IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009 243
Optimization of Microchannel Heat Sinks Using
Entropy Generation Minimization Method
Waqar Ahmed Khan, J. Richard Culham, Member, IEEE, and M. Michael Yovanovich
Abstract—In this paper, an entropy generation minimization
(EGM) procedure is employed to optimize the overall perfor-
mance of microchannel heat sinks. This allows the combined
effects of thermal resistance and pressure drop to be assessed
simultaneously as the heat sink interacts with the surrounding
flow field. New general expressions for the entropy generation rate
are developed by considering an appropriate control volume and
applying mass, energy, and entropy balances. The effectof channel
aspect ratio, fin spacing ratio, heat sink material, Knudsen num-
bers, and accommodation coefficients on the entropy generation
rate is investigated in the slip flow region. Analytical/empirical
correlations are used for heat transfer and friction coefficients,
where the characteristic length is used as the hydraulic diameter
of the channel. A parametric study is also performed to show the
effects of different design variables on the overall performance of
microchannel heat sinks.
Index Terms—Analytical model, entropy generation minimiza-
tion, microchannel heat sinks, optimization.
Total heating surface area [mm ].
Cross-section area of a single fin [mm ].
Hydraulic diameter [mm].
Volume flow rate [m /s].
Equality and inequality constraints.
Channel height [m].
Specific enthalpy of the fluid [J/kg].
Average heat transfer coefficient [W/m K].
Number of imposed constraints.
Thermal conductivity of solid [W/m K].
Sum of entrance and exit losses.
Manuscript received March 20, 2006; revised July 26, 2006. Current ver-
sion published July 22, 2009. This work was supported by the Centre for Mi-
croelectronics Assembly and Packaging and Natural Sciences and Engineering
Research Council of Canada. This work was recommended for publication by
Associate Editor B. Sammakia upon evaluation of the reviewers comments.
W. A. Khan is with the Department of Engineering Sciences, National
University of Sciences and Technology, PN Engineering College, PNS Jauhar,
Karachi 75350, Pakistan.
J. R. Culham and M. M. Yovanovich are with the Microelectronics Heat
terloo, Waterloo, ON N2L 3G1 Canada (e-mail: email@example.com).
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TCAPT.2009.2022586
Ratio of thermal conductivity of fluid to solid
Thermal conductivity of fluid [W/m K].
Length of channel in flow direction [mm].
Fin parameter [m].
Total mass flow rate [kg/s].
Total number of microchannels.
Number of design variables.
Nusselt number based on hydraulic diameter
Peclet number based on hydraulic diameter
Heat transfer rate from the base [W].
Heat transfer rate from the fin [W].
Heat flux [W/m ].
Reynolds number based on hydraulic diameter
Total entropy generation rate [W/K].
Specific entropy of fluid [J/kg K].
Absolute temperature [K].
Average velocity in channels [m/s].
Specific internal energy [J/kg].
Specific volume of fluid [m /kg].
Width of heat sink [mm].
Half of the channel width [mm].
Half of the fin thickness [mm].
Pressure drop across microchannel [Pa].
Thermal diffusivity [m /s] or constant defined by
Channel aspect ratio
Heat sink aspect ratio
Fin spacing ratio
Ratio of specific heats
1521-3331/$25.00 © 2009 IEEE
244IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009
,Mean free path [m] or Lagrangian multipliers.
Absolute viscosity of fluid [kg/m s].
Kinematic viscosity of fluid [m /s].
Fluid density [kg/m ].
Tangential momentum accommodation coefficient
Energy accommodation coefficient.
Slip velocity coefficient.
Temperature jump coefficient.
Contraction and expansion.
trend in the electronic industry toward denser and more pow-
erful products requires a higher level of performance from
cooling technology. After the pioneering work of Tuckerman
and Pease , microchannels have received considerable
attention especially in microelectronics. Microchannel heat
sinks provide a powerful means for dissipating high heat flux
with small allowable temperature rise. Due to an increase
in surface area and a decrease in the convective resistance
at the solid/fluid interface, heat transfer is enhanced in mi-
crochannels. These heat sinks can be applied in many important
fields like microelectronics, aviation and aerospace, medical
treatment, biological engineering, materials sciences, cooling
of high-temperature superconductors, thermal control of film
deposition, and cooling of powerful laser mirrors. The two
important characteristics of microchannel heat sinks are the
high heat transfer coefficients and lower friction factors.
In this paper, an entropy generation minimization (EGM)
criterion is used to determine the overall performance of a
microchannel heat sink, which allows the combined effect of
thermal resistance and pressure drop to be assessed through the
NE of the important aspects of electronics packaging
is the thermal management of electronic devices. The
simultaneous interaction with the heat sink. A general expres-
sion for the entropy generation rate is obtained by using the
conservation equations for mass and energy with the entropy
II. LITERATURE REVIEW
The concept of “microchannel heat sinks” was first intro-
duced by Tuckerman and Pease . Following this work, sev-
eral experimental, numerical, and theoretical studies on the op-
timization of microchannel heat sinks have been carried out.
These studies are reviewed in this section.
Steinke and Kandlikar  presented a comprehensive review
of conventional single-phase heat transfer enhancement tech-
niques. They discussed several passive and active enhancement
techniques for minichannels and microchannels. Some of their
proposed enhancement techniques include fluid additives, sec-
ondary flows, vibrations, and flow pulsations.
Kandlikar and Grande  explored the cooling limits of the
plain rectangular microchannels with water cooling for high
heat flux dissipation and illustrated the need for enhanced mi-
crochannels. They described a simplified and well-established
fabrication process to fabricate 3-D microchannels. They also
demonstrated the efficacy of the fabrication process in fabri-
cating complex microstructures within a microchannel.
Knight et al. , , Perret et al. , , Kim , Upadhye
and Kandilkar , Kandlikar and Grande , and Liu and
Garimella  developed theoretical optimization procedures
to minimize the overall thermal resistance of microchannel heat
sinks for a given pressure drop, whereas Singhal et al. , and
Kandlikar and Upadhye  carried out analyses to optimize
the channel configuration that yields a minimum pressure drop
for a given heat load. They presented solution procedures for
laminar/turbulent flow and generalized their results in terms of
different heat sink parameters.
Kleiner et al. , Aranyosi et al. , and Harris et al. 
investigated theoretically and experimentally the performance
of microchannel heat sinks. They modeled the performance in
terms of thermal resistance, pressure drop, and pumping power
as a function of heat sink dimensions, tube dimensions, and air
flow rate. Their results show an enhancement in heat removal
capability compared to traditional forced air-cooling schemes.
Garimella and Singhal  and Jang and Kim  analyzed
experimentally the pumping requirements of microchannel heat
sinks and optimized the size of the microchannels for minimum
pumping requirements. Jang and Kim  showed that the
cooling performance of an optimized microchannel heat sink
subject to an impinging jet is enhanced by about 21% compared
to that of the optimized microchannel heat sink with parallel
flow under the fixed-pumping-power condition.
Choquette et al. , Zhimin and Fah , Meysenc et al.
, Chong et al. , Liao et al. , Ryu et al. , Wei and
timizations of thermal performance of microchannel heat sinks
for given pumping power/pressure drop. Zhimin and Fah 
considered laminar, turbulent, developed, and developing flow
and heat transfer in the analysis. Using self-developed software,
they investigated the effects of heat sink channel aspect ratio,
fin-width-to-channel-width ratio, and the channel width on the
KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 245
performance of heat sink. They found that the channel width is
the most important parameter and governs the performance of
a microchannel heat sink. Min et al.  showed numerically
that the tip clearance can improve the cooling performance of
a microchannel heat sink when tip clearance is smaller than a
channel width. Delsman et al.  performed a CFD study for
the optimization of the plate geometry to reach the design target
regarding the quality of the flow distribution.
Haddad et al.  investigated numerically the entropy
generation due to steady laminar forced convection fluid flow
through parallel plate microchannels. They discussed the effect
of Knudsen, Reynolds, Prandtl, and Eckert numbers and the
nondimensional temperature difference on entropy generation
within the microchannel. The entropy generation within the
microchannel is found to decrease as Knudsen number in-
creases, and it is found to increase as Reynolds, Prandtl, and
Eckert numbers and the nondimensional temperature difference
increase. The contribution of the viscous dissipation in the total
entropy generation increases as Knudsen number increases
over wide ranges of the flow controlling parameters.
It is obvious from the literature survey that all the studies re-
imize thermal resistance for a given pressure drop or to min-
imize pumping power for a specified thermal resistance. No
study exists to optimize both thermal and hydraulic resistances
simultaneously. In this study, an EGM method is applied as a
unique measure to study the optimization of thermal and hy-
draulic resistances simultaneously. All relevant design parame-
ters for microchannel heat sinks, including geometric parame-
ters, material properties, and flow conditions are optimized si-
multaneously by minimizing the entropy generation rate
subject to manufacturing and design constraints.
III. MODEL DEVELOPMENT
The geometry of a microchannel heat sink is shown in
Fig. 1(a). The length of the heat sink is
The top surface is insulated and the bottom surface is uniformly
heated. A coolant passes through a number of microchannels
along the -axis and takes heat away from the heat dissipating
electronic component attached below. There are
and each channel has a height
of each fin is
whereas the thickness of the base is
temperature of the channel walls is assumed to be
inlet water temperature of
. At the channel wall, the slip
flow velocity and temperature jump boundary conditions were
applied to calculate friction and heat transfer coefficients.
Taking advantage of symmetry, a control volume is selected
for developing the entropy generation model, as shown in
Fig. 1(b). The length of the control volume is taken as unity for
convenience and the width and height are taken as
and, respectively. This control volume includes half
of the fin and half of the channel along with the base. The side
surfaces AB and CD and the top surface AC of this CV can be
regarded as impermeable, adiabatic, and shear free (no mass
transfer and shear work transfer across these surfaces). The heat
flux over the bottom surface BD of the CV is . The ambient
and the surface temperature of the channel
. The bulk properties of the coolant are represented
and the width is.
and width . The thickness
Fig. 1. Microchannel heat sink: geometry, control volume, and definitions.
at the outlet, respectively. The irreversibility of this system
is due to heat transfer across the finite temperature difference
and to friction. Heat transfer in the control volume
[Fig. 1(b)] is a conjugate, combining heat conduction in the fin
and convection to the cooling fluid. To simplify the analysis,
the following assumptions were employed.
1) Uniform heat flux on the bottom surface.
2) Smooth surface of the channel.
3) Adiabatic fin tips.
4) Isotropic material.
5) Fully developed heat and fluid flow.
6) Steady and laminar flow.
7) Slip flow (i.e.,
9) Negligible axial conduction in both the fin and fluid.
10) Changes in kinetic and potential energies are negligible.
The mass rate balance for the steady state reduces to
,, andat the inlet and by, , and
) with negligible creep
Assuming negligible changes in the kinetic and potential ener-
can be written as
as specific enthalpy; therefore, the energy rate balance reduces
From the second law of thermodynamics
246IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009
For steady state,
from the bottom of the heat sink
rate balance reduces to
, and the total heat transferred
, so the entropy
Integrating Gibb’s equation from inlet to the outlet gives
represents the absolute temperature of the heat sink
Combining (3), (5), and (6), the entropy generation rate can be
Rearranging the terms and writing
, we have
, the entropy generation rate can be written as
pressure drop across the channel. This expression describes the
entropy generation rate model completely and it shows that the
entropy generation rate depends on the total thermal resistance
and the pressure drop across the channel, provided that
the heat load and ambient conditions are specified. If
volume flow rate, then the total mass flow rate can be written as
is the total thermal resistance andis the total
The average velocity in the channel
is given by
is the number of channels given by
IV. THERMAL RESISTANCE
The total thermal resistance is defined as
fluid temperature. These temperatures are given by
is the heat sink base temperature andis the bulk
resistances can be written as
, the convective and fluid thermal
is the total heat surface area and is given by
Using slip flow velocity and temperature jump boundary condi-
tions, Khan  solved the energy equation and developed the
following theoretical model for the dimensionless heat transfer
coefficient for a parallel plate microchannel:
Using (14)–(21), (13) can be written as
KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 247
V. PRESSURE DROP
The pressure drop associated with flow across the channel is
where the friction factor
channel aspect ratio, and slip velocity coefficient , and can
be written as
depends on the Reynolds number,
Kleiner et al.  used experimental data from Kays and
London  and derived the following empirical correlation
for the entrance and exit losses
and fin thickness:
in terms of channel width
VI. ENTROPY GENERATION RATE
Substituting (28) and (37) into (9), we get
to heat transfer and fluid friction, respectively, and
andshow the entropy generation rates due
VII. OPTIMIZATION PROCEDURE
The problem considered in this study is to minimize the
entropy generation rate, given by (9) or (40), for the optimal
overall performance of microchannel heat sinks. If
resents the entropy generation rate that is to be minimized
subject to equality constraints
mathematical formulation of the optimization problem may be
written in the following form:
, then the complete
ASSUMED PARAMETER VALUES
subject to the equality constraints
and inequality constraints
andare the imposed equality and inequality
denotes the vector of the design variables
. The objective function can be redefined
by using the Lagrangian function as follows:
positive or negative but the
consideration, is that the Hessian matrix of
andare the Lagrange multipliers. The
to be a local minimum of the problem, under
. The necessary
should be positive
of the Hessian matrix should be
A system of nonlinear equations is obtained, which can
be solved using numerical methods such as a multivariable
248 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009
RESULTS OF OPTIMIZATION
Newton–Raphson method. This method has been described in
 and applied by Culham and Muzychka  and Culham
et al.  to study the optimization of plate fin heat sinks, and
by Khan et al. ,  for pin fin heat sinks and tube banks.
In this paper, the same approach is used to optimize the overall
performance of a microchannel heat sink in such a manner
that all relevant design conditions combine to produce the best
possible heat sink for the given constraints. The optimized
results are then presented in graphical form.
VIII. CASE STUDIES AND DISCUSSION
as the default case to determine the overall performance of mi-
crochannel heat sinks. The fluid properties are evaluated at the
Microchannel width, height, and fin thickness are optimized
in terms of channel aspect ratio and fin spacing ratio for three
different volume flow rates and Knudsen numbers in the slip
flowregion. Thecorrespondingvalues ofthetotalthermal resis-
tance, pressure drop, and entropy generation rate are tabulated.
It is observed that both channel aspect and fin spacing ratios in-
crease with the increase in volume flow rate and Knudsen num-
bers in the slip flow region. The total thermal resistance and the
pressure drop are found to decrease with the increase in volume
flow rates and increase with the decrease in Knudsen numbers.
A parametric study is carried out to investigate the depen-
dence of the optimal dimensions of the microchannel heat sink
onthethermal and hydraulicperformance. Thecurrentstudyat-
temptstofindanoptimal flowrate, channelwidth,heat sinkma-
terial, and the effect of tangential momentum and energy coeffi-
cients. Fig. 2 shows the variation of the total entropy generation
rate as a function of volume flow rate for three different values
of Knudsen numbers. This figure shows that as the Knudsen
number increases the optimal entropy generation rate decreases
Fig. 2. Effect of volume flow rate on ?
due to the increase in velocity slip and temperature jump at the
wall that lead to reduced heat transfer and momentum transfer
from the wall to the fluid. It also shows that the optimal flow
rate increases with the increase in Knudsen number.
The friction factor and the Nusselt number, in the laminar
this aspect ratio is shown in Fig. 3 for three different Knudsen
numbers in the slip flow region. The increase in aspect ratio in-
creases the cross-sectional area available for flow and the total
surface area available for convective heat transfer, which re-
duces thermal and hydraulic resistances. This figure also shows
the decrease in the optimum entropy generation rate with an in-
crease in Knudsen number.
KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 249
Fig. 3. Effect of channel aspect ratio on ?
Fig. 4. Effect of fin spacing ratio on ?
The fin spacing ratio
transfer analysis. It should be greater than 1 to ensure that there
is flow in the microchannel. The effect of this ratio is shown in
Fig. 4 for the same Knudsen numbers in the slip flow region.
For each Knudsen number, the fin spacing ratio is optimized
to get a minimum entropy generation rate. It can be observed
that the optimum fin spacing ratio is a very weak function of
the Knudsen number, however, optimal entropy generation rate
depends upon the Knudsen number and decreases with an in-
crease in the Knudsen number. It shows that lower fin spacing
ratios are appropriate in the case of microchannel heat sinks.
The total entropy generation rate
tions due to heat transfer and viscous dissipation. The thermal
conductivity of the heat sink material affects only the contribu-
tion due to heat transfer
, whereas the contribution due to
remains unchanged. Fig. 5 shows the
plays an important role in the heat
includes the contribu-
Fig. 5. Effect of heat sink material on ?
Fig. 6. Effect of accommodation coefficients on ?
effect of thermal conductivity on the
generation rate decreases with the increase in Knudsen num-
bers. This figure shows that the entropy generation rate due to
heat transfer decreases sharply from 25 to 180 W/m K and then
becomes almost constant.
The accommodation coefficients model the momentum and
energy exchange of the gas molecules impinging on the walls.
They are dependent on the specific gas and the surface quality
and are tabulated in . Very low values of
number flows, due to the 2-
figure shows that for high Knudsen numbers, in the slip flow re-
ratebutasthe Knudsennumberdecreases, thelosses due to heat
transfer and fluid friction become negligible.
. Again the entropy
factor in (26) and (27). The ef-
250 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009
Based on the results of case studies and parametric optimiza-
tion, we have the following conclusions.
1) Thermal resistance and pressure drop across the mi-
crochannel decrease with an increase in volume flow rate
and increase with a decrease in Knudsen numbers in the
slip flow region.
with the volume flow rate to allow the decrease in thermal
resistance and pressure drop.
3) The optimum entropy generation rate decreases with the
increase in Knudsen numbers in the slip flow region.
4) Due to slip flow and temperature jump boundary condi-
tions, fluid friction decreases and heat transfer increases in
the microchannels which decreases the total entropy gen-
5) A low thermal conductivity heat sink with a large number
entropy generation rate.
6) Lower tangential momentum and energy accommodation
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KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 251 Download full-text
Waqar Ahmed Khan received the Ph.D. degree
from the University of Waterloo, Waterloo, ON,
ical Engineering at the National University of Sci-
ences and Technology, Karachi, Pakistan. He has de-
flow and heat transfer across single cylinders (cir-
cular/elliptical), tube banks and pin-fin heat sinks to
Newtonian and non-Newtonian fluids. His research
interests include modeling of forced convection heat
transfer from complex geometries, microchannel heat sinks, thermal system op-
and conjugate heat transfer in air and liquid cooled applications. He has more
than 28 publications in refereed journals and international conferences.
He is a member of the American Society of Mechanical Engineers (ASME),
the American Institute of Aeronautics and Astronautics (AIAA), and the Pak-
istan engineering council.
J. Richard Culham (M’98) received the Ph.D. de-
gree from the University of Waterloo, Waterloo, ON,
Currently, he is a Professor of Mechanical and
Engineering for Research and External Partnerships
at the University of Waterloo. He is the Director
and a Founding Member of the Microelectronics
Heat Transfer Laboratory. Research interests include
modeling and characterization of contacting inter-
faces and thermal interface materials, development
of compact analytical and empirical models at micro- and nanoscales, natural
and forced convection cooling, optimization of electronics systems using
entropy generation minimization, and the characterization of thermophysical
properties in electronics and optoelectronics materials. He has more than 100
publications in refereed journals and conferences in addition to numerous
technical reports related to microelectronics cooling.
Prof. Culham is a member of the Professional Engineers Ontario.
M. Michael Yovanovich received the Sc.D. degree
from the Massachusetts Institute of Technology,
Currently, he is a Distinguished Professor Emer-
itus of Mechanical Engineering at the University
of Waterloo, Waterloo, ON, Canada, and is the
Principal Scientific Advisor to the Microelectronics
Heat Transfer Laboratory. His research in the field
of thermal modeling includes analysis of complex
heat conduction problems, external and internal
natural and forced convection heat transfer from and
in complex geometries, and contact resistance theory and applications. He
has published more than 350 journal and conference papers, and numerous
technical reports, as well as three chapters in handbooks on conduction and
thermal contact resistance. He has been a consultant to several North American
nuclear, aerospace and microelectronics industries and national laboratories.
Dr. Yovanovich has received numerous awards for his teaching and his
significant contributions to science and engineering. He is the recipient of the
American Institute of Aeronautics and Astronautics (AIAA) Thermophysics
Award and the American Society of Mechanical Engineers (ASME) Heat
Transfer Award. He is a Fellow of AAAS, AIAA, and ASME.