Proceedings of the 14th International Heat Transfer Conference
IHTC14
August 8-13, 2010, Washington, DC, USA
IHTC14-
NUMERICAL INVESTIGATION OF HEAT TRANSFER EFFECTS IN MICROCHANNELS UNDER
H2 BOUNDARY CONDITION
V. V. Dharaiya
Rochester Institute of Technology
vvd3159@rit.edu
R. R. Srivastava*
Rochester Institute of Technology
rrs2837@rit.edu
S. G. Kandlikar
Rochester Institute of Technology
sgkeme@rit.edu
ABSTRACT
Study of fluid flow characteristics at microscale level is
gaining importance with shrinking device sizes. Better
understanding of fluid flow and heat transfer in microchannels
will have important implications in biomedical industry,
MEMS, electronic chip cooling, heat exchangers, and
microfluidic devices. Also, due to short lengths employed in
microchannels, entrance header effects can be significant and
needs to be investigated. In this work, three dimensional model
of microchannels, with aspect ratios (α=a/b) ranging from 0.1
to 10, are numerically simulated using CFD software,
FLUENT. Heat transfer effects in the entrance region of
microchannel are presented by plotting average Nusselt number
as a function of non-dimensional thermal entrance length x*.
The numerical simulations with both circumferential and axial
uniform heat flux (H2) boundary conditions were performed for
four wall, three wall and two wall cases. Large numerical data
sets, generated in this work for rectangular cross sectional
microchannels for three walls and two walls H2 boundary
condition, can provide better understanding and insight into the
transport processes in the microchannel.
INTRODUCTION
With rapid advancements in fabrication of MEMS (microelectro mechanical systems), highly compact and effective
cooling technologies are required for the dissipation of heat
generated by microelectronic devices. Microchannels are the
most efficient way of high heat removal from small areas. High
surface area to volume ratio, smaller volume and mass, and
high convective heat transfer coefficient are the features of
microchannels due to which high removal of heat flux is made
*Present address: Department of Mechanical Engineering, University of
Minnesota, Minneapolis.
possible. Due to their inherent advantages, microchannel heat
sinks have found applications in automotive heat exchangers,
cooling of high power electronic devices, aerospace industry,
etc. Fundamental understanding of fluid flow and heat transfer
in microchannels is essential in design of microfluidic devices.
Use of microchannels as high performance heat sink for
cooling of electronic devices was first demonstrated by
Tuckerman and Pease [1] in 1981. Fabricating rectangular
microchannel heat sink in silicon wafer and using water as
coolant, they showed that microchannel heat sink was able to
dissipate 790 W/cm2 with substrate to coolant temperature
difference being kept as 71˚C. Deviations from classical theory
of conventional channels have been reported for heat transfer in
microchannels by various investigators. In their experiments
with rectangular microchannels, Wu and Little [2] found the
Nusselt number to be higher than conventional channels. In
their work, Choi et al. [3] showed that Nusselt number in
laminar flow depended upon Reynolds number unlike
conventional channels where Nusselt number was constant for
fully developed flow. Adams et al. [4] performed experiments
on single phase turbulent flow and found the Nusselt number to
be higher than the predicted values for large sized channels. In
turbulent regime, Yu et al. [5] found the Nusselt number to be
higher than those predicted by Dittus-Boelter equation. In the
recent years, Celata et al. [6] and Bucci et al. [7] have also
found the Nusselt number to exceed the theoretical predictions
for conventional channels.
Many researchers have also found the Nusselt number to
lie below the predicted theoretical values for conventional
channels. Peng et al. [8] and later Peng and Peterson [9]
performed experiments on rectangular microchannels of
different dimensions and found the Nusselt number to lie below
the predicted values of Dittus-Boelter equation. Based on their
1
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experiments, Qu et al. [10] showed the Nusselt number to be
lower than the predicted values in the laminar region. Rahman
and Gui [11, 12] found the Nusselt number to be higher than
theory for laminar flow whereas lower than that predicted by
theory in the case of turbulent flow.
There have also been various studies in which heat transfer
in microchannels was found to be accurately predicted by
macroscale equations. Lee et al. [13] performed numerical
analysis on copper microchannels over a range of Reynolds
numbers and found that heat transfer can be predicted using
classical theory for conventional sized channels. Performing
experiments on array of microchannels, Harms et al. [14] found
the Nusselt number to be similar to the macroscale predictions.
Qu and Mudawar [15] carried out numerical and experimental
work on microchannels in the laminar region and found the
heat transfer results to have good agreement with theory.
Owhaib and Palm [16] performed experiments on
microchannels in turbulent region and found the heat transfer
results to be accurately predicted by conventional correlations.
Deviations in heat transfer predictions by classical theory
have been mainly attributed to surface roughness, entrance and
exit effects, thermal and flow boundary conditions, etc. Lee et
al. [13] showed that flow in microchannels is generally
thermally developing with considerable entrance effects by
numerically simulating microchannels with inlet and exit
manifolds. Applying correct thermal boundary conditions is
also very important in determining accurate heat transfer
coefficients. Different thermal boundary conditions generally
applied in fluid domain are: H1 (axially constant wall heat flux
and circumferentially constant wall temperature), H2 (uniform
wall heat flux, axially and circumferentially) and T (uniform
wall temperature, axially and circumferentially) [17]. Using
generalized integral transform technique, Aparecido and Cotta
[18] analytically solved for bulk temperature and Nusselt
number in thermally developing laminar flow in rectangular
ducts with uniform wall temperature (T) as the boundary
condition. Montgomery and Wibulswas [19] performed
numerical analyses of thermally developing flow in rectangular
microchannels with constant wall temperature and constant
wall heat flux (H1) as the boundary condition. Lee and
Garimella [20] numerically investigated heat transfer in
thermally developing flow in microchannels having aspect
ratios ranging from 1 to 10 with the H1 thermal boundary
condition.
However, there is little information available for the
entrance region heat transfer under the H2 boundary condition.
Kandlikar [21] emphasizes the need to generate entrance region
effects for smooth microchannels under H2 boundary
condition.
different boundary conditions: (1) uniform heat flux on all four
walls of microchannels, (2) uniform heat flux on three walls of
microchannels, and (3) uniform heat flux on two walls of
microchannels. Fully developed Nusselt number (NuH2) found
from the numerical results of four wall heating conditions are
compared with the published data of Shah and London [17]. In
order to study the effect of entrance conditions on Nusselt
number, microchannels having inlet and outlet headers are
modeled and simulated. Numerical results of this geometry are
compared with the findings of microchannels of the previous
case having abrupt entrance (plain microchannels with no
headers). In the next step rectangular microchannels with aspect
ratio ranging from 0.1 to 10 are simulated to generate large data
sets for Nusselt number with uniform heat flux H2 boundary
condition, circumferentially and axially at three walls and two
walls. In addition, the heat transfer effects in entrance region of
microchannels are studied by plotting Nusselt number as a
function of non-dimensional length x*.
NOMENCLATURE
a
b
l
Dh
ṁ
𝐴𝑐
Ah
P
Ph
T
Q
q''
Cp
k
h
Re
Nu
xth
x*
Subscripts
w
f
H2
fd
avg
x
m
OBJECTIVE
In the present work, microchannels are numerically
simulated using commercial CFD software FLUENT for
constant wall heat flux H2 (constant axial and circumferential
wall heat flux) boundary condition and different heated wall
configurations. The microchannels with aspect ratio ranging
from 0.1 to 10 are simulated with constant wall heat flux
boundary conditions. The geometries are simulated for three
height of rectangular microchannel, (μm)
width of rectangular microchannel, (μm)
length of rectangular microchannel, (mm)
hydraulic diameter, (μm)
mass flow rate (kg/s)
cross-sectional area of microchannel, (m2)
area of heated walls (m2)
perimeter of microchannel (m)
perimeter of heated walls (m)
temperature (K)
heat transfer rate (W)
total heat flux (W/m2)
specific heat of fluid (J/kg-K)
thermal conductivity of fluid (W/m-K)
heat transfer coefficient (W/m2-K)
Reynolds number
Nusselt number
thermal entrance length (m)
Non-dimensional thermal entrance length
x*= x / (DhRePr)
wall
fluid
uniform heat flux boundary condition
fully developed flow
average
local
mean
Greek
α
ρ
µ
2
channel aspect ratio (refer to Fig. 1)
density of water (kg/m3)
dynamic viscosity (N-s/m2)
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MATHEMATICAL FORMULATION
The continuity and flow energy equations in their steady
state, with associated H2 boundary conditions for all the cases
were solved using finite-volume approach using following
assumptions:
Table 1: Channel dimensions used for numerical
simulations for each cases of rectangular microchannel with
varying aspect ratios of 0.1 to 10.
(1) Laminar Steady state flow behavior,
(2) Incompressible fluid flow conditions,
(3) Fluid properties remain constant along the length of
microchannel,
(4) Neglect viscous dissipation,
(5) Neglect radiative effects.
q''
wall1
a
wall 4
wall 3
q''
wall2
b
Figure 1: Cross-sectional area of rectangular microchannel
with uniform heat flux on all four walls.
Table 1 shows the dimensions of rectangular
microchannels used for numerical simulations with varying
aspect ratios of 0.1 to 10. Figure 1 represents the crosssectional area of rectangular microchannel with constant heat
flux boundary condition on all the four walls. The length (l) and
height (a) of rectangular microchannels were kept constant for
all the computations and width (b) was varied in accordance
with change in aspect ratios as shown in Table 1. The aspect
ratio and hydraulic diameter for all the three different boundary
conditions of rectangular microchannels were defined by
equation (1);
𝑎
𝛼=𝑏
and
4∙𝑎∙𝑏
𝐷𝑕 = 2∙(𝑎+𝑏)
(1)
For numerical simulations of uniform heat flux on three
walls of microchannels, wall 1 of Figure 1 was considered as
adiabatic wall. Similarly, for two wall heating boundary
conditions, wall 3 and 4 were assumed as adiabatic walls. A
finite volume approach is employed to investigate the thermally
developing flow regime in microchannels. The local and
average Nusselt numbers are calculated numerically as a
function of non-dimensional axial distance and channel aspect
ratio. The heat transfer coefficient and Nusselt number for
rectangular microchannels can be calculated using
𝑕=𝑇
𝑞′′
𝑤 ,𝑎𝑣𝑔 −𝑇 𝑓,𝑎𝑣𝑔
and
𝑁𝑢𝐻2 =
𝐷𝑕
𝑘𝑓
𝑕(𝑥)
(2)
In equation (2), the fluid average temperature along the length
of microchannel is calculated using the energy balance equation
as follows
𝑇𝑓,𝑥 =
𝑞 ′′ ∙𝑃𝑕 ∙𝑥
𝑚 ∙𝐶𝑝
+ 𝑇𝑓,𝑖𝑛
(3)
α
a
b
l
Dh
(a/b)
(µm )
( µm )
( mm )
( µm )
0.10
150
1500
100
272.7
0.25
150
600
100
240.0
0.33
150
450
100
225.1
0.50
150
300
100
200.0
0.60
150
250
100
187.5
0.75
150
200
100
171.4
1
150
150
100
150.0
2
150
75
100
100.0
5
150
30
100
50.0
10
150
15
100
27.3
The averaged wall temperature, obtained from the
computational results, was calculated using temperature at
different nodes on each heated walls along the length of
microchannel. Moreover, each respective wall temperature was
averaged by considering five nodes on each wall. However for
each wall of rectangular microchannel, temperature peak is
observed at corners and hence the temperature values at each
corner nodes were halved while calculating average
temperature for each wall. The average wall temperature is
defined differently for all three different wall boundary
conditions. For four-wall uniform heat flux boundary condition,
𝑇𝑤 ,𝑎𝑣𝑔 =
𝑇𝑎 ∙𝛼+𝑇𝑏
where, 𝑇𝑎
=
1+𝛼
𝑇3 +𝑇4
2
and
𝑇𝑏 =
𝑇1 +𝑇2
2
(4)
For three-wall uniform heat flux boundary condition,
𝑇𝑤 ,𝑎𝑣𝑔 =
𝑇𝑎 ∙ 2𝛼 + 𝑇𝑏
where, 𝑇𝑎
1 + 2𝛼
=
𝑇3 +𝑇4
2
and
𝑇𝑏 = 𝑇2
(5)
For two-wall uniform heat flux boundary condition,
𝑇𝑤 ,𝑎𝑣𝑔 =
𝑇1 +𝑇2
2
(6)
MODEL DESCRIPTION
The fluid flow and heat transfer effects in rectangular
microchannels with varying aspect ratio from 0.1 to 10 were
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investigated using commercial CFD software, FLUENT.
GAMBIT was used as pre-solver software for designing
geometric models, grid generation and boundary definition.
Water enters the rectangular microchannels with a fully
developed velocity profile at inlet temperature of 300K. The
numerical simulation was initially performed for rectangular
microchannels with four-side constant wall heat flux boundary
condition. This case was tested for three different aspect ratios
of 0.1, 0.5 and 1, and the results were verified by the scheme
proposed by Shah and London [17]. The only published data
understand the effect of entrance types on heat transfer
characteristics in entrance region.
The pre-solver GAMBIT was used for grid generation in
microchannels. Figures 4 and 5 show the meshed geometry of
rectangular channel having aspect ratio of 1 with abrupt
entrance type and smooth entrance type respectively.
Hexwedge cooper scheme was used for generating mesh in
geometries. The mesh spacing was kept as 0.005. Finer meshes
were created to predict the fluid flow and heat transfer effects
more accurately. There were approximately 2 million grid
Figure 2: Schematic of geometric model with abrupt
entrance type.
Figure 4: Schematic of meshed model with abrupt
entrancetype having aspect ratio of 1.
Figure 3: Schematic of geometric model with smooth
entrance type.
Figure 5: Schematic of meshed model with smooth entrance
type having aspect ratio of 1.
available for NuH2 for varying aspect ratios are for four-wall
heating conditions given by Shah and London [17]. Hence, the
data sets for uniform heat flux (H2) boundary condition for
three-walls and two-walls are numerically generated. The
rectangular channels used in their proposed scheme were
having abrupt entrance type as shown in Figure 2, and in
general channels always have some type of headers for
practical applications. Hence, new geometric models were
designed with same rectangular microchannel dimensions as
earlier along with a smooth entrance type (with inlet and outlet
headers) as shown in Figure 3. This comparison can help us to
elements in each of the rectangular geometries and the
processing time for the simulation was noted to be around 7-8
hours (Intel Core 2 Duo processor). Grid independence test was
carried out in order to ensure the best mesh spacing for the
geometrical model.
The numerical simulations were performed using
commercial CFD software package FLUENT. Reynolds
number was kept constant as 100 for all cases in order to
confirm laminar flow through rectangular microchannels.
Pressure-based solver was used for simulation in order to
achieve steady state analysis. The SIMPLE (Semi-Implicit
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Method for Pressure-Linked Equations) algorithm was used for
introducing pressure into the continuity equation. The energy
equation was activated during the analysis to predict heat
transfer effects in microchannels and check the effect of
entrance type on heat transfer. The flow momentum and energy
equations were solved with a first-order upwind scheme. The
simulations were performed for a convergence criterion of E-6.
microchannels with and without abrupt entrance are shown for
two different aspect ratios of 0.1 and 0.5. From these plots it
can be seen that entrance type does not have any significant
impact on Nusselt number in entrance region of thermally
developing flow.
16
14
The first objective was to validate the results of numerical
simulation with available published data of Shah and London
[17] for uniform heat flux boundary condition (H2) on all the
four walls of rectangular microchannel. Table 2 shows the fully
developed Nusselt number, found by numerical simulation as
well as available published data, for three different aspect ratios
of 0.1, 0.5 and 1 with H2 boundary condition. It can be seen
from the Table 2 that fully developed Nusselt number found
from simulation results are slightly higher than the fully
developed Nusselt number for macroscale ducts.
Table 2: Comparison of NuH2,fd of numerical work with
proposed values by Shah and London [17]
NuH2,fd
Aspect
ratio,
From Shah and
Current Numerical
α
London, 1971
work
0.1
2.950
3.083
0.25
2.940
2.901
0.33
2.970
3.023
0.5
3.020
3.110
1
3.091
3.299
Effects of entrance type on Nusselt number:
Abrupt entrance type (AR=0.5)
Smooth entrance type (AR=0.1)
12
Smooth entrance type (AR=0.5)
10
8
6
4
2
0
0
2
4
6
8
10
Length along the microchannel, x (mm)
Figure 6: Plot of Nu along the length of the microchannel
having smooth and abrupt entrance with H2 boundary
conditon on all the four walls.
16
Abrupt entrance type (AR=0.1)
14
Nusselt Number, Nu
Validation of numerical model:
Nusselt Number, Nu
RESULTS AND DISCUSSIONS
Initially the simulations were performed for H2 boundary
conditions on all four walls for three different aspect ratios of
0.1, 0.5 and 1. The results of fully developed Nusselt number
were then compared with the values reported by Shah and
London [17]. Further simulations were performed on all the
three cases of boundary conditions; four-wall, three-wall and
two-wall for two different aspect ratios of 0.1 and 0.5 for two
different cases of abrupt and smooth entrance types. The results
from the simulations were used to predict the effect of entrance
type on thermally developing entrance region in rectangular
microchannels. Large data set was generated for thermally
developing Nusselt number for three-wall and two-wall
uniform heat flux boundary condition (H2) for whole range of
aspect ratio varying from 0.1 to 10. The aspect ratio was
defined as a/b for all the numerical simulations with four wall,
three wall, and two wall H2 boundary conditions.
Abrupt entrance type (AR=0.1)
Abrupt entrance type (AR=0.5)
Smooth entrance type (AR=0.1)
12
Smooth entrance type (AR=0.5)
10
8
6
4
2
0
0
2
4
6
8
Length along the microchannel, x (mm)
10
Figure 7: Plot of Nu along the length of the microchannel
having smooth and abrupt entrance with H2 boundary
conditon on three walls.
Figure 6, figure 7 and figure 8 are the plots showing
variation in Nusselt number along the length of the
microchannel having uniform heat flux on 4 walls, 3 walls and
2 walls respectively. These cases were plotted only for first
10mm length of rectangular microchannel in order to see the
effect of entrance headers on Nusselt number. Results for
5
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20
18
Abrupt entrance type (AR=0.5)
16
Smooth entrance type (AR=0.1)
14
Smooth entrance type (AR=0.5)
AR=0.25
AR=0.33
AR=0.5
AR=0.6
AR = 0.75
AR = 1
AR = 2
AR = 5
AR = 10
18
16
Nusselt Number, Nu
Nusselt Number, Nu
20
Abrupt entrance type (AR=0.1)
12
10
8
6
14
12
10
8
6
4
4
2
2
0
0
0
2
4
6
8
Length along the microchannel, x (mm)
0
10
20
40
60
80
100
Length along the microchannel, x (mm)
Figure 8: Plot of Nu along the length of the microchannel
having smooth and abrupt entrance with H2 boundary
conditon on two walls.
Figure 10: Plot of Nu along the length of the microchannel
for whole range of aspect ratios with H2 boundary conditon
on two walls.
Results for 3-wall & 2-wall H2 boundary condition:
Moreover, for three wall H2 boundary condition the values of
fully developed Nusselt number converges and are
approximately around 3 for the whole range of aspect ratio
from 0.1 to 10. Also, the trend of Nusselt number along the
length of microchannels for three walls H2 condition was
similar to that observed for four wall cases. The values of
Nusselt number did not seem to vary much for the whole range
of channel aspect ratio. Whereas in case of two wall H2
boundary condition, the fully developed Nusselt number values
vary within a wide range.
Moreover, the fully developed Nusselt number obtained for
four wall and three wall H2 boundary condition seem to have
increased with increase in channel aspect ratio. The similar
trend was observed by Shah and London [17] for four wall
cases. This trend was opposite to that observed by investigators
for four wall and three wall cases for H1 and T boundary
conditions [22].
Figure 10 shows local Nusselt number along the length of
rectangular microchannel for two wall H2-constant wall heat
flux boundary condition. The fully developed Nusselt number
for two wall uniform heat flux cases seem to have decreasing
value with increase in aspect ratio. This trend was similar to the
published data for H1 and T boundary condition for two wall
cases [22].
In order to study heat transfer effects in the entrance region
of microchannel, Nusselt number is plotted as a function of non
dimensional length x*. Figures 11 and 12 show results for
rectangular microchannels for the whole range of aspect ratios
under H2 boundary condition on three and two walls
respectively. From these figures, it can be seen that a larger
channel aspect ratio will have a larger dimensionless thermal
entrance length x*, and its value decreases with simultaneous
decrease in aspect ratio. The results show that the value of
Nusselt number starts high and decreases rapidly along the
length of microchannel.
Next important step in this work was to generate large data
set for Nusselt number in thermally developing flow with H2
boundary condition. Microchannels having aspect ratios
ranging from 0.1 to 10 were simulated with H2 boundary
condition. Figure 9 and figure 10 are the plots showing Nusselt
number along the length of microchannel with uniform heat
flux on three walls and two walls respectively. The graphs
show that the Nusselt number is very high at the beginning of
the entrance region of microchannel and thereafter decreases
exponentially and achieve the fully developed Nusselt number.
20
AR=0.25
AR=0.33
AR=0.5
AR=0.6
AR = 0.75
AR = 1
AR = 2
AR = 5
AR = 10
18
Nusselt Number, Nu
16
14
12
10
8
6
4
2
0
0
20
40
60
80
Length along the microchannel, x (mm)
100
Figure 9: Plot of Nu along the length of the microchannel
for whole range of aspect ratios with H2 boundary conditon
on three walls.
6
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Nusselt number, Nu
10
Table 3: NuH2,fd for fully developed laminar flow, for three
walls and two walls uniform heat flux boundary condition.
AR=0.25
9
AR=0.33
8
AR=0.5
7
AR = 0.6
6
AR = 0.75
NuH2,fd
3-wall
condition
4
3
0.10
0.25
0.33
0.50
0.60
0.75
1
2
5
10
2
1
0
0
0.2
0.4
0.6
0.8
1
Non-dimensional length, x*
1.2
Figure 11: Plot of Nu as a function of x* for different aspect
ratios with H2 boundary conditon on three walls.
16
AR=0.25
14
Nusselt number, Nu
2-wall
boundary
condition
Aspect ratio,
α (a/b)
AR = 1
5
boundary
AR=0.33
AR=0.5
12
AR = 0.6
10
AR = 0.75
AR = 1
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
Non-dimensional length, x*
Figure 12: Plot of Nu as a function of x* for different aspect
ratios with H2 boundary conditon on two walls.
Generated numerical data set for 3-wall and 2-wall H2
boundary condition:
Table 3 represents the values of fully developed Nusselt
number found from the results of numerical simulations of
three walls and two walls uniform heat flux (H2) boundary
condition for rectangular microchannels for the whole range of
aspect ratios. Hence the large data set for constant wall heat
flux, both circumferentially and axially has been generated by
performing CFD computations for wide range of cases.
2.463
2.401
2.533
2.724
2.762
2.835
3.148
3.243
3.770
4.632
6.803
6.183
6.086
5.626
5.235
4.844
4.611
3.201
1.785
1.065
CONCLUSIONS
The numerical simulations were carried out with uniform
wall heat flux boundary condition for three different cases; four
walls, three walls and two walls heating. The numerical model
for four wall H2 boundary condition was validated with
published data by Shah and London [17] and the results
obtained for microchannels seem to have good accordance with
the findings of majority of researchers. The effect of entrance
type on the heat transfer in the thermally developing entrance
region was studied. The results predict insignificant effect of
entrance type on Nusselt numbers for all the three cases. Large
data sets were generated for wide range of aspect ratios varying
from 0.1 to 10 for rectangular microchannels for three walls
and two walls uniform heat flux boundary condition. The
numerical data obtained for three wall H2 boundary condition
followed similar trend as observed for four wall uniform heat
flux condition for different aspect ratios. The heat transfer
results for four wall and three wall H2 condition did not vary
much with increase in aspect ratio contrary to that observed by
Schmidt [22] for H1 and T boundary condition. On the other
hand, fully developed Nusselt number values for two wall H2
boundary condition in rectangular microchannels for aspect
ratios ranging from 0.1 to 10 followed the similar trend as
compared to H1 and T boundary condition. Higher channel
aspect ratio predicted larger values of non-dimensional thermal
entrance length, x*. A comprehensive numerical data set was
generated for heat transfer aspects in rectangular microchannels
under more practical H2 boundary condition.
ACKNOWLEDGMENTS
The work was carried out in Thermal Analysis and
Microfluidics and Fuel Cell Laboratory (TAµFL), Rochester
Institute of Technology, Rochester, NY.
7
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