Optimal Thermal Design of Forced
Conwection Heat Sinks-Analytical
R. W. Knight
Assistant Professor.
J. S. Goodling
Professor.
D. J. Hall
Graduate Student.
Mechanical Engineering Department,
Auburn University,
Auburn, AL 36849
For fully developed flow in closed finned channels used to augment heat transfer,
there exists an optimal geometrical design of the size and number of cooling channels.
In this paper, the problem is generalized with a statement of dimensionless thermal
resistance in terms of
9
the number of channels
9
a fin to channel thickness ratio
8
the length to width {planar dimensions) ratio of the heat source, and
9
a specified fin efficiency or fin length
9
a fluid to fin thermal conductivity ratio
9
the Prandtl Number of the coolant
8
a dimensionless pressure term, which incorporates the maximum allowable pressure
drop through the cooling channels or alternatively,
9
a dimensionless work rate term, which incorporates the maximum allowable coolant pumping power required,
An optimization scheme is described and used for comparison with two previously
published cases wherein both designs were restricted to afixedfin to channel thickness
ratio and laminar flow; one by Goldberg (1984) using air and copper and a second
one only by Tuckerman and Pease (1981) for water-cooled Silicon wafers. Results
from the present optimization scheme show that upon reexamination of the first
study by Goldberg, significant reduction of thermal resistance can be obtained
by using fin/channel dimensions other than unity. A similar reduction is found in
the second instance (Tuckerman and Pease) with the relaxation of the laminar
limitation.
1
Introduction
For over a decade, efforts have been expended to provide
innovative methods of heat removal from increasingly powerful electronic circuits. The methods used and being investigated are summarized in the recent book by Bar-Cohen and
Kraus (1990).
The present work is inspired by a technique devised by Tuckerman and Pease in 1981, which used very narrow channels
etched onto the backside of a silicon wafer. Designed for optimal performance subject to some constraints (pressure drop,
planar dimensions, fin efficiency, etc.) with laminar flow in
mind and for water as the coolant, these authors made a test
section which achieved a flux level of 790 W/cm2 with a maximum temperature rise of 71 °C. Their pioneering work is invariably referenced in successive papers on microchannel
cooling.
Goldberg (1984) built and tested an air cooled narrow channel heat sink. Three different fin thicknesses were considered,
with the channel thickness always made equal to the fin thickness. All cases were restricted to laminar flow. The pressure
drop across each device was adjusted to provide an air flow
rate of 30 liters per minute. The lowest thermal resistance was
Contributed by the Electrical and Electronic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received by the
EEPD August 27, 1990; revised manuscript received June 25, 1991. Associate
Editor: W. Z. Black.
found from the design with the smallest channel width and the
highest pressure drop.
Sasaki and Kishimoto (1986) optimized the dimensions of
channels at a given pressure loss through the water cooled fins
in a silicon chip. Again, the criterion of fin to channel thickness
ratio of unity was invoked. The analysis matched the experimental results well, however, the former is not presented. The
optimal channel thickness was found to be at 340 /xm for a
pressure drop of either 200 or 2000 kg/m2.
Hwang et al. (1987) designed a novel cooling package which
places the cooling channels just beneath and parallel to the
heat source. This design, which differs considerably from the
fin concept of Tuckerman and Pease where the channels are
perpendicular to the source, was suggested in an earlier work
by Tuckerman (1984, Fig. 2-7, p. 36). For Hwang's design,
the fluid dynamics of the channel flow dominate the fin effects.
A two-dimensional conduction analysis was performed with
boundary conditions at the solid/liquid interface which used
either laminar or turbulent convective correlations. Channel
dimensions were systematically varied over a limited range of
Reynolds numbers (1100 to 1600 for laminar flow and 12,000
to 13,000 for turbulent flow) and over a large pressure range
from 13.1 to 682.4 kPa (1.9 to 99 psi).
Nayak, Hwang et al. (1987) followed the previous cited work
with experiments using the designs chosen above for a multichip module. Coolant flows were regulated so the convection
Journal of Electronic Packaging
SEPTEMBER 1991, Vol. 113 / 313
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Ap
thermal conductivity to present appropriate dimensionless parameters and to examine trends for the simplest of geometries.
This is followed by an analysis for the more realistic case of
finite fins, where the problem cannot be optimized analytically.
/I
insulation
^ ^ ^ ^ ^ ^
2
Fig. 1
Schematic of heat spreader with infinitesimally thin fins
in the channels was clearly in either the turbulent or laminar
flow region and, as expected, the former gave lower thermal
resistance than the latter.
Phillips (1990) recently authored an article in which he reviews recent works as well as his own on microchannel heat
sinks. Extensive discussion is offered on the influencing effects
(simplifying assumptions, properties, friction factors, viscous
dissipation, developing flow, etc.) and quantifies many of these
parameters with a computer solution of the governing thermal
resistance equation. Most results are presented in the form of
thermal resistance as a function of channel width with all other
parameters predetermined and specified, including the fin to
channel thickness ratio. For the test case discussed, turbulent
flow is shown to provide a lower value for thermal resistance
than laminar flow.
This paper generalizes the optimization method for sizing
coolant channels for any scale heat sink (spreader), be it microscopic or macroscopic in size. Furthermore, the restrictions
of laminar flow and fixed fin to channel thickness ratio are
lifted. The first section deals analytically with a highly simplified model that includes infinitesimally thin fins with infinite
Model
The structure under study is shown in Figs. 1 and 8. It
consists of a flat rectangular energy source whose cooling is
enhanced by the addition of multiple parallel fins closed at the
tips with a cover plate and with a coolant forced through the
array. In a design setting, the size and circuit power are constrained. Therefore, in addition to the physical dimensions of
width (W) and length (L), the rate of thermal energy to be
removed (q) is fixed. Furthermore, the pressure drop across
the fin array (Ap) would be a predetermined value due to
specified pump or air handler. The usual assumptions for this
type of analysis are made (steady state, constant properties,
adiabatic end plate, two-dimensional analysis, and fully developed). It is recognized that the last item is not usually true
for microchannels and that the heat transfer and frictional
losses are larger for developing flow. For very narrow channels
when n is large, the flow could be laminar. Conversely, the
flow could be turbulent. The problem here is to design channel
dimensions so the thermal resistance is a minimum.
The thermal spreader can be analyzed as two-dimensional
flow through narrow channels with the thermal boundaries
held at either constant temperature or constant flux. Figures
2 and 3 display the temperature profiles through the heat sink
for the two cases analyzed here. It is probable that the true
solution lies somewhere between these two boundary conditions. However, results shown later indicate that the two solutions yield quite similar results.
Using standard descriptors and nomenclature of heat exchangers where the wall temperature is constant in the streamline direction, the following equations are applicable (Incropera
and DeWitt, 1990):
q = mcp[Tft0-Tfi,]
(1)
Nomenclature
A
area
cp
constant pressure specific
heat
C\, C->
coefficients defined by
Eqs. (57) and (65)
D = depth of heat sink, see
Fig. 1
D„ = hydraulic diameter of
fluid flow channel
/ = friction factor, (Ap/L)Dh/
(pU2m/2)
G = a parameter defined by
Eq. (38)
h = heat transfer coefficient
k = thermal conductivity
I = channel width
L = length of heat sink in the
direction of fluid flow
flP,fin 1/2
m =
Kfin^4c,fin
m = total mass flow rate of
coolant through channels
n = number of cooling channels
N&p = pressure difference number,
(Ap/L)W3/(pv2)
•Nwork = work rate n u m b e r ,
wW/(pvl)
3 1 4 / V o l . 113, SEPTEMBER 1991
7i
Nu = Nusselt number, hDh/kan[i
Ap = pressure drop through the
heat sink channels
P = perimeter
Pr = Prandtl number, v/a
<7 - heat source power
Rec„ = Reynolds number based
on hydraulic diameter
^ t l a m = laminar Stanton number,
Nu(L/W)/(NApPr)
Stturb = turbulent Stanton number,
(L/W)/(N)&?T2n)
T = temperature
AT = largest temperature difference between coolant and
source
fluid velocity
uVm == mean
volumetric flow rate
w = pumping power
W = width of heat sink
a = thermal diffusivity of
fluid
n> 73 = coefficients defined by
Eqs. (37), (49), and (59)
of fin thickness to
r = ratio
channel width
i) = fin efficiency
V = kinematic viscosity of
fluid
mass density
thermal resistance, AT/q
Q = dimensionless thermal reAT
sistance, — : —
q
P
=
e=
fcfluidW
Subscripts
c = cross sectional available
for flow
c,fin = cross sectional of fin
/ . ' = fluid inlet
f,o = fluid outlet
h = hydraulic
H = constant flux case
lam = laminar
opt = optimal
optJam = optimal laminar case
opt-turb = optimal turbulent case
^ = surface available for heat
transfer
s,i = surface at the fluid inlet
face
S,0 = surface at the fluid outlet
face
turb = turbulent
T = constant temperature case
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CI)
V—
•*-•
T
s
T
s
crj
1—
CD
Q.
T
E
CD
T
fo
CD
*™
td
i_
AT
AT
CD
Q.
E
'f,o
CD
f,i
\i
Length
Fig. 2
Length
Constant temperature walls
(T s - 7),0) = (T s - 7),,)exp( - hAs/mcp)
Fig. 3
(2)
Upon defining thermal resistance as the largest temperature
difference between source and coolant (TSi0 - 7},,) divided by
the electrical power of the source, these two equations are
combined to form thermal resistance as
A dimensionless thermal resistance 0 =
q = hAs[TSJ- 7),,] = hAs[Ts,0 - 7>,0]
q = mcp[Tf,0-Tfi,]
(4)
(5)
Using the same procedure to define thermal resistance in terms
of the largest temperature difference between source and coolant (Ts<0- Tf,d, Eqs. (4) and (5) are combined to form thermal
resistance as
0 = A7 y^
*+J_
(6)
hAs mCp
2.A
Infinitesimally Thin Fins (Partitioning Walls)
2.A.1 Laminar Flow. With the use of several common
heat transfer groups, the equations for thermal resistance are
simplified.
• In this problem, no characteristic velocity is specified.
However, the friction factor based on hydraulic diameter
can be manipulated to define one as:
2(Ap/L)Dh
2Ap/LDh 1/2
Va,b)
/=
Um =
pU2m '
pf
For very narrow channels, the laminar friction factor for
fully developed flow is 9 6 / R e ^ .
• The Nusselt number (hDh/kaaii). For fully developed laminar channel flow, it achieves constant values of 8.24 and
7.54 for the constant flux and constant temperature cases,
respectively.
• The Prandtl number {v/a).
Further, when two groups are introduced, Eqs. (3) and (6)
are made dimensionless.
• A laminar Stanton number with L/W included
I Stiam =
— ) . The Stanton number is normally de\
NApPr Wj
fined as Nu/RePr, but in this case NAp (defined below)
takes the place of the Reynolds number .(L/W) is included
for the sake of notational brevity.
A dimensionless pressure drop number, NAp
(Ap/L) W3
. This group is similar to the friction factor.
pv
It arises since, rather than velocity, the pressure drop is
dictated by the fluid handler (pump or fan).
Journal of Electronic Packaging
AT
Using this definition of dimensionless thermal resistance, Eqs.
(3) and (6) are now concisely rewritten respectively as:
1
• AT/q = —
(3)
mcp[\ - exp( - fiAs/mcp)]
This heat sink can also be modeled as though the coolant
experiences a nonvarying flux of energy as it progresses through
the channel. Again using the nomenclature of constant flux
heat exchangers,
Constant flux walls
a.
\2rf
NApPr(D/W)
[l-exp(-12Stian,«4)]-'
for constant temperature
(8)
\2n
[1 +1/(12 St l a m « 4 )]-'
~'NApPr(D/W)
for constant flux
(9)
l
©la
The minima of (8) and (9) are found to occur, respectively,
when
12 Stlam/z4= 1.256
for constant temperature
(10)
12 Stiamn4 = 1.000
for constant flux
for
(11)
It is noteworthy that the two models yield values of n
for lowest thermal resistance which differ by only [1.256/
1.000]<1/4) or about 6 percent. This suggests that the choice of
the model (constant temperature or constant flux) is of little
consequence.
For the case of laminar flow through narrow channels, the
lowest temperature rise of the hottest portion of the circuit is
obtained by partitioning the width approximately into n channels where
(12)
« = (12St lam ) ( - 1 / 4 )
This means that once the physical (W, D, L) and system
(<y, A/?) parameters and the fluid (with properties p, v, a, k,
cp) are chosen, if the flow is constrained to be laminar, which
sets the Nusselt number, then the minimum thermal resistance
occurs when n is determined by Eq. (12). Once n is determined,
Reynolds number based on hydraulic diameter must be calculated and shown to be less than the critical Reynolds number
(~ 2300) for laminar flow. For the problem at hand, this means
that
Reiam = N A / /(6« 3 )<2300
(13)
2.A.2 Turbulent Flow. The procedure above is now repeated for the turbulent analysis. Here the friction factor,/,
for fully developed flow in smooth channels (Incropera and
DeWitt, 1990) becomes
(/ = 0.316 Rej,"
1/4)
(14)
The Nusselt number is no longer a constant but its value can
be obtained through the Chilton-Colburn analogy
Nu
/
(15)
1
8 RePr,1/3
'
The dimensionless group hAs/mcp becomes for the turbulent
case
SEPTEMBER 1991, Vol. 113 / 315
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1.5
Optimal
Laminar Solution
•
-
Optimal
/ ^
Turbulent Solution
0.5
Turbulent
'%
Transilion
-
-10'°
20
40
60
Laminar
,
I
80
100
120
n
n!?mh-, ^ll^LTil*!°nlefs'h0ermal
resistanpes as a function of the
1.5
Optimal
Turbulent Solution
by choosing n from Eqs. (19) or (20). Reynolds number based
on hydraulic diameter must then be found and shown to be
greater than the Reynolds number (~ 4000) which assures turbulent flow and allows proper use of these equations. For the
problem at hand, this means that
Optimal
Laminar Solution
\
^
1
~~~~—-—_!_____
optjaminar
Fig. 6 R a t i o o f d i m e n S ionless thermal resistances as a function of the
number of channels for W i p = 1012
Ret„rh =
9
2A
- 4,12/7
^4000
(21)
•
0.5
Turbulent
N
1
AP= °
Transition
where 9.42 is (2 16 /0.316 4 ) 1/7 .
In cases where the resulting Reynolds number is less than
4000, Eq. (21) with Re turb = 4000 must be solved for n to find
the best available turbulent solution.
Laminar
8
10
i
i
20
30
40
n
Fig. 5 Ratio of dimensionless thermal resistances as a function of the
number of channels for AVAp= 108
0.045 {L/W)nh
(16)
< 7 Pr 2/3
Again for the sake of notational brevity, Stturb, defined as (L/
W)/(NXH Pr 2/3 ), is introduced and Eqs. (3) and (6) simplify
after non-dimensionalization for constant temperature and
flux, respectively, to:
hAs/mcp
-
„5/7
«*°~
0.21,1'
[1 - e x p ( - 0.045 St turb H 10/7 )r
N%Vr(D/W)
0.21« 5
©turb —
NfJPr(D/W)
[1 + 1/(0.045 St t u r i y 0 / 7 )]
(17)
(18)
Noting that 0.316 is the coefficient of the friction factor Eq.
(14), the constant 0.21 is determined from (0.316 4 /2 9 ) 1/7 while
0.045 is found from (0.316 8 /2 18 ) 1/7 . Optimization of thermal
resistance with respect to n for the turbulent cases yields
0.045 Stturbn10 = 1.256 for constant temperature
(19)
and
0.045 St turb n 10/7 = 1.000 for constant
flux
(20)
These four equations ((17) through (20)) are similar to their
counterparts for the laminar development above, both in form
and the exponential power of n in the two competing terms
(one is the square of the other). The latter comes about as a
result of the surface and cross sectional area dependencies on
n. What was said about the significance of these equations for
laminar flow applies here also.
As for laminar flow, once the physical and system parameters and the fluid are chosen and the flow is constrained to
be turbulent, then the minimum thermal resistance is obtained
316 / Vol. 113, SEPTEMBER 1991
2.A3
Laminar or Turbulent Flow. It is now possible to
combine results from the constant flux cases above and provide
a procedure which minimizes thermal resistance without imposing either the laminar or turbulent flow condition. This
procedure is shown by use of an example in which 7VAp is fixed
at a realistic value of 1010. This value corresponds to the cooling
of a 5 cm by 5 cm heat source with room air at a pressure
drop of about 1 kPa (or 5 inches of water) through the heat
spreader. Equations (9) and (18) for laminar and turbulent
cases are normalized against the thermal resistance occurring
for the optimal laminar case (Eq. (9) with the solution to Eq.
(11) inserted) and plotted as a function of discrete number of
channels in Fig. 4. Several observations are made:
• turbulent flow occurs at 63 or fewer channel (n found
from Eq. (21) for Re = 4000)
• laminar flow occurs at 90 or greater channels (« found
from Eq. (13) for Re = 2300)
• 6oPt_turbuient °r optimal turbulent thermal resistance occurs
at n = 70 channels (solution of equation (20))
• ©opt_iaminar or optimal laminar thermal resistance occurs
at n = 92 channels (solution of equation (11))
Designing a device for operation in the transition zone of
Reynolds numbers between 2300 and 4000 (63 < n < 90) should
be avoided since the flow is not characterized as being either
laminar or turbulent. For the case at hand, a heat sink with
70 channels should not be used. The best design incorporates
63 channels where the flow is certainly turbulent. Lifting the
constraint of laminar flow from the problem gives a turbulent
thermal resistance which is 12% lower than that for the best
laminar case (« = 92).
Two other examples are presented for the same values of
L/W and Prandtl number but differing NAp. For the case of
a low NAp (108), Fig. 5 shows that the best laminar case (n = 29)
produces a thermal resistance which is 30 percent better than
the best available turbulent case (« = 13). At7V Ap =10 12 , Fig.
6 indicates the best laminar solution is not available. Equation
(12) yields n = 293 for the best laminar case, but Eq. (13) reveals
that this occurs at Re = 6626, well into the turbulent region.
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1000
:
:
1E-03
. . - • " • ' "
" " ^
^_^^J 1E-04
^ \
opt lam
100
'opt
/I
c
500
200
Ap-
insulation
^^^S^^^^^^^m^^^^^
I
1E-05
20
optjurb
10
y.
^ ^ ^ \ •'
Laminar Turbulent
5
1E+08
i
1E+09
1E+10
1E+11
1E-06
1E+12
W
flow
Fig. 8 Schematic of heat spreader with finite fins
Fig. 7 Optimal number of channels and dimensionless thermal resistances as a function of W4n
The trend here is that for a given L/WanA fluid, there exists
a value of NAp above which turbulent flow thermal resistance
is lower than that for laminar flow and below which the opposite is true. If the design is restrained to laminar flow for
some reason such as noise or erosion, there is an optimal
number of channels which minimizes thermal resistance. If
turbulence is allowed, the optimal design may be either in the
laminar or turbulent regime depending on NAp. This point is
expressed by use of Fig. 7 where the optimal number of fins
is shown as a function of NAp over the realizable range of
values for fixed L/W=\,
D/W=\ and Pr = 0.71. The discontinuity in the nopt line which occurs at about 109 is due to
the avoidance of solution in the transition zone
(2300<Re<4000). Below NAp = 2x 1010, the optimal n found
from the solution of Eq. (20) results in Re< 4000. Hence, Eq.
(21) is solved for «opt_tUrb with Re = 4000. For NAp>2x 10fo
Eq. (20) yields the optimal number of channels for Reynolds
number greater than 4000. Here Eqs. (19) and (20) can be
solved to show that «0pt_turb is proportional to NApw,°, whereas
it is proportional to NApl when Eq. (21) is used. The best
dimensionless thermal resistance is also plotted on that figure.
It is noted that four orders of magnitude change in NAp results
and only two orders of magnitude change in 9 .
2.B
Finite Fins (Partitioning Walls)
Overview. A more realistic model includes fins of finite
thickness. Figure 8 defines the geometric parameters of this
model. For this configuration there are n channels and n— 1
fins. The two effects resulting from the use of the finitely thick
fins are a reduction of cross sectional flow area for fixed overall
geometry (D and W) and the introduction of an influential
fin efficiency. The same assumptions are made here regarding
properties, steady state, entrance effects, etc. as were made in
Section 2.
To the previous list of fixed quantities must now be added
the fin thermal conductivity, k(in. The variables to be determined by optimization are the number of channels and the
channel (/) and fin (IV) thicknesses. As in the previous problem, the problem is constrained by specifying the pressure drop
through the heat spreader. In addition, the fin length or fin
efficiency is limited due to space considerations. If the problem
is constrained solely by maximum pressure drop, resulting
optimal designs could require, due to very high volumetric
flow rates, pumping power comparable in magnitude to the
rate of thermal energy to be removed from the heat source.
Therefore, the problem could be further inhibited to a specified
maximum pumping power. As before the formulation is made
concise and general by non-dimensionalizing the thermal resistance equation and expressing it in terms of dimensionless
parameters.
Journal of Electronic Packaging
Geometrical Factors. Figure 8 shows the geometry of the
heat spreader being analyzed. V is the ratio of fin to gap
thickness. The following four Eqs. (22)-(25) are strictly geometric. They describe respectively the hydraulic diameter of
one channel, the cross-sectional area available for flow in the
system, the aspect ratio for one channel, and the surface area
available for heat transfer.
D„ =
2W
T(n-\)+W/D
n+
nWD
n + T(n-l)
W/D
l/D =
n + T(n-V)
Ae =
nWL
+
As = n + Y(n-\)
2r,DL(n-l)
(22)
(23)
(24)
(25)
The first term of (25) is the area available for heat transfer at
the base and between the channels; the second is the effective
fin area, with fin efficiency accounted for. The tip ends of the
fins are assumed to be insulated, as are the two outer sides of
the array.
Dimensionless Groups.
group introduced earlier,
In addition to the pressure drop
NAp =
(Ap/L)W3
(26)
pv1
a dimensionless form of pumping work is also required.
Nwork=
wW
pv
f
(27)
This problem can be driven by specifying a maximum pressure
loss value, NAp. With that limitation alone, it is possible that
the best solution will occur when the ratio of pumping work
to heat source power is larger than unity, a clearly undesirable
solution. The use of Nmrk will be demonstrated in Section 3.
The relationship between NAp and N work is
W work
knmiW
Ap
WPr
(28)
One Dimensional Fins. Invoking the usual notation associated with one dimensional fins with a constant heat transfer
coefficient, the following relation defining m is useful.
hPn
kfm ^4c,fin
2h
(29)
%rT7
For the approximate equality, it has been assumed that the
length of the fin is much more than its thickness, L> >Tl.
From the definition of N u ^
SEPTEMBER 1991, Vol. 113 / 317
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NUj,, kg
(30)
h=-
Dh
Use of Eqs. (22), (24), and (30) in Eq. (29) yields the following.
n + Tin
imDf^u^
ZT
W/D
kf\n
W/Dn+ {
l TW/D
:^ l)
(3D
An infinitely long fin of constant cross-sectional area, with
constant heat transfer coefficient and heat transfer in one
dimension will transport heat through it equal to q!\n,«<7fin,oo = [h Pfia ^fin^cfin]
[Tbase
—
Tfluid]
(32)
A fin of finite length D with an insulated tip, still assuming
one dimensional heat transfer and constant h, will transfer
heat through it equal to <jfi„.
?fin= [hPtinknnACi[in]W2tanh(mD)[Tbaxrnuid]
(33)
Under these constraints, an infinitely long fin will transfer the
greatest amount of heat possible, for a given h, PSin, Acjin, kSm
and T b a s e - 7>luid. Division of Eq. (33) by Eq. (32) reveals that
a fin of finite length with an insulated tip will transfer tanh
(m D) of the heat that an infinitely long fin will transfer.
The efficiency of a fin with one-dimensional heat transfer,
constant convective heat transfer coefficient and an insulated
tip is
tanh (m D)
V=
n
(34)
m D
In this analysis, the percent of infinite fin efficiency is chosen
as a design constraint. It should be recognized that when fins
are in the 90 percent efficiency range, an increase of several
percent efficiency could require a considerable lengthening of
the fin.
Dimensionless Thermal Resistance. For an array of channels and fins as seen in Fig. 8, where the heat source is one of
constant flux at the base of the fins, the dimensional thermal
resistances are the same as for the array with infinitesimally
thin fins, equation (6).
hA
mc„
(6)
mc„
(35)
Traditionally, the first term in 8 is known as the convective
resistance and the second term has been called the caloric
resistance. The latter will be referred to as the capacity term,
a phrase more appropriate to modern heat exchanger terminology. These two terms will now be represented by the above
named dimensionless parameters for laminar and turbulent
flow.
Flow Characterization
2.B.1
Laminar Flow
Capacity Term. For fully developed flow in a channel, the
friction factor is defined by
/=
2(Ap/L)D„
pU2m
(36)
For laminar f l o w , / i s given by
/=
7i
Re Dh
(37)
The value of 71 is determined by the aspect ratio of the rectangular channel. A parameter G is defined as suggested by
Bejan, 1984.
3 1 8 / V o l . 113, SEPTEMBER 1991
Equation (39) yields results which agree with exact values (Kays
and Crawford, 1980) to within 3 percent. For a fixed l/D, Eq.
(38) gives G, (39) gives 71, a n d / i s determined from (37).
Combination of Eqs. (36) and (37) with the definition of
Reynolds number and solution for mean velocity yields the
following relationship.
U„ =
2ApD,,
(40)
jiLvp
From the definition of Reynolds number, Eq. (40) and Eq
(22), the Reynolds number for laminar flow is
2 ApW3
7i Lpv2 n +
Re
Dh-
(41)
Y(n-l)+W/D
Use of the definition presented in Eq. (26) yields
Re'Dh
7i
n+
(42)
T(n-l)+W/D
for laminar flow.
Since the mass flow rate, m, is equal to p Ac Um, use of
Eqs. (22), (23) and (40) reveals the capacity thermal resistance
in laminar flow
yt(n+T{n-\)+W/D)2(n
NApVrD/Wn
^fiuid^
mc„
+
T(n-\))
(43)
Substitution of Eq. (43) into Eq. (28) yields the relationship
between geometry, pressure drop and pumping work for laminar flow.
=
With the definition of dimensionless thermal resistance as before, (6) becomes
hA,
(l/D)2+\
(38)
(i/D+iy
Note that G is invariant to an / to D transformation. This
means that an aspect ratio of l/D gives the same G as an aspect
ratio of D/l, as it should. Performance of least squares fit of
a straight line in G to available values for 71 yields the result
that
(39)
7 l = 18.80 + 78.57 G.
G=
i
work
7l
N\P(D/W)n
[n + Y(n-\)+W/D]2[n
+ T(n-\)]
L_
W
(44)
Convective Term. The Nusselt number for fully developed
laminar flow in a rectangular channel is also a function only
of aspect ratio of the channel. Use of the same parameter G
defined in Eq. (38), a least squares fit to the exact values
available gives
Nu Z 3 / „ / / =-1.047 +9.326 G
(45)
Nu Dh,T = -1.681 + 9.139G
(46)
NuDjiiH is the Nusselt number which results from a boundary
condition of constant heat flux around the channel and Nu BA>r
results from a constant temperature of the channel wall (Kays
and Crawford, 1980). These equations agree with analytical
results to within 3 percent. For this study, Nu flfr// is used to
be consistent with the thermal resistance model.
Equation (31) can be rewritten as follows.
(D/W)2+(D/W)
n+
1
T(n-l)
(mD)2T
2=0
NuflA knuili/kfm[n + T (n - 1)]
(47)
NuD/] in the above equation is found from Eq. (45), and it
should be noted that NuflA is a function of n, T, and D/W.
The convective component of thermal resistance is found
from the definition of Nu fl/| and Eq. (25) as
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AW
hAs
2.B.2
HuDh (n + T(n-l)+W/D)[n
n + T(n-\)
+ 2r1(D/W)(n-l)(n
Low Reynolds Number Turbulent Flow
Capacity Term.
factor is given by
2.B.3
For turbulent flow in channels, the friction
/=72Re5 A
1/
(49)
where y2 is 0.316. Equation (49) is valid, according to Incropera
and DeWitt (1990), for fully developed flows with turbulent
Reynolds numbers up to 20,000. This relation is valid, according to White (1974), for ducts of any cross section, as long
as the ducts are not too thin. The mean velocity in turbulent
flow in a channel is found by combination of Eqs. (36) and
(49).
2 Ap
72 L n +
u„,=
2W
T(n-l)+W/D
1
(50)
pv
From the definition of Reynolds number
Re
<
Dh-
1
T(n-l)+W/D
?
n+
(51)
Use of the relationship between mass flow rate and mean
velocity yields the capacity component of thermal resistance
in low Reynolds number turbulent flow.
l5/7
5
[n + T(n-l)][n + T(n-l)+
W/D]
9
U1
{2 /y$) Ntp Pr n (D/W)
knuidW
m cp
(52)
^ Mvork
(2V^)'
[n + T(n-l)][n
/7
~.
< n (D/W)
+ T(n \)+W/Df
W
/ ^ p „ l / 3
7=73Re^1/
1
Y(n-\)+W/D
n+
hA„ " "M/2
Pr'
(55)
(mDfT
m
?r
(k
nuii/klin)ln
NTP
n
+
T(n-\)\
(57)
Combination of the definition of Nu/)A, Eq. (55) and Eq. (25)
yields the convective component of thermal resistance for low
Reynolds number turbulent flow.
iW
1
T(n-\)+W/D
(61)
[n +
T(n-l)][n+T(n-l)+W/D)2n
(2 /yl) N5A/p9Prn(D/W)
,1
i/9
(62)
Convective Term. From the Chilton-Colburn analogy, Eq.
(54), an expression for the Nusselt number in high Reynolds
number flows is found.
1/9
A
1
n+
T(n-Y)+W/D
Substitution of Eq. (63) into Eq. (31) yields
NuDft =
<
2"
9
(D/W)1 - (D/W)[n + T(n-\)]C2-C2
Pr 1
(63)
=Q
(64)
where
(mD)2T
n U9
hl/2 ] K
9
P r " 3 ( W * n n ) [n + T (n - 1)]
(65)
The convective component of thermal resistance for high Reynolds number flows can also be found.
+
T(n-l)])]
(66)
Solution Procedure
Generally, the overall size and configuration of the heat
source to be cooled is known. The material from which the
fins are to be made is usually known, from weight, economic
and other considerations. As discussed above, the fraction of
infinite fin performance can be specified and space or weight
considerations can set a maximum allowable value for fin
length. Identification of a cooling fluid and a nominal oper-
[[n +
V(n~l)]+W/Df
' [y2/29}ulN\npWn(L/W)[7,(n-l)(D/W)+n/(2[n
Journal of Electronic Packaging
n+
3
(56)
where
hAs
N Ap
mc„
{[n +
T(n-l)]+W/D]1
(L/W)[v(n-l)(D/W)+n/(2[n
(D/W)s - (D/W) [n + T (n - l)]Q - Q = 0
(y\/2y
^ =
knMW
11,1/99 ».,4/9p r l i
' N:
2W
T(n-\)+W/D
The relation between mean velocity and m reveals that
C2 =
Substitution of Eq. (55) into Eq. (31) gives
C,=
2_Ap
73 L n +
1
(60)
1/5
pv
From the definition of Reynolds number it is found that ReDh
takes the following form for high Reynolds number flows.
Um =
Substitution of Eqs. (49) and (51) into (54) yields
N Ap
(59)
The value of 73 is 0.184. The use of Eq. (49) for Reynolds
numbers up to 20,000 and the use of Eq. (59) for Reynolds
numbers greater than 20,000 leads to a discontinuity in / at
20,000. This was considered to be unacceptable. Equations
(49) and (59) have identical / values at a Reynolds number of
49,820. In this study, Eq. (49) was used to determine / for
Reynolds numbers up to 49,820, and Eq. (59) was used for
values greater than that. The maximum difference in/values
predicted by the two equations for Reynolds numbers between
20,000 and 49,820 is less than 5 percent.
As in low Reynolds number flow, Eq. (59) and Eq. (37) are
combined to find the following expression for mean velocity.
(53)
(54)
Nu^-Re^Pr
(48)
Capacity Term. Equation (49) is traditionally cited as being
valid for turbulent flows whose Reynolds numbers are less
than 20,000 (Incropera and DeWitt, 1990). For high Reynolds
numbers, the following relationship is given.
/7
Convective Term. To find the heat transfer coefficient in
turbulent flow, the Chilton-Colburn analogy between friction
factor and heat transfer coefficient is used. The analogy is
given by Incropera and DeWitt (1990) in the following form.
Nu Dh~
T(n-l))](L/W)
High Reynolds Number Turbulent Flow
R e
Substitution of Eq. (52) into Eq. (28) gives the relationship
between geometry, pressure drop and pumping work for a
given cooling fluid in low Reynolds number turbulent flow.
+
+ T(n- 1)])]
(58)
SEPTEMBER 1991, Vol. 113 / 319
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ating temperature range gives fluid properties. A maximum
pressure drop available across the array is often known. Additionally, the amount of pumping work to be expended to
cool a device is prescribed. Therefore values of NAp and A^ork
can be computed from Eqs. (26) and (27), respectively. The
dimensionless quantities oiL/W, D/W (maximum), k!m/kf\u\i,
Pr and {m D) can also be computed. Once these parameters
are known, the governing equations above can be used to find
the optimal design in terms of the smallest thermal resistance.
The number of channels, n, and the ratio of fin thickness
to channel thickness, Y, are systematically varied through a
wide range of values. For a given n and Y:
3.A
3.A.I.
Solution in the Laminar Region
Solve Eq. (44) for NAp and Eq. (47) for D/W simultaneously. If either NAp or D/W is greater than the
maximum allowable value, use trje appropriate maximum.
3.A.2. Calculate the Reynolds number using Eq. (42). If ReD/i
is greater than 2300, the geometry under examination
will not yield laminar flow, so skip to 3.B.
3.A.3. Calculate the thermal resistance using Eq. (43) and
(48) in Eq. (35). If this 9 is lower that any previously
calculated G, save n, Y, 0 , D/W, NAp and other appropriate values.
3.B Solution in the Low Reynolds Number Turbulent
Region.
3.B. 1. Solve Eq. (53) for NAp and Eq. (56) for D/W simultaneously. If either NAp or D/W is greater than the
maximum allowable value, use the appropriate maximum.
3.B.2. Calculate the Reynolds number using Eq. (51). If ReD^
is less than 4000, the geometry under examination will
not yield completely turbulent flow, so go to 3.B.4.
If ReC/i is greater than 49,820, the geometry under
examination will not yield low Reynolds number turbulent flow, so skip to 3.C. If R e ^ is between 4000
and 49,820, proceed to 3.B.3.
3.B.3. Calculate the thermal resistance using Eqs. (52) and
(58) in Eq. (35). If this 0 is lower than any previously
calculated 0 , save n, Y, 0 , and other appropriate
values.
3.B.4. Change n or Y and return to step 3.A.
3.C Solution in the High Reynolds Number Turbulent
Region:
3.C1. Solve Eq. (63) for NAp and Eq. (65) for D/W simultaneously. If either NAp or D/W is greater than the
maximum allowable value, use the appropriate maximum.
3.C.2. Calculate the Reynolds number using Eq. (61). If ReflA
is less than 49,820, the geometry under examination
will not yield high Reynolds number flow, so skip to
3.C.4. If ReD/i is greater than 49,820, proceed to 3.C.3.
3.C.3. Calculate the thermal resistance using Eqs. (62) and
n
Case I
Goldberg (/ = 0.127 mm)
Present Study
Same pressure drop, Ap
Same pumping power, w
Case II
Goldberg (/ = 0.254 mm)
Present Study
Same pressure drop, Ap
Same pumping power, w
Case III
Goldberg (/ = 0.635 mm)
Present Study
Same pressure drop, Ap
Same pumping power, w
320 / Vol. 113, SEPTEMBER 1991
r
(66) in Eq. (35). If this 0 is lower that any previously
calculated 0 , save n, Y, 0 , and other appropriate
values.
3.C.4. Change n or Y and return to step 3.A.
In this study, n values ranging from 2 to 500 channels and
F values ranging from 0.01 to 2.0 are considered. In this manner, the optimal design for laminar and turbulent flow are
found.
4
Results
The results obtained here are applied to two previous studies:
one by Goldberg (1984) with a copper thermal spreader and
air as the coolant; and another work by Tuckerman and Pease
(1981) with water-cooled silicon fins. In both studies, the fin
to channel thickness ratio was set to unity and only laminar
flow was considered. Use of the present optimization scheme
shows that upon reexamination of the cases studied by Goldberg, significant reduction of thermal resistance can be obtained by using fin/channel dimensions other than unity. A
similar reduction is found to be true for the case investigated
by Tuckerman and Pease with the relaxation of the laminar
limitation.
4.A Goldberg (1984)
Goldberg designed, built and tested systems fashioned from
the design procedure of Tuckerman and Pease (1981). The size
of the square heat source was 0.635 x 0.635 cm (1/4 x 1/4 in.)
with fins fixed at 1.27 cm (1/2 in.) length and a fin to channel
thickness ratio equal to unity. The Nusselt number was chosen
to be constant at 8 and the flow constricted to laminar. Air
was the coolant and the material for the heat spreader was
copper. Properties were evaluated at room temperature. Futher,
the design by Goldberg hinged on a constant volumetric flow
rate of 30 liters/min., thus fixing the capacitance component
of thermal resistance. An "average" value of 0 c a p(=l/
2mcp) was used, whereas the total capacitance value (0CaP= 1/
mcp) is used here. For a fixed flow rate of 30 liters/min,
Goldberg's value of 0cap (0.9 C/W) corresponds to 1.8 C/W
for comparison purposes in this paper. Goldberg did not optimize the design, but rather chose three values of fin and
channel thickness of 0.127, 0.254 and 0.635 mm (5, 10 and 25
mils) for his experiments. In the present study, all conditions
and properties were maintained the same as those of Goldberg;
however, the fin to channel thickness ratio and the nature of
the flow were allowed to vary. Comparisons are shown below
for all the three cases investigated by Goldberg. In each instance, the optimization scheme described above was used by
fixing either the pressure drop through the device or the power
consumed by the fan at the same value set by Goldberg. In
every case for these low pressures, the minimum thermal resistance was found in the laminar regime. NAp varied from
5 . 5 x l 0 6 to 1.34 x10 s for Cases I and III. As identified in
Section 4. A.3, laminar solutions are better than turbulent ones
in this low NAp range. The last column quantifies the improvement in thermal resistance, A6.
Ap
kPa(in H 2 0)
vv,
Watts
V
1/min
25
1
1.17(4.68)
0.583
30
24
20
0.435
0.39
1.17(4.68)
0.73(2.92)
1.747
0.583
89.9
48.2
12.5
1
0.29(1.17)
0.146
30
18
20
0.316
0.39
0.29(1.17)
0.26(1.06)
0.239
0.146
49.2
32.8
5
1
0.047(0.19)
0.024
30
12
15
0.25
0.32
0.047(0.19)
0.075(0.30)
0.018
0.024
22.4
19.1
Ad,
%
32.4
15.4
18.4
11.4
38.6
46.2
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In the three cases above, the optimal design occurs with T
considerably below a value of unity: the fins are thin compared
to the channel width. In case I and case II, there is some penalty
for optimization in the form of added flow rate, yet the total
pumping power still is relatively small in comparison with the
heater power. In case III, for the same pressure drop, a reduced
flow rate of about 50 percent leads to a reduced thermal resistance of nearly 40 percent.
ixm. wide, separated by fins of 191 /*m thick and 442 pm in
depth. These data are not shown in the above table.
It is recognized that entrance effects on both friction and
heat transfer could influence the results presented herein. Care
should be exercised to restrict the use of these results or procedures to cases where fully developed flow exists in the channels.
4.B Tuckerman and Pease (1981)
A comparison is made below between the results of Tuckerman and Pease (1981) and the present analysis. In the former,
very narrow channels were etched onto the backside of a silicon
wafer for use as coolant passageways. Designed for optimal
performance subject to laminar flow and for water as the
coolant, these authors made a test section with a designed
thermal resistance of 0.086 C/W.
*
It can be seen from the calculated values of the capacity and
convective thermal resistance terms, that relaxation of the Y
5
Constraints
Size, Length (L) by Width (W)
pressure drop, Ap
fin efficiency, 77
Coolant
Fin Material
fin to channel thickness ratio, V
Nusselt Number
type of flow
71, laminar friction factor
Dimensionless Groups
L/W
maximum N&p
maximum jVwork
Calculated Results
n, number of channels
Depth, D, (im
Fin thickness, fim
Channel thickness, y.m
Reynolds Number
Volumetric Flow Rate, cm3/sec
Aspect Ratio
Nusselt Number
Yi, laminar friction factor
NAp
^"work
Capacity Thermal Res, C/W
Convective Thermal Res, C/W
Total Thermal Resistance, C/W
Reduction in thermal resistance
Tuckerman
and Pease
1 cm x 1 cm
206.8 kPa (30 psi)
76%
water
Silicon
1
6
laminar
96
Present Study
same
same
same
same
same
unrestricted
unrestricted
laminai• or turbulent
a funct ion of aspect ratio
1
2.82 X 10'°
3.62 x 10'3
same
same
unrestricted
88
365
57
57
730
11
6.4
6
96
2.82 x 10'°
3.62 x 10'3
0.022
0.064
0.086
Laminar
83
357
60
61
834
12.4
5.8
5.9
77.8
2.82 X 10'"
4.08 X 10'3
0.019
0.058
0.077
10.5%
constraint results in a reduction of both terms for the laminar
case. When turbulent flow is allowed, these terms are reduced
by 40 and 75 percent from those for the laminar analysis. The
wide channels found for the best turbulent solution allow, for
fixed pressure drop, a greatly increased mass flow, thereby
reducing the capacity term. Comensurately, the heat transfer
coefficient increases due to the presence of turbulence.
The overall thermal resistance for the turbulent solution is
reduced by 34 percent from that of Tuckerman and Pease
whose design was confined to laminar flow. A maximum pressure drop of 206.8 kPa (30 psi) is common to all three cases,
but there was no need to place an upper limit on pumping
power in the present case. By doing so, the power consumed
is raised by a factor of 2.5 over that of Tuckerman and Pease,
from 2.27 Watts (corresponding to 11 cmVs flow rate at 206.8
kPa, or 30 psi) to 5.77 Watts. When compared to the maximum
power of the electronic device (-790 Watts), the pumping to
circuit power ratio is still less than 1 percent.
If instead of the pressure drop through the device, the pumping power is limited to a maximum of 2.27 Watts, a reduction
in thermal resistance of 13 percent is realized by lifting the
laminar restriction. However, the pressure drop through the
channels for the optimal turbulent solution is found to be 39
percent of the Tuckerman and Pease laminar case, or 79.2 kPa
(11.7 psi). For this solution, there are 24 channels, each 234
Journal of Electronic Packaging
Future Work
The authors are currently testing two microchannel heat
spreaders designed along the guidelines of Tuckerman and
Pease; one optimized for performance in the laminar zone and
one for peak performance with turbulent flow. In addition,
the same effort is being made for macrochannel heat spreaders
of dimensions 5 cm by 5 cm using aluminum and water and
large heat sinks designed for optimal use with air. The modeling
equations are also being modified to include entrance effects.
Results will be reported in later publications.
Turbulent
33
319
134
173
4006
27.9
1.85
35.8
not applicable
2.82 x 10'°
9.19 x 1013
0.009
0.048
0.057
33.7%
References
Bar-Cohen, A., Kraus, A. D., 1990, Advances in Thermal Modeling of Electronic Components and Systems, Volume 2, ASME Press, New York, N.Y.
Bejan, Adrian, 1984, Convection Heat Transfer, Wiley, New York, N.Y.,
pp. 75-82.
Goldberg, N., 1984, "Narrow Channel Forced Air Heat Sink," IEEE Transactions, Components, Hybrids and Manufacturing Technology, Vol. CHMT,
No. 1, pp. 154-159.
Hwang, L. T., Turlik, I., and Reisman, A., 1987, " A Thermal Module Design
for Advanced Packaging," Journal of Electronic Materials, Vol. 16, No. 5, pp.
347-355.
Incropera, F. P., and DeWitt, D. P., 1990, Fundamentals of Heat and Mass
Transfer, Wiley, New York.
Kays, W. M., and Crawford, M. E., 1980, Convective Heat and Mass Transfer,
McGraw-Hill, New York.
Nayak, D., Hwang, L. T., Turlik, I., and Reisman, A., 1987, "A HighPerformance Thermal Module for Computer Packaging," Journal of Electronic
Materials, Vol. 16, No. 5, pp. 357-364.
Phillips, R. J., 1990, "MicroChannel Heat Sinks," Chapter 3 of Advances in
thermal Modeling of Electronic Components and Systems, Volume 2, Ed. by
A. Bar-Cohen and A. D. Kraus, ASME Press, New York, N.Y.
Sasaki, S., and Kishimoto, T., 1986, "Optimal Structure for Microgrooved
Cooling Fin for High-Power LSI Devices," Electronics Letters, Vol. 22, No.
25, pp. 1332-1334.
Tuckerman, D. B., 1984, "Heat Transfer Microstructures for Integrated Circuits," S.RC Technical Report No. 032, SRC Cooperative Research, Box 12053,
Research Triangle Park, NC 27709.
Tuckerman, D. B., and Pease, R. F. W., 1981, "High-Performance Heat
Sinking for VLSI," IEEE Electron Device letters, Vol. EDL-2, No. 5, pp. 126129.
White, F. M., 1974, Viscous Fluid Flow, McGraw-Hill, New York.
SEPTEMBER 1991, Vol. 113 / 321
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