of the heat sink base.
A simple version of base conduction was implemented by setting the
size of each finite volume equal to the volume of the base under each
fin, i.e., S T × S L × tb . The base temperature T0 for a fin is now the base
Other types of fins such as interrupted plate, elliptical, wing shaped
etc. can be modeled using the same heat sink model by simply providing
the applicable correlation equations for Nusselt number and friction factor, e.g. those described by Muzychka and Yovanovich [19] for offset
strip fin arrays.
temperature at the center of the finite volume below that fin. Thermal
conductance terms connect the temperature in a control volume to adjacent control volumes in the x and y directions and to the boundary conditions specified on the lower surface of the heat sink base. The heat
conduction equation for the base control volume below the ith pin in the
jth row can be written as follows:
The heat sink model was compared against the experimental data of
Jonsson and Moshfegh [13] for a heat sink consisting of a 9 x 9 array of
10 mm long 1.5 mm diameter pin fins arranged on a square grid with a
pitch of 6.5 mm. We assumed a base thickness of 4 mm since this was
not provided in [13]. Pressure drop and thermal resistance predictions
were made for a fully ducted configuration for approach velocities ranging from 1.7 to 9 m/s corresponding to a Reynolds number range from
about 200 to 1100.
(14)
k
– C x T i – 1, j, 0 – C x T i + 1, j, 0 – C y T i, j – 1, 0 – C y T i, j + 1, 0
q
q
= S i, j Q i, j + ( 1 – S i, j )C z Tb i, j
CFD simulations were conducted using Icepak [14] for the above
heat sink geometry using three levels of grid refinement. Even at the finest grid which used ~450,000 control volumes the CFD results were not
grid independent. Further grid refinement was not conducted because
the computational array sizes exceeded the physical memory on the
desktop PC (~512 MB). Each CFD simulation took 44 minutes to complete.
3
200
R-Data [13]
λ m S T tb
λ m S L tb
where the conductance terms are C x = -----------------, C y = ----------------- , and
SL
ST
2λ m S L S T
-.
C z = --------------------tb
Thermal Resistance
(C/W)
tb
1  is added to the term
An additional resistance Rc  = ----------------------- + ------------- 2λ S S

2λ
m T K
mD
zl 
 ----------- in equation (13) to account for conduction between the position
 λ m A f
at which T0 is now computed and the base of the actual fin. This term
includes the constriction resistance at the base of the fin. The heat transfer from the top surface of the heat sink base around the fins is included
in the convective heat transfer expression for the first segment of the fin.
The heat transfer coefficient on the base area was assumed to be 0.5
times the heat transfer coefficient on the surface of the fins based on data
reported by Metzger et. al [18]. The constant of proportionality will
mainly affect the thermal resistance predictions for heat sinks with
sparse pin fin arrays where heat transfer from the wall area is significant.
This wall heat transfer coefficient is very difficult to measure experimentally and its dependence on the array geometry, aspect ratio (L/D) of
the pins, Reynolds number etc. is not known sufficiently and is a candidate for further study, maybe using CFD. The heat transfer from the fin
tips has been left out of the present model but it can be easily included in
the expression for the last segment of the fin if the fin tips are exposed to
air flow.
R-Model
R-CFD 449,320 grids
DP-Data [13]
DP-Model
DP-CFD 449,320 grids
2.5
2
160
120
1.5
80
1
40
0.5
0.00
3.00
6.00
Pressure Drop (Pa)
q
( 2C x + 2C y + S i, j C z )T i, j, 0 + ∑ q i, j, k
MODEL RESULTS AND DISCUSSION
0
9.00
Approach Velocity Ua (m/s)
Figure 10. Model predictions for a pin fin heat sink
compared to experimental data of [13]
Figure 10 shows that the predictions from the model are in excellent
agreement with the experimental data while the CFD simulations show
considerable differences. For the model, the excellent agreement with
the pressure drop data is to be expected since the data itself was used to
determine the coefficients for the friction factor model (shown in Table
3). However, the good agreement over the whole velocity range shows
the good correlation that is achieved between the Reynolds number and
the friction factor through equation (4).
Extensions of the Heat Sink model
Although we have focused on a simple transverse flow aligned with
in-line rows of pin fins to develop the model, the approach presented
The excellent agreement between the model predictions and experimental data for thermal resistance validates the Nusselt number correla-
7
tion used and provides confidence in the overall model including the
choice of the proportionality constant for the heat transfer coefficient on
the base surface around the fins. If the proportionality constant were set
to 1.0 instead of 0.5, the predicted thermal resistance would decrease by
6.5% at 1.7 m/s and by 10% at 9 m/s. The influence of the thermal wake
function in this case is more significant. If the adiabatic temperature rise
were replaced by the bulk temperature rise (by setting θ i – l = 1 ) in the
[5] Teertstra P.M., Yovanovich, M. M., Culham, J. R. and Lemczyk, T.
F., 1999, “Analytical forced convection modeling of plate fin heat
sinks,” in Proc. 15th Annu. IEEE Semicon. Thermal Meas. Manag.
Symp., San Diego, CA, March 9-11, pp. 34-41.
[6] S. Lee, “Optimum design and selection of heat sinks”, in Proc. 11th
IEEE SEMI-THERM Symposium, 1995, pp. 48-54.
calculation of heat transfer from the fins, the predicted thermal resistance would be reduced by 18% at 1.7 m/s and by 12% at 9 m/s air
velocity.
[7] Holahan, M. F., Kang, S. S., and Bar-Cohen, A. 1996, “A flowstream based analytical model for design of parallel plate heatsinks”, in Proc. 31st National Heat Transfer Conference, ASME
HTD-Vol. 329-7, pp 63-71.
CONCLUSIONS
[8] Chen,C.L., Peppi, K., Day, S., Liao, C., 1998, “Thermal analysis
design tool for parallel plate heatsinks”, in Proc. Sixth Intersoc.
Conf. on Thermal Phenomena, ITHERM 1998, pp371-377.
A physics based model for pin fin heat sinks in forced convection
has been developed. Key empirical data and correlations for the pressure
drop and heat transfer in in-line arrays of circular pin fin arrays were
validated through CFD simulations. New correlation equations were
developed to represent the empirical friction factor data. New knowledge about the thermal wake within the pin fin arrays was developed
from the CFD simulations. Correlations to represent the thermal wake
were developed. An attempt was made to accurately represent within the
model framework the most relevant mechanisms governing heat transport within the heatsink structure including heat transfer from the fins
and the fluid wetted wall of the base, heat spreading in the base and thermal wake effects due to fluid routing through the fin array. The model
was implemented using the commercial mathematics software Mathcad.
Predictions from the model were in good agreement with experimental
data, much better than grid-limited CFD results for the full heat sink.
[9] H. Shaukatullah, W. R. Storr, B. J. Hansen, M.A. Gaynes, 1996,
“Design and optimization of pin fin heat sinks for low velocity
applications”, 12th IEEE SEMI-THERM Symposium, pp.151-163.
[10] Jonsson H. and Palm, B. 1996, “Experimental comparison of different heat sink designs for cooling of electronics,” in Proc. 31st
National Heat Transfer Conference, ASME HTD-Vol. 329-7, pp
27-34.
[11] Dogruoz M. B., Urdaneta, M., Ortega, A. and Westphal, R. V.,
2002, “Experiments and modeling of the hydraulic resistance of inline square pin fin heat sinks with top bypass flow,” in Proc. Eighth
Intersociety Conference on Thermal Phenomena, ITHERM 2002,
pp. 251-260.
[12] Zukauskas, A., 1987, “Convective heat transfer in cross flow,” in:
Handbook of Single-Phase Convective Heat Transfer, S. Kakac,
R.K. Shah and W. Aung, eds., Wiley, New York, Chap. 6.
The model provides a framework for putting existing literature correlations and information developed from detailed fine-grid CFD simulations to work in the design of pin fin heat sinks. At a minimum, the
model can be expected to predict heat sink performance similar to a
detailed CFD model with a fraction of the data entry and computational
effort. Better yet, the incorporation of empirical information allows the
present model to capture complex physical effects which would require
huge detailed CFD models to simulate from scratch each time. Last but
not least, the model provides detailed but easily interpreted results
regarding heat transport and temperatures within the heat sink structure.
It is anticipated that the model will become a valuable tool to enable
consistent performance comparisons and selection between various
types of pin fin and plate fin heat sinks.
[13] H. Jonsson, B. Moshfegh, 2001, “Modeling of the thermal and
hydraulic performance of plate fin, strip fin, and pin fin heat sinks Influence of flow bypass”, IEEE Trans. Components and Packaging Technologies, Vol. 24, No. 2, pp. 142-149.
[14] Icepak 4.0, 2002, Fluent Inc., Lebanon NH.
[15] Churchill, S. W. and Usagi, R.,1972, “A general expression for the
correlation of rates of transfer and other phenomena,” J. American
Institute of Chemical Engineers, Vol 18, pp. 1121-1128.
[16] Mathcad 2001i, 2001, MathSoft Engineering and Education, Inc.,
Cambridge, MA.
[17] Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow,
Hemisphere, Washington, DC.
REFERENCES
[1] Bejan, A., and Sciubba,E., 1992, “The optimal spacing of parallel
plates cooled by forced convection,” Int. J. Heat Mass Transfer,
Vol. 35, pp. 3259-3264
[18] Metzger, D. E., Fan, C. S., and Haley, S. W., 1984, “Effects of Pin
Shape and Array Orientation on Heat Transfer and Pressure Loss in
Pin Fin Arrays,” Journal of Engineering for Gas Turbines and
Power, Vol. 106, pp. 252-257.
[2] Bejan, A. and Morega, A. M., 1993, “Optimal arrays of pin fins and
plate fins in laminar forced convection,” ASME Journal of Heat
Transfer, Vol. 115, pp. 75-81.
[19] Muzychka, Y. S. and Yovanovich, M. M., 1999, “Modeling the f
and j Characteristics of the Offset Strip Fin Array,” ASME HTDVol. 364-1, pp 91-100.
[3] Culham, J.R. and Muzychka, Y. S., 2001, “Optimization of plate fin
heat sinks using entropy generation minimization,” IEEE Trans.
Comp. Packag. Technol., Vol 24, pp. 159-165.
[4] Iwasaki, H., Sasaki, T., and Ishizuka, M., 1994, “Cooling performance of plate fins for multichip modules,” in Proc. Fourth Intersoc. Conf. Thermal Phenomena in Electronic Systems, ITHERM
1994, pp. 144-147.
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