Proceedings of ICMM2005
3rd International Conference on Microchannels and Minichannels
June 13-15, 2005, Toronto, Ontario, Canada
Paper No. ICMM2005-75114
SINGLE-PHASE LIQUID HEAT TRANSFER IN MICROCHANNELS
Mark E. Steinke
Satish G. Kandlikar
Thermal Analysis and Microfluidics Laboratory
Mechanical Engineering Department
Kate Gleason College of Engineering
Rochester Institute of Technology, Rochester, NY USA
[email protected]
Thermal Analysis and Microfluidics Laboratory
Mechanical Engineering Department
Kate Gleason College of Engineering
Rochester Institute of Technology, Rochester, NY USA
[email protected]
ABSTRACT
The development of advanced microchannel heat
exchangers and microfluidic devices is dependent upon the
understanding of the fundamental heat transfer processes that
occur in these systems. Several researchers have reported
significant deviation from the classical theory used in
macroscale applications, while others have reported general
agreement, especially in the laminar region. This fundamental
question needs to be addressed in order to generate a set of
design equations to predict the heat transfer performance of
microchannel flow devices.
A database is generated from the available literature to
critically evaluate the reported experimental data. An in-depth
comparison of previous experimental data is performed to
identify the discrepancies in the reported literature. It is
concluded that the classical theory is applicable to
microchannel and minichannel flows. The literature reporting
discrepancies do not account for developing flows, fin
efficiency, erros in channel geometry measurements and
experimental uncertainties. It is further concluded that if all
these factors are accounted for, the available data have good
general agreement with macroscale theories.
A similar
approach is presented for pressure drop in microchannels in an
accompanying conference paper, Steinke and Kandlikar (2005).
INTRODUCTION
The validity of the conventional heat transfer theories in
microchannel passages is a subject of major interest. Several
researchers have focused on many aspects of this complex
problem.
However, there is still a need for careful
experimentation related to the fundamental physics associated
with microchannel heat transfer.
Several early researchers have reported significant
deviations from conventional theory. At the same time, a
number of researchers have reported good agreement with
conventional theories. These discrepancies must be reconciled
and addressed to arrive at a proper conclusion about the validity
of the conventional theories.
Heat transfer in the microscale is a very complex issue due
to challenges in microchannel fabrication as well as in
performance characterization. The determination of heat
transfer parameters in microchannel flow is often very difficult.
There are physical size considerations, surface to fluid
interaction concerns, and experimental uncertainties that can
have drastic effects upon the heat transfer parameters. A clear
picture of these issues is required in order to develop suitable
correlations to predict the performance of a microchannel heat
exchanger.
Therefore, the present work is undertaken to identify the
previous literature focusing on single-phase liquid heat transfer
in microchannels. The experimental setup and data acquisition
techniques are critically evaluated. The various aspects, such
as entrance region effects, channel dimension effects, boundary
conditions, etc. are reviewed. Finally, the work required to
fulfill our understanding of heat transfer in microchannels is
identified.
OBJECTIVES OF PRESENT WORK
Following the needs addressed in the preceding section, the
objectives of the present work are set as follows:
• Identify the available literature on single-phase liquid
flow in microchannels including heat transfer
experiments.
• Determine the sources of errors in estimating the heat
transfer performance of microchannel geometries.
• Conduct non-destructive and destructive testing of the
channels to identify the deviations from the idealized
geometry used for the fabrication process.
• Provide an explanation for the reported deviation in
published experimental data from conventional
theories.
1
Copyright © 2005 by ASME
Table 1: Selected Literature for Single-Phase Liquid Flow in Microchannel Passages.
Dh
Fluid /
q”
αc
αf
Author
Year
Shape*
( µm )
=a/b
=s/b
Re
( W/cm2 )
Nu
j
L / Dh
Account
Losses
Agree
Laminar
Tuckerman & Pease
Missaggia et al.
1981
1989
water / R
water / R
92 - 96
160
0.17 - 0.19
0.25
0.14 - 0.17
0.25
291 - 638
2350
187 - 790
100
ID
ID
ID
ID
104 - 109
6
N
N
N
N
Riddle et al.
Gui & Scaringe
1991
1995
water / R
water / Tr
86 - 96
338 - 388
0.06 - 0.16
0.73 - 0.79
0.06 - 0.17
ID
96 - 982
834 - 9955
100 - 2500
12 - 112
4.9 - 17.7
9 - 31
ID
6.19x10 - 5.77x10-3
156 – 180
119 – 136
N
N
N
N
Peng et al.
Peng & Peterson
1995
1995
methanol / R
water / R
311 - 646
311
0.29 - 0.86
0.29
0.2 - 10
5 - 45
0.2 - 4.3
0.6 - 1.6
4.38x10-5 - 6.40x10-4
1.10x10-3 - 3.68x10-3
70 – 145
145
N
N
N
N
Cuta et al.
Peng & Peterson
1996
1996
R124 / R
water / R
425
133 - 200
0.27
0.5 - 1.0
ID
21.50 - 22.00
101 - 578
136 - 794
ID
ID
4.1 - 12.8
0.2 - 0.7
8.97x10-3 - 3.22x10-2
1.69x10-4 - 1.39x10-3
48
25 – 338
Y
N
Y
N
Tso & Mahulikar
Vidmar & Barker
1998
1998
water / C
water / C
728
131
NA
NA
NA
NA
16.6 - 37.5
2452 - 7194
0.5 - 0.8
506 - 2737
0.3 - 1.1
ID
7.35x10-3 - 1.59x10-2
ID
76 – 89
580
Y
Y
Y
NA
Adams et al.
Qu et al.
1999
2000
water / Tr
water / Tr
131
62 - 169
ID
2.16 - 11.53
ID
66.44 - 225
3899 - 21429
94 - 1491
ID
ID
15.7 - 91.7
0.6 - 2.3
ID
6.29x10 - 5.70x10-3
141
178 – 482
Y
N
NA
N
Lee et al.
Qu & Mudawar
2002
2002
water / R
water / R
85
349
0.25
0.32
0.45
7.22
119 - 989
137 - 1670
35
100 - 200
3.2 - 9.9
402 - 1788
5.54x10-3 - 1.69x10-2
5.65x10-1 - 1.63x100
118
128
Y
Y
Y
Y
Bucci et al.
Lee & Garimella
2003
2003
water / C
water / R
172 - 520
318 - 903
NA
0.17 - 0.22
NA
ID
2 - 5272
558 - 3636
ID
ID
4.8 - 50.6
7.1 - 35.4
9.33x10-3 - 1.08x100
3.19x10-3 - 8.03x10-3
ID
28 - 80
Y
Y
Y
Y
Wu & Cheng
2003
water / Tr
169
1.54 - 26.20
ID
16 - 1378
ID
0.2 - 4.1
1.01x10-3 - 1.73x10-2
192 - 467
N
N
Owhaib & Palm
2004
R134a / C
800 - 1700
NA
NA
1262 - 16070
ID
11.5 - 922.2 9.23x10-3 - 1.05x10-1
191 – 406
Y
Y
NA = Not Applicable
ID = Insufficient Data
3.40 - 5.71 1530 - 13455
21.50 - 22.00 214 - 337
* C = circular, R = rectangular, Tr = trapezoid
Figure 1: Reynolds Number vs. Hydraulic Diameter for the Selected Data
Sets.
2
-4
-4
** Y = Yes, N = No
Figure 2: Distribution of Data Sets and Data Points for the Range of
Hydraulic Diameters.
Copyright © 2005 by ASME
LITERATURE REVIEW OF EXPERIMENTAL DATA
For the present study, the papers containing heat transfer
experimental data are reviewed in depth. There are
approximately 40 papers that report the detailed experimental
data. Selected papers are shown in Table 1.
The ranges of the important microchannel fluid flow
parameters are also reported in Table 1. However, not every
parameter necessary to evaluate the heat transfer performance
is reported by the authors; some of the parameters are
calculated by the present authors using the data from the
respective papers.
All authors have reported the channel geometry and aspect
ratio for rectangular channels. For rectangular channels, there
have been several different definitions for channel aspect ratio
in literature. The channel aspect ratio for the present work is
given by Eq. (1).
αc =
a
b
(1)
where a is the microchannel width and b is the microchannel
height.
The orientation of the geometry and the applied boundary
conditions are crucial for heat transfer. It would be desirable to
have an a αc < 1.0 to get several deep, narrow microchannels
with than to have a few wide, shallow microchannels from a
heat transfer perspective. The fin aspect ratio is defined as the
fin thickness divided by the fin height, Eq. (2).
αf =
s
b
(2)
where s is the fin thickness. This ratio is a measure of the
geometry of the fin structure. It is desirable to have tall, thin
fins as opposed to short, wide fins.
A wide range of hydraulic diameters and Reynolds
numbers are shown. The hydraulic diameters range from 62
µm to 1,700 µm and the Reynolds numbers range from 2 to
21,000. The Reynolds number versus the hydraulic diameter is
plotted for the entire literature database in Fig. 1. The previous
literature contains circular, rectangular, trapezoidal and
triangular shaped microchannels. The general trend is that as
the hydraulic diameter decreases, the Reynolds number also
decreases. This makes sense because the pressure drop
increases sharply as the hydraulic diameter is reduced and
lower flow rates are employed.
The laminar flow is the dominate flow regime for microchannel
flows. Unfortunately, there are very few data points in the
turbulent regime. This is due to the very large pressure drops
occurring in that regime.
In many papers, the authors report data for several
different hydraulic diameters. Therefore, a single occurrence of
a diameter is considered to be an individual data set.
Approximately 220 data sets reported in these papers have been
reviewed. The total number of data points is over 5,000.
The total number of data sets and data points can be used
to gain insight into these respective quantities. Figure 2 shows
the percentage of data sets and data points reported for each
range of hydraulic diameter. The most common range of
hydraulic diameter is 100 – 200 µm, with 60 data sets. The
most data points also fall within the 100 – 200 µm hydraulic
diameter range. Also, the Reynolds numbers in the laminar and
transition regions have undergone the most study as this seems
to be the primary regime of interest from a heat transfer
perspective.
HEAT TRANSFER IN MICROCHANNELS
The previous literature is further reviewed to determine the
important parameters for microchannel heat transfer. There are
several non-dimensional parameters that have been suggested
and used in microchannel heat transfer. The common numbers
include Reynolds, Nusselt, Prandtl, Stanton, and Brinkman
numbers.
The hydraulic diameter is used to compare the data and is
given by Eq. (3).
4 ⋅ Ac
(3)
Dh =
P
where Ac is the cross sectional area of the passage and P is the
wetted perimeter. The Nusselt number is given by Eq. (4)
h ⋅ Dh
(4)
Nu =
kf
where kf is the thermal conductivity of the fluid.
The Colburn j factor is used mainly in compact heat
exchanger design and could provide an insight into
microchannel heat exchangers. It is given in Eq. (5), Colburn
(1933).
1
Nu ⋅ Pr 3
(5)
Re
where St is the Stanton number. It is based upon the similarity
between the heat and momentum transfer theories. The
Colburn j factor is one half the friction factor. The fanning
friction factor for a smooth tube is shown in Eq. (6).
ρ ⋅ ∆p ⋅ D h
(6)
f =
2⋅ L ⋅G 2
where f is the Fanning friction factor, ρ is the density, ∆p is the
pressure drop, L is the microchannel length, and G is the mass
flux. Steinke and Kandlikar (2005) present further discussion
on friction factor theory in microchannels.
The papers listed in Table 1 are carefully reviewed to
determine the Nusselt numbers for each data set. The Nusselt
number versus the Reynolds number plot from the data is
shown in Fig. 3. The conventional theory predicts a constant
Nusselt number in the laminar range. This does not appear to
be the case with the reported data. It is interesting to note that
there are no data sets that show the constant Nusselt number
behavior in the laminar regime and a few data sets that show a
decreasing trend. It is very difficult to provide a direct
comparison between data sets because of different boundary
conditions, such as three or four side heating or constant heat
flux or temperature.
In the transition regime, there seems to be a better general
agreement. The conventional theory predicts a dependency
upon Reynolds number in the turbulent regime. The actual
slope of the line is dependent upon which correlation is used.
There are several different trends seen in the data shown in Fig.
3. However, some data sets present good general agreement
with conventional theory. The data in Fig. 3 represents data
collected in test section where the developing regions might be
quite long or covering the majority of the flow length.
Therefore, the data sets that do not account for or adress the
developing length issue are removed from the database.
j = St ⋅ Pr
3
2
3
=
Copyright © 2005 by ASME
Figure 4 shows the Nusselt number for the data sets that
have been corrected by including only those that account for
developing lengths. There is still a trend that has some
1.E+03
experiments. The theoretical Nusselt numbers for constant
surface temperature or constant heat flux boundary conditions
are calculated and the boundary condition that gives the best
agreement is chosen to represent that data set, if it was not
specified in the paper.
1.E+02
1.E+01
1.E+01
A*
Nusselt Number, Nu
1.E+02
1.E+00
1.E+00
1.E-01
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
1.E-01
Reynolds Number
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
Figure 3: Nusselt Number versus Reynolds Number for All
Reported Data Sets.
Reynolds Number
Figure 5: A* versus Reynolds Number for All Reported
Data Sets, Laminar Regime.
1.E+03
1.E+02
1.E+01
1.E+01
A*
Nusselt Number, Nu
1.E+02
1.E+00
1.E+00
1.E-01
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
1.E-01
Reynolds Number
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
Figure 4: Nusselt Number versus Reynolds Number for
Corrected Data Sets.
dependence upon Reynolds number in the laminar region. The
trend has a much larger slope in the turbulent region.
A parameter is used to provide a metric for how well the
experimental data compares to the predicted value. A similar
parameter C* is utilized in comparing the theoretical and
experimental friction factors.
C* is the ratio of the
experimental fRe product to the predicted fRe product. The
same method is utilized in the present work. The ratio of the
experimental Nusselt number to the theoretical Nusselt number
is given by A*, Eq. (7).
A* =
Nu exp
Nu ttheory
(7)
The proper Nusselt number for each data set is carefully
determined from the information contained in the respective
paper. This is a difficult process for many data sets as the
boundary conditions are not clearly defined in many
Reynolds Number
Figure 6: A* versus Reynolds Number for Reduced Data
Sets, Laminar Regime.
The present work mainly focuses on laminar flow.
Therefore, the A* ratio is calculated using the appropriate
laminar Nu for each data set. Figure 5 shows the A* ratio for
all of the data sets. A value of 1.0 would indicate a perfect
match between experimental data and theoretical prediction.
The sharp rise in A* in the transition region is due to the use of
laminar Nusselt numbers and is expected.
Once again, only the data sets that adress the developing
flows are used in plotting Fig. 6. A large number of data sets is
removed and only a few data sets remain. There seems to be
good general agreement in the laminar region.
Bucci et al. performed experiments with water in circular
tubes and conclude that there is good agreement with Hausen
(1959) for the laminar region and Gnielinski (1976) for the
transition region. Furthermore, the critical Reynolds number
4
Copyright © 2005 by ASME
occurs at the conventional location but the actual values is
greatly dependent upon flow and heat transfer conditions. There
is a suggested dependency upon Reynolds number in the
laminar region.
However, the reported experimental
uncertainty for the Nusselt number is 22%. This allows the
majority of the laminar data to fall into agreement with the
constant Nusselt number theory, as seen in Fig. 7.
7.0
6.0
where q is the heat transfer rate, h is the average heat transfer
coefficient, Aht is the heat transfer area, Ts is the surface
temperature, and Tm is the bulk mean temperature. This
equation can be rewritten in terms of the local parameters to
give the local heat transfer coefficients. The heat transfer rate
can be more accurately described using a log mean temperature
difference, as is used in conventional heat exchangers, Eq. (9).
(9)
q = h Aht ∆TLMTD
where ∆TLMTD is the log mean temperature difference given by
Eq. (10).
(T − T ) − (Ts − To )
(10)
∆TLMTD = s i
Ts − Ti
ln
Ts − To
where Ts is the surface temperature, Ti is the inlet fluid
temperature, and To is the outlet fluid temperature.
5.0
A*
4.0
3.0
2.0
1.0
0.0
1.E+00
The starting point for this discussion is the basic
convective heat transfer equation shown in Eq. (8).
(8)
q = h Aht (Ts − Tm )
1.E+01
1.E+02
1.E+03
1.E+04
Reynolds Number
Figure 7: A* versus Reynolds Number for Bucci et al.
(2003) Including Reported Uncertainties.
There still can be some major errors associated with heat
transfer measurements in microchannels.
Some of the
deviations seen in the data can be attributed to a number of
factors. The following is a list of the possible complications
that are believed to be the source of the remaining discrepancy.
• More complex developing flows and entrance
conditions
• Fin efficiency not accounted for
• Large experimental errors
• Hard to determine boundary conditions due to
conjugate heat transfer
Another source of error can come from the separating
walls in a microchannel heat exchanger. Typically, these
would be referred to as fins. The fin efficiency can have an
effect on the experimentally derived heat transfer coefficients
in the microchannels. Very few authors have discussed this
issue and any correction seems to be missing to account for this
aspect. Finally, the microchannel heat exchanger is a complex
system. More accurate modeling could involve describing the
conjugate heat transfer that is occurring in these systems.
THEORETICAL CONSIDERATIONS
The present work focuses on the flow of single-phase
liquids in microchannel passages. The compressibility effects,
slip boundary condition, and the rarefied flow concerns do not
apply for these single-phase liquid flows in microchannels.
Also, no electrokinetic effects are seen at the scales considered
in the reported experiments. For the present work, it will be
assumed that the continuum assumption is valid for
microchannels with hydraulic diameters larger than 10 µm.
Boundary Conditions
As mentioned in the previous section, the theoretical
predictions are greatly dependent upon identifying the proper
boundary condition. The boundary conditions used in the heat
transfer analysis are critical in solving the problem. The
boundary conditions will cause the solution to take on a
specific form. This is no different in the microchannel regime.
For the present work, three boundary conditions will be
considered, as defined in Kakaç et al. (1987). The constant
temperature boundary condition, T, is defined as the wall
having a constant temperature that is both circumferentially and
axially uniform and constant. There are two constant heat flux
boundary conditions used here. First, the H1 boundary
condition is defined as having a circumferentially constant heat
flux, constant wall temperature, and an axially constant heat
flux. Secondly, the H2 boundary condition is defined as having
both a circumferentially and axially uniform heat flux.
All three boundary conditions have applications in
microchannels. If one is modeling a microprocessor, the H1
boundary condition would allow for local “hot spots” where the
local heat flux is higher than the mean. The T boundary
condition could be applicable to a micro-total analysis system
or a lab-on-a-chip system, where the wall temperature is
desired to be at a constant temperature to aid in a chemical
reaction. However, the H2 boundary condition is more
common across a wide variety of test sections and will be the
primary focus here.
Furthermore, the location of the boundary conditions is a
major complication in microchannel heat transfer. A four side
heated passage is the most common assumption when deriving
correlations. However, this case is not that common in
practical microchannels. The microchannels employed by the
investigators are only heated from three sides with an adiabatic
top. This leads to some difficulty in comparing results from
many researchers and predicted results. Fin effects on the two
sidewalls of the microchannels cause further deviations from
any idealized boundary conditions.
5
Copyright © 2005 by ASME
n
µb
Nu
(11)
=
Nu cp
µw
where Nucp is the constant property Nusselt number, Nu is the
property corrected Nusselt number, µb is the viscosity of the
fluid calculated at the bulk fluid temperature, µw is the viscosity
of the fluid calculated at the wall temperature, and n = 0.14 for
laminar heating in a fully developed, circular duct.
Flow Regimes
The classical flow regimes are laminar, transitional, and
turbulent apply to microchannel passages. The transition point
for the microchannel passages has been under much debate. In
a separate paper, Steinke and Kandlikar (2005) address this
issue and conclude that there is no early transition in
microchannels. Therefore, the critical Reynolds number is still
considered to be approximately 2,300.
There are four distinct flow conditions one must consider
in microchannel flow analysis. They are hydrodynamically
developing flow, fully developed hydrodynamically but
thermal developing flow, simultaneously developing flow, and
fully developed flow. Each case presents its own challenges.
An effort is made to address each flow case.
The heat transfer behavior changes with any number of the
parameters mentioned previously and therefore it is very
difficult to adress every single case. As a result, the present
work will discuss the cases that have the greatest applicability
in microchannel heat transfer, including rectangular geometries.
Fully Developed Laminar Flow
This flow case is usually the beginning point for most
analysises and is the simplest of all flow cases. The most
commonly accepted result for this flow case is that the Nusselt
number is constant in the laminar region and a function of
Reynolds number in the turbulent regime.
The Nusselt number is constant in this flow case. The values
for a circular tube are; 3.66 for T and 4.36 for H2. The Nusselt
number for a rectangular channel has a dependency upon
channel aspect ratio. Shah and London (1978) presented data
for that dependency. Figure 8 shows the fully developed Nu
for the three boundary conditions with four sided heating.
Equation (12, 13, 14), Shah and London (1978) shows the
Nusselt number for T, H1, and H2, respectively.
(12)
1 − 2.610α c + 4.970α c − 5.119α c
2
NuT = 7.541
+ 2.702α c − 0.548α c
4
3
5
(13)
1 − 2.0421α c + 3.0853α c − 2.4765α c
2
Nu H 1 = 8.235
+ 1.0578α c − 0.1861α c
4
3
5
(14)
1 − 10.6044α c + 61.1755α c − 155.1803α c
2
Nu H 2 = 8.235
+ 176.9203α c − 72.9236α c
4
3
5
If the channel aspect ratio is greater than 1.0, then the
reciprocal of the channel aspect ratio is used in the above
equations. Remember that these equations are for fully
developed, laminar flow with four sided heating.
Nu,T
9.0
Nusselt Number, Nu
Temperature and Temperature Difference
Measurements
The heat transfer occurring in microchannels is very
efficient. In the microchannel, a very small ∆T between the
surface and bulk mean temperature can occur.
Using
conventional temperature measurement techniques, this leads to
significant uncertainties when the ∆TLMTD is in general smaller
than 2 °C, assuming a 0.2 °C accuracy on the thermocouple.
In addition to measuring the temperatures of desired
quantities, the property used to calculate the parameters are
temperature dependent. There are sometimes large temperature
gradients in the flow and this can greatly affect the fluid
properties as well as axial conduction, which is not considered
here. A fluid property correction method shown in Eq. (11),
Kakaç et al. (1987), will correct the Nusselt number for the
variations in fluid viscosity. The liquid viscosity has a much
greater dependency upon temperature than the other physical
properties, such as thermal conductivity.
Nu, H1
8.0
Nu, H2
7.0
6.0
5.0
4.0
3.0
2.0
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Aspect Ratio Channel, α c
Figure 8: Fully Developed Nusselt Number versus Channel
Aspect Ratio with Four Sided Heating.
Fully Developed Turbulent Flow
There are several correlations available for heat transfer in
the turbulent flow regime. One of the first correlations began
with Reynolds (1874), Eq. (15).
f
(15)
Nu =
Re⋅ Pr
2
for Pr = 1.0, Re > 2,300.
Dittus and Boelter (1930) presented a correlation with two
different exponents for heating and cooling, Eq. (16).
4
(16)
Nu = 0.023 Re 5 Pr n
where n is 0.4 for heating and 0.3 for cooling, for 0.7 Pr
120 and 10,000 Re 1.24x106. Gnielinski (1976) extended
the valid Reynolds number range to the critical Reynolds
number of 2,300, seen in Eq. (17).
6
Copyright © 2005 by ASME
f
Nu =
2
⋅ (Re − 1,000 ) ⋅ Pr
(17)
1
2
f 2
⋅ Pr 3 − 1
2
for 0.5 Pr 106 and 2,300 Re 5x106, where f is given by
Filonenko (1954) in Eq. (18), for 0.5 Pr 106 and 4,000 Re
5x106.
1
(18)
= 1.58 ⋅ ln(Re) − 3.28
f
The previous equations are among the most commonly
accepted correlations for turbulent Nusselt number. There are
many other correlations that other researchers have presented.
Sobhan and Garimella (2001) presented an excellent
summation of the Nusselt number correlations available in
literature. It is interesting to note that the correlations are
irrespective of the boundary conditions. Therefore, they are
valid for any of the boundary conditions discussed. Since the
vast majority of correlations have been developed using
circular passages, it would be important to investigate their
application to non-circular passages.
1 + 12.7 ⋅
Hydrodynamically Developing Flow
The flow is considered to be hydrodynamically fully
developed when the maximum velocity reaches 99% of the
fully developed value. The definition of a non-dimensional
flow distance is defined in Eq. (19).
x
(19)
x+ =
Dh ⋅ Re
where x+ is the non-dimensional flow distance and x is the axial
flow direction location. It is commonly accepted that for a
value of x+ = 0.05, the flow can be considered fully developed.
However, it is more complex for microchannel flows, as
reported by Steinke and Kandlikar (2005), but we will use the
commonly accepted criterion until further research is conducted
to address this issue.
Thermally Developing Flow
In this flow regime, the flow is considered to be
hydrodynamically fully developed but thermally developing. A
flow can be considered thermally fully developed when the
local Nusselt number reaches 1.05 times the fully developed
Nusselt number. The non-dimensional, thermal axial flow
length is given in Eq. (20).
x
(20)
x* =
Dh ⋅ Re⋅ Pr
where x* is the non-dimensional, thermal axial flow location.
As with x+, the flow is fully developed when x* is 0.05.
Simultaneously Developing Flow
This is the most complex flow case. The flow is
simultaneously developing hydrodynamically and thermally.
The Nusselt number is dependent upon the hydrodynamic and
thermal axial locations.
In addition, the Prandtl number
influences the value of Nusselt number as well. It is very
difficult to arrive at a general equation for this case, due to the
dependency upon Re, Nu, x+, x*, and Pr. Much more data is
needed in this flow regime.
Fin Efficiency
In a practical microchannel heat exchanger, there are many
parallel microchannels to increase the heat transfer and flow
areas. The complication here is that there will be a temperature
gradient within the fin. Therefore, the fin efficiency will have
an influence upon the heat transfer coefficient. It must be
addressed in order to generate fundamental data. In the
previous literature, many researchers did not take this effect
into account.
The fin geometry should be considered when designing the
microchannel heat exchanger. Let us assume that the fin has a
rectangular geometry and an adiabatic tip. The fin efficiency
would be determined by Eq. (21).
η=
tanh (m ⋅ b )
m⋅b
(21)
where η is the fin efficiency and m is given by Eq. (22)
m=
h ⋅ Pfin
(22)
k fin ⋅ Ac , fin
The parameter of the fin is given by twice the length plus the
thickness. Finally, the actual heat transfer area that includes the
fin efficiency would be found by Eq. (23)
AHT ' = 2 ⋅ η ⋅ b + a ⋅ L ⋅ n
(23)
This can have a major impact upon the heat transfer coefficient
and could account for further discrepancies in the data
presented. At present, it would be very difficult to reconcile
the data and determine the fin efficiency for the previous
literature because of the inadequate information reported in the
papers. The effect of fin efficiency on the thermal boundary
conditions has not been considered by any researchers.
(
)
EXPERIMENTAL UNCERTAINTIES IN
MICROCHANNELS
The experimental uncertainties can become quite large for
a microchannel heat exchanger. Some of the challenges
include the physical size of the system being measured and the
magnitudes of the measurements.
The heat transfer occurring in microchannels is very
efficient. Therefore, the temperature differences between the
liquid and the walls can be very small. The ∆T can be only a
few degrees or less.
Fortunately, several of the standards for experimental
uncertainties still apply at the microscale. The two best
standards for determining experimental uncertainties are ASME
PTC 19.1 (1998) and NIST Technical Note 1297 (1994). There
are many similarities between these standards and many
published works. In general, the total uncertainty is comprised
of two parts - systematic error and random error as given by:
U =2
B
2
2
+
σ
N
2
(24)
where U is the total uncertainty, B is the bias error, σ is the
standard deviation, and N is the number of samples. The bias
error is a measure of the systematic error and the precision error
is a measure of the random errors in the system.
When propagating errors, Eq. (25) gives the uncertainty of
a calculated parameter based upon the measured variables.
7
Copyright © 2005 by ASME
Up =
2
∂p
u ai
∂a i
n
i =1
(25)
where p is the calculated parameter. The uncertainty in any
parameter is the sum of the uncertainties of the components
used to calculate that parameter.
The uncertainty of the thermocouples and the temperature
resistors is a function of the calibration. When more points are
used, the fit of the calibration equation is improved.
Consider the uncertainty of the Colburn j factor, as an
example. The uncertainty in the Colburn j factor can be
calculated using the fractional uncertainties method, Eq. (26).
Uj
j
U Nu
Nu
=
2
2
1 U
+ ⋅ Pr
3 Pr
U Re
+
Re
1/ 2
2
(26)
The total uncertainty is dependent upon the parameters used to
calculate it. It would be more beneficial to determine the
uncertainty based upon the parameters that are actually
measured and not upon calculated parameters like Reynolds
number. Rewriting Eq. (26) in terms of typical measured
values gives Eq. (27).
2
Uρ
+
ρ
Uj
U
= + 2⋅ b
j
b
2
2
+
Nu
=
+
2
+
2
UI
I
U
+ b
b
2
+
U Tso −To
s,i
2
Ub
+
a+b
1/ 2
2
2
(27)
Microchannel
Ua
+
a+b
2
UL
L
2
Ub
+
a+b
Ua
a + 2 ⋅η ⋅ b
+
+ 2⋅
− Ti ) − (Ts , o − To )
U
+ 2⋅ a
a
TEMPERATURE MEASUREMENTS IN
MICROCHANNELS
The experimentally derived heat transfer coefficient
depends upon properly measuring temperatures at the inlet and
outlet of the fluid, the heater, and the surface. As seen in Eq.
(28), the Nusselt number is greatly dependent upon proper
temperature measurements. This can become quite difficult for
a microchannel heat exchanger. One method is to measure the
inlet and outlet fluid temperatures. A complication here is that
traditional thermocouples can be large by comparison and must
be placed a certain distance from the actual inlet and outlet of
the microchannel reducing their accuracy.
Micro-Thermocouple
2
2
(a + 2 ⋅η ⋅ b )
(T
2
U Nu
+
Nu
b ⋅ Uη
+ 2⋅
U Nu
U
+ a
a
UV
V
Q
Ua
+
a+b
2
2
UQ
+
µ
1 U
+ ⋅ Pr
3 Pr
Uk
kf
2
Uµ
technique. The average Nusselt number for a laminar flow,
with a constant heat flux boundary condition, average heat
transfer coefficient, with fin efficiency included, and a ∆T log
mean temperature difference measurement the uncertainty is
given by Eq. (28), where I is the current of the power supply, V
is the voltage of the power supply, η is the fin efficiency, and T
is the temperature. The fin efficiency is also included in the
uncertainty calculations. It can be seen that the Nusselt number
is greatly dependent on the temperature measurements.
However, the microchannel geometric measurements can also
greatly affect the uncertainty. Therefore, great care must be
taken not only with temperature measurements, but also with
the microchannel geometric measurements.
η ⋅U b
(a + 2 ⋅η ⋅ b )
2
+
(T
s ,i
1/ 2
2
2
2
U Tsi −Ti
Inlet
Plenum
2
− Ti ) − (Ts , o − To )
+
(T
(T
Tso −To
(T − T )
− Ti )
⋅ ln s ,i i
(Ts ,o − To )
s , o − To )
s ,i
(28)
The dependency upon the flow rate and the microchannel
geometry are shown. The Nusselt number dependency is also
shown. However, the uncertainty of the Nusselt number based
upon actual measured parameters is more complex.
The uncertainty in the Nusselt number is dependent upon
the boundary conditions used and the temperature measurement
Outlet
Plenum
Developing
Region
2
U Tsi −Ti
Fully
Developed
Figure 9: Micro-thermocouples Embedded in the Inlet and
Outlet Plenums of a Microchannel.
This problem can be overcome by incorporating microthermocouples in the inlet and outlet plenums. Figure 9 shows
two very small thermocouples embedded in the plenums. The
sensing bead on the thermocouple can be the approximate size
of the microchannel however. This method would be more
appropriate for a multiple pass manifold versus a single
channel. In addition, extreme care must be taken to ensure that
the bead does not interfere with the flow distribution in the
manifold, and that the flow is properly mixed at the
measurement location.
8
Copyright © 2005 by ASME
An alterative to both of these scenarios is to incorporate
thermocouples into the test section. Several researchers have
been exploring this possibility. Typically, these test sections
utilize silicon microchannels and take advantage of the
microelectronics fabrication technology to incorporate on-chip
temperature sensing devices. One device is simply a p-n
junction, where the voltage across the p-n junction is a function
of the temperature.
Another method is to use a metal or a metal oxide that
relies on the thermal coefficient of resistance. The resistance of
the metal is a function of temperature. An example of this for
use with microchannels is shown in Steinke et al. (2005). They
describe the fabrication of a silicon microchannel test section
with copper resistors for temperature measurements intermixed
amongst a heater. However, the complication for this method
is that the surface temperatures must be calculated using the
backside heater temperatures, accounting for the temperature
drop across the silicon base.
MICROCHANNEL GEOMETRY EVALUATION
The channel dimensions have a major effect on the heat
transfer calculations as seen from Eq. (28). To assess the
geometry effects, a silicon microchannel was fabricated using
the dry reactive ion etching (DRIE) process. The dimensions
measured from non-destructive optical measurement techniques
yield a microchannel width of 201 ± 5 µm and a microchannel
depth of 247 ± 5 µm and a rectangular profile. Then, the
microchannel test section was cleaved in order to measure
accurate cross sectional measurements. Upon destruction, the
profile of the microchannels is actually found to be trapezoidal
in shape with rounded corners.
Figure 10: Cross Section of Silicon Microchannel. a = 194
µm, b = 244 µm, θ = 85 degrees, Dh = 227 µm.
Figure 10 shows an SEM image of the microchannel
geometry. It can be seen that there is significant undercutting
of the fin area in these microchannels. The width at the top is
194 ± 1 µm. The depth is 244 ± 1 µm with a sidewall angel of
85 degrees. This profile could not be obtained from nondestructive measurement techniques. Careful measurements
are made to determine the cross sectional area and the proper
hydraulic diameter is calculated.
Unfortunately, there is no available correlation that gives
the Nusselt number for a three side heat trapezoid with rounded
corners. The four side heated trapezoid correlation should be
utilized in determining the Nusselt number until improved
correlations are available.
Figure 11: Fin Cross Section of Silicon Microchannel. stop =
94 µm, sbottom = 54 µm, θ = 85 degrees.
The fin geometry is also of concern with heat transfer
calculations. Figure 11 shows the fin geometry for the
fabricated microchannels. The fin thickness is narrower at the
base of the fin than at the top (93.9 m at the top against 54.3
m at the fin base). The undercutting resulting from the
fabrication can become substantial. The heat flux through the
fin area will not be uniform. This leads to a severe reduction in
the fin efficiency. It is therefore recommended that the
thickness of the fin at the base should be used in determining
fin performance. More work is needed in this area to
characterize the thermal performance of microchannels
fabricated in silicon.
SURFACE ROUGHNESS MEASUREMENTS
Finally, the surface roughness of the microchannel walls is
another important parameter. In conventional sized channels,
the surface roughness has been identified to play a dominate
role only in the turbulent region. The data reduced from
previous literature suggest the same trend for microchannels.
However, this is still an open topic of discussion. It would be
important for researchers to report the surface roughness in
their work in order to build confidence in that statement.
Figure 12 shows the nature of the surface roughness in the
silicon microchannel test section. The pictures of the side wall
and bottom wall surfaces showing surface roughness features
are obtained using scanning electron microscope (SEM) and are
shown in the figure. The majority of the surface has a very
small e/Dh ratio, approximately 0.002. The bottom wall does
have some roughness features seen in the bottom inset. The
average size of those features is 1.5 µm, making the e/Dh ratio
approximately 0.007. The side wall has larger roughness
9
Copyright © 2005 by ASME
features due to the method of fabrication. These features are an
artifact of the deep reactive ion etching processes used. The
average roughness feature for the side wall is 2.5 µm and the
resulting e/Dh ratio is 0.01. The measurement of the surface
roughness for future microchannel works should be carefully
evaluated.
confirmed with a profilometer measurement as well. This is a
very rough tube. The roughness ratio is approaching the limits
of the Moody diagram. Once again, the change in area
resulting from these roughness structures can become
significant and will affect the heat transfer performance as well
as the fluid flow. Much more work is required to characterize
the effect the roughness has on the heat transfer.
Side
Bottom
Figure 12: Surface Roughness in the Microchannels. Side
wall e/Dh = 0.01, Bottom wall e/Dh = 0.007.
Figure 14: Surface Roughness
Microtube. D = 203 µm, e/D = 0.045.
for
Stainless
Steel
CLOSING REMARKS
The available experimental data needs to be evaluated
carefully in light of the considerations presented in this paper.
However, it is an arduous task since not all the relevant
information needed to correctly determine the Nusselt from the
data is reported by the investigators. There is a definite need to
carefully design an experimental setup that addresses the issues
raised in this paper and then generate accurate and consistent
data sets. Such an effort is being reported by Steinke et al.
(2005) in a separate paper dealing with microchannels built
directly in silicon substrates. Other researchers are encouraged
to address the issues raised here in generating reliable
experimental data.
Figure 13: Surface Roughness
Microtube. D = 413 µm, e/D = 0.032.
for
Stainless
Steel
The surface roughness of commercial hypodermic needles
gives a good indication of the typical surface roughness for
commercial stainless steel microtubes. Figure 13 shows the
cross section of a nominal 406 µm diameter tube. The
measured diameter is 413 µm. The roughness structures are
visible and are approximately 13 µm in height. The resulting
e/Dh is 0.032 which is a fairly rough tube. Although, this does
not have a significant impact on fluid flow, as most flows are
laminar. However, this could create a significant increase in
heat transfer area and thus heat transfer performance.
The cross-section for a smaller diameter stainless steel
microtube is shown in Fig. 14. The nominal and measured
diameter is in agreement with a value of 203 µm. The
roughness structures are more prevalent with the smaller
diameter microtube. The measured e/D ratio is 0.045, as
CONCLUSIONS
The following conclusions are drawn from the present
work.
• There are over 150 papers specifically addressing the
topic of fluid flow and heat transfer in microchannels.
Approximately 40 of those papers present useful data
on heat transfer in microchannels. A database of over
5,000 data points has been generated.
• In order to resolve the previous discrepancies in
literature regarding heat transfer in microchannels, a
critical review of the data and experimental setups has
been conducted. An explanation for the deviation
from the classical theory has been presented for the
first time. The papers that do not account for the
entrance and exit losses or the developing flow in the
microchannel are the same papers that report
significant deviations.
10
Copyright © 2005 by ASME
•
•
•
•
The reduced experimental data sets show good general
agreement with the classical theory. The laminar
region shows agreement when the experimental
uncertainties are considered. There is no evidence that
supports an early transition to turbulent flow in
smooth channels.
The experimental uncertainties for heat transfer in
microchannel flows are very important.
The
uncertainty in Nusselt number is most affected by the
microchannel geometry measurements.
The
temperature measurements are also very important,
especially when small temperature differences below a
few degrees are encountered.
Proper measurements of the microchannel geometries
are critical. A measurement of the actual profile is
highly recommended. Some measurement techniques
can lead to false assumptions of geometry. A crtical
evaluation of the passage profile and surface
roughness is required to get an accurate picture of the
heat transfer performance
Much more heat transfer data is required to properly
determine microchannel heat transfer performance.
More studies are required to develop heat transfer
correlations for practical microchannel heat
exchangers.
FUTURE WORK
It would be beneficial to develop additional techniques for
measuring local surface temperatures. More experimental data
for simultaneously developing flow would allow for more
accurate correlations for that boundary condition. Improve
correlations for different practical boundary conditions. In
addition, heat transfer data for turbulent flow needs to be fully
explored.
NOMENCLATURE
a
channel width, m
A
area, m2
A*
normalized Nusselt number = Nuexp / Nutheory
b
channel height, m
B
systematic error
Br
Brinkman number
Cp
specific heat, J kg-1 K-1
Dh
hydraulic diameter, m
e
roughness height, m
f
fanning friction factor
G
mass flux, kg m-2 s-1
h
heat transfer coefficient, W m-2 K-1
h
I
j
k
L
m
n
N
Nu
p
average heat transfer coefficient, W m-2 K-1
current, A
Colburn j factor
thermal conductivity, W m-1 K-1
channel length, m
value used in Eq. (22)
exponent used in Eq. (12) = 0.14 for present work
number of samples
Nusselt number
pressure, Pa ; parameter
P
Po
Pr
q
q”
Q
rh
Re
s
St
T
U
V
perimeter, m
Poiseuille number, = f * Re
Prandtl number
heat transfer, W
heat flux, W m-2
volumetric flow rate, m3 s-1
hydraulic radius, m
Reynolds number
fin thickness, m
Stanton number
temperature, °C
total uncertainty
voltage, V
V
mean velocity, m s-1
axial location, m
non-dimensional flow distance
non-dimensional thermal flow distance
x
x+
x*
Greek
αc
αf
∆
η
µ
ρ
σ
τw
channel aspect ratio = a / b
fin aspect ratio = s / b
change in
fin efficiency
viscosity, N s m-2
density, kg m-3
standard deviation
wall shear stress, Pa
Subscripts
app apparent
b
bulk
c
cross section
cp
constant property
f
fluid
fin
fin
FD fully developed
H1 H1 boundary condition
H2 H2 boundary condition
ht
heat transfer
HT’ heat transfer with fin effect
i
inlet
LMTD log mean temperature difference
m
mean
o
outlet
s
surface
T
temperature boundary condition
tot
total
w
wall
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