NUMERICAL OPTIMIZATION FOR DESIGN OF MICRO-CHANNEL
1
COMBINED NUMERICAL OPTIMIZATION AND CONSTRUCTAL
THEORY FOR THE DESIGN OF MICRO-CHANNEL HEAT SINKS
by
Bello-Ochende T*., Meyer J.P., Ighalo F.U
*Author for correspondence
Department of Mechanical and Aeronautical Engineering,
University of Pretoria,
Pretoria, 0002,
South Africa,
E-mail: tbochende@up.ac.za
Abstract
This study deals with the geometric optimization of a silicon based micro-channel heat sink using a
combined numerical optimization and constructal theory. The objective is to minimize the wall peak
temperature subject to various constraints. The numerical simulations are carried out with fixed
volumes ranging from 0.7 mm
3
to 0.9 mm
3
and pressure drop between 10 kPa to 60 kPa. The effect of
pressure drop on the optimized aspect ratio, solid volume fraction, hydraulic diameter and the
minimized peak temperature are reported. Results also show that as the dimensionless pressure drop
increases the maximized global thermal conductance also increases.
Keywords: Micro-channel, geometric optimization, thermal conductance, gradient based algorithm
NUMERICAL OPTIMIZATION FOR DESIGN OF MICRO-CHANNEL
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Nomenclature
A [m
2
] Channel cross-sectional area
B [m] Channel width
Be [-] Bejan number
C [-] Global thermal conductance
C
p
[J/Kg] Specific heat capacity
D [m] Diameter
f(x) [-] Objective function
g
i
(x) [-] i-th equality constraint function
G [m] Computational domain width
h
j
(x) [m] j-th equality constraint function
H [m] Computational domain height
k [W/m.K] Thermal conductivity
L [m] Channel axial length
Nu [-] Nusselt number
P [Pa] Pressure
P[k] [-] Successive sub-problem
q” [W/m
2
] Heat flux
n
[-] n-dimensional real space
T [K] Temperature
t
1
[m] Half thickness of vertical solid
t
2
[m] Channel base thickness
t
3
[m] Channel base to height distance
U [m/s] Velocity
V [m
3
] Volume
W [m] Heat sink width
x,y,z [m] Cartesian coordinates
Greek symbols
α
[m
2
/s] Thermal diffusivity
D
[-] Difference
m
[kg/m.s] Dynamic viscosity
r
[kg/m
3
] Density
f
[-] Volume fraction
¶
[-] Step limit
NUMERICAL OPTIMIZATION FOR DESIGN OF MICRO-CHANNEL
3
Subscripts
c Channel
f Fluid
h Hydraulic
max Maximum
min Minimum
opt Optimum
s Solid
Introduction
The impact of the new generation drive for high processing speed, sophisticated and compact
electronic devices is the rising transistor density and switching speed of microprocessors. This
challenge results in an increase in the amount of heat flux dissipation which is predicted to be in the
excess of 100 W/cm
2
in the near future [1,2]. With this comes the need for advanced cooling
techniques and micro-channels have recently generated great interest by researchers as it proves to
yield high heat transfer rates.
As far back as 1981, Tuckerman and Pease [3] proposed that single phased microscopic heat
exchangers using water as the coolant could achieve power density cooling of up to 1000 W/cm
2
and
with experimentation; the cooling water could dissipate a heat flux of about 790 W/cm
2
. Dirker and
Meyer [4] developed correlations that predict the cooling performance of heat spreading layers in
rectangular heat generating electronic modules. They discovered that the thermal performance was
dependent on the geometric size of the volume posed by the presence of thermal resistance.
Investigation into the heat transfer characteristics and fluid flow behaviour of micro-channels show
that shape and geometric parameters such as the aspect ratio and hydraulic diameter of a micro-
channel greatly influences the cooling capabilities of these heat sinks [5]. Wu and Cheng [6] showed
experimentally that the friction factor of micro-channels may differ if their geometric configurations
are different though their hydraulic diameters are the same. The trend of their experimental results
showed an increase in the friction factor as the aspect ratio of the heat sink increased. Koo and
Kleinstreuer [7] found out that the aspect ratio influences the viscous dissipation in micro-channels
which in turn affects the heat transfer rate. Their work showed that as the aspect ratio deviated from
unity, the dissipation effect increases. Chen [8] conducted an investigation into forced convection heat
transfer within a micro-channel, it was reported that the heat transfer was influenced mainly by the
aspect ratio and effective thermal conductivity of the heat sink. Muzychka [9] developed approximate
expressions for the optimal geometry for various fundamental duct shapes. In his work, he showed
NUMERICAL OPTIMIZATION FOR DESIGN OF MICRO-CHANNEL
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that the dimension of an optimal duct is independent of its array structure. Bello-Ochende et al. [10]
presented a three dimensional geometric optimization of a micro-channel heat sink using scale
analysis and intersection of asymptotes method. They used the constructal design theory to determine
optimal geometric configurations that maximize the global thermal conductance in a dimensionless
form. Ambatipudi and Rahman [11] numerically investigated the heat transfer within a micro-channel
heat sink using silicon substrate. In their study, they explored the effects of channel depth and width
across a range of Reynolds number. It was reported that at higher Reynolds number, higher heat
transfer can be achieved. Their work also documented that optimum channel depth exists for various
Reynolds number. Experimental and numerical techniques has been utilised to further investigate and
maximize the cooling abilities of micro-channel heat sinks in recent researches [12-18]. In this work,
an optimal geometry for a micro-channel heat sink is numerically determined which minimizes the
peak wall temperature using mathematical optimization and constructal design theory.
Computational Model
Figure 1 shows the physical model and fig. 2 the computational domain for a micro-channel
heat sink. The computational domain is an elemental volume selected from a complete micro-channel
heat sink by the use of the symmetrical property of the heat sink. Heat is supplied to a highly
conductive silicon substrate with known thermal conductivity from a heating area located at the
bottom of the heat sink. The heat is then removed by a fluid flowing through a number of micro-
channels. The heat transfer in the elemental volume is a conjugate problem that combines heat
conduction in the solid and convective heat transfer in the liquid.
Figure 3 shows the computational domain and its grid sizes. The following assumptions were
made to model the heat transfer and fluid flow in the elemental volume:
· The hydraulic diameter of the micro-channel under analysis is greater than 10 μm
· For water, the continuum regime applies hence the Navier-Stokes and Fourier equations can still
be used to describe the transport processes
· Steady-state conditions for flow and heat transfer
· Incompressible flow
· Constant solid and fluid properties
· Negligible heat transfer due to radiation and natural convection
· Negligible buoyancy and viscous heating
· Large number of micro-channels
Based on the assumptions listed above, the continuity, momentum and energy equations
governing the fluid flow and heat transfer for the cooling fluid within the heat sink are given in eqs.
(1), (2) and (3) respectively.
NUMERICAL OPTIMIZATION FOR DESIGN OF MICRO-CHANNEL
5
(
)
0
r
Ñ=
U (1)
(
)
2
0
P
rm
Ñ+D-Ñ=
.UUU (2)
(
)
2
pf
0
C TkT
r
Ñ-Ñ=
.U
(3)
For the solid material, the momentum and energy governing equations are given by eq. (4)
and eq. (5) below.
0
=
U (4)
2
s
0
kT
Ñ=
(5)
The conjugate heat transfer problem is modelled with heat being supplied to the bottom wall
of the heat sink at 1 MW/m
2
. Water at 20°C is pumped through the channel across the axial length
with a pressure drop ranging between 10 kPa and 60 kPa. A vertex-centred finite volume code was
used to solve the continuity, momentum and energy equations using the appropriate boundary
conditions. A second order upwind scheme was used in discretizing the momentum equation while a
SIMPLE algorithm was used for the pressure-velocity coupling. Convergence criteria was set to less
than 1x10
-4
for continuity and momentum residuals while the residual of energy was set to less than
1x10
-7
.
Code Validation
To ensure accuracy of the results, mesh refinement was performed until a mesh size with
negligible changes in thermal resistance was obtained. The grid dependence test was conducted using
five different mesh sizes having 19 200, 25 920, 57 600, 88 000 and 110 880 grid cells. The
computational volume whose dimensions are given in table 1 was used for the analysis. From the
results given in table 2a and table 2b (for inlet pressures of 30 kPa and 60 kPa respectively), it follows
that a mesh of 57 600 cells assures a less than 1% change in the thermal resistance with increasing
mesh size. Thus a mesh with 57 600 cells was chosen for the numerical simulation as it will guarantee
results which are independent of mesh size.
The numerical code was then evaluated by comparing the results generated with available
widely accepted analytical results. Figures 4 and 5 show the numerical and analytical dimensionless
velocity profile for fully developed flow within the micro-channel along the x and y axis respectively.
The velocity profile for the numerical solution was generated at the centre of the channel. Shah and
London [19] analytical solution was used to compare with the numerical prediction obtained and an
excellent agreement was found.
