Experimental Thermal and Fluid Science 37 (2012) 12–18
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Experimental Thermal and Fluid Science
journal homepage: www.elsevier.com/locate/etfs
Heat transfer and friction characteristics of air flow in microtubes
Chien-Yuh Yang a,⇑, Chia-Wei Chen a, Ting-Yu Lin b, Satish G. Kandlikar b
a
b
National Central University, Jhong-Li, Taoyuan, Taiwan
Rochester Institute of Technology, Rochester, NY, USA
a r t i c l e
i n f o
Article history:
Received 6 May 2011
Received in revised form 6 September 2011
Accepted 6 September 2011
Available online 29 September 2011
Keywords:
Microtube
Heat transfer
Liquid Crystal Thermography
a b s t r a c t
Several researches dealing with the single-phase forced convection heat transfer inside microchannels
have been published in the past decades. The performance of liquid flow has been proved that agrees
with the conventional correlations very well. However, owing to the low heat transfer coefficient of gaseous flow, it is more difficult to eliminate the effects of thermal shunt and heat loss than water flow while
measuring its heat transfer performance. None of the heat transfer performance experimental results
have been published in the literature. This study provides an experimental investigation on the pressure
drop and heat transfer performance of air flow through microtubes with inside diameter of 86, 308 and
920 lm. The Liquid Crystal Thermography method was used to measure the tube surface temperature for
avoiding the thermocouple wire thermal shunt effect. The experimental results show that the frictional
coefficient of gas flow in microtube is the same as that in the conventional larger tubes if the effect of
gaseous flow compressibility was well taken consideration. The conventional heat transfer correlation
for laminar and turbulent flow can be well applied for predicting the fully developed gaseous flow heat
transfer performance in microtubes. There is no significant size effect for air flow in tubes within this
diameter range.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
Owing to the fabrication technology development during the
past decades, the so-called microtubes with internal diameters
smaller than 1 mm can be easily made and used for increasing
the compactness of heat exchangers. These kinds of heat exchangers are able to attain extremely high heat transfer surface area per
unit volume, high heat transfer coefficient and low thermal resistance. However, the conventional forced convection heat transfer
correlations were derived from tubes with diameter much larger
than those used in microchannels. They have not been verified to
work well for predicting the heat transfer coefficient for flow inside
small diameter tubes. The study on heat transfer performance in
microchannels has become more important due to the rapid
growth of the application for high heat flux electronic devices
cooling.
Several researches dealing with the single-phase friction and
forced convection heat transfer in microtubes have been published
in the past years. Wu and Little [1,2] measured the flow friction
and heat transfer characteristics of gases flowing through trapezoid silicon and glass microchannels of hydraulic diameters from
45 to 165 lm. They observed that the friction coefficients for silicon channels are in agreement with that for smooth tubes shown
⇑ Corresponding author. Tel.: +886 3 4267347; fax: +886 3 4254501.
E-mail address: [email protected] (C.-Y. Yang).
0894-1777/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.expthermflusci.2011.09.003
in the Moody chart but the results for glass channels are not. They
concluded that for microchannels, the relative surface roughness is
high for so-called smooth channels. The methods of manufacture
and shape of the channels are all factors which affect the value
of the friction coefficient in small channels.
Choi et al. [3] measured the friction factors and convective heat
transfer coefficients for flow of nitrogen gas in microtubes with inside diameters ranged from 3 to 81 lm in both laminar and turbulent flow regime. The experimental results indicated significant
departures from the thermofluid correlations used for conventional
sized tubes. They concluded that the Colburn analogy was not
applied for microtubes having inside diameters less than 80 lm.
Yu et al. [4] studied the fluid flow and heat transfer characteristics
of nitrogen gas and water in circular tubes with diameters of 19, 52
and 102 lm and Reynolds numbers ranging from 250 to near
20,000. The measured friction factors were slightly lower than the
Moody chart values for both laminar and turbulent regimes. However, the Nusselt numbers for cooling of water in the turbulent
regime were considerably higher than those would be predicted
for larger tubes, suggesting that the Reynolds analogy does not hold
for micro-channel flow. Adams et al. [5] investigated turbulent single-phase forced convection of water in circular microchannels
with diameters of 0.76 and 1.09 mm. Their data suggested that
the extent of enhancement increases as the channel diameter decreases and Reynolds number increases. Based on the data they
obtained, along with earlier data for small circular channels by
C.-Y. Yang et al. / Experimental Thermal and Fluid Science 37 (2012) 12–18
13
Nomenclature
A
cp
di
do
Dh
f
G
h
kf
L
Lh
Lm
LCT
_
m
Nud
heat transfer area (m2)
specific heat (J/kg K)
tube inside diameter (m)
tube outside diameter (m)
hydraulic diameter (m)
friction coefficient, dimensionless
mass velocity (kg/m2 s)
heat transfer coefficient (W/m2 °C)
water conductivity (W/m °C)
tube length
tube heating length (m)
wall temperature measuring position (m)
Liquid Crystal Thermography
mass flow rate (kg/s)
Nusselt number, dimensionless
Yu et al. [4], they developed a correlation for the Nusselt number for
turbulent, single-phase, forced convection in circular microchannels with diameters range from 0.102 mm to 1.09 mm.
Mala and Li [6] investigated water flow through microtubes with
diameters ranging from 50 to 254 lm. The experimental results
indicate that at high Reynolds number laminar flow condition, the
friction factor is higher than that given by the conventional Poiseuille flow theory. Celata et al. [7] reported the results of refrigerant
R-114 flowing in capillary tubes with a diameter of 130 lm. They
found that the friction factor was in good agreement with the
Poiseuille theory for Reynolds number below 600 but higher than
that for higher Reynolds number. Li et al. [8] tested the frictional
characteristic of water flowing in glass, silicon and stainless steel
microtubes with diameters ranging from 79.9 to 205.3 lm. They
concluded that for smooth tubes, the friction factor is consistent
with the results in macro tubes, while the value of fRed in rough
tubes is 15–37% higher than 64. Yang et al. [9] provided a systematic test of friction characteristic for air, water, and liquid refrigerant R-134a in 10 tubes with inside diameters from 0.173 to
4.01 mm including the laminar and turbulent flow regime. The test
results show that the conventional correlations for large tubes may
be adequately used to estimate the friction factors for water, refrigerant, and laminar air flow in microtubes. For turbulent airflow,
however, the friction coefficients are lower than the values predicted by Blasius equation. The discrepancy increased with increasing Reynolds number.
Yen et al. [10] measured heat transfer performance of laminar
refrigerant R-123 flow in 0.3 mm diameter tube by direct attaching
K-type thermocouple on the tube wall. The results are in reasonable
agreement with the analytical laminar constant heat flux value
(Nud = 4.36). However, the Nusselt number data have a very high
scattering distribution from around 2–5. Lelea et al. [11] investigated developing and laminar distilled water flow in microtubes
with diameter 0.1, 0.3 and 0.5 mm. The experimental results confirm that, including the entrance effects, the conventional or classical theories are applicable for water flow through microtubes of the
sizes tested. Grohmann [12] measured the heat transfer coefficient
of liquid argon at around 120 K in microchannels with diameter 250
and 500 lm. The results revealed that there is no physical difference in heat transfer mechanisms between macrotubes and microtubes. The enhancement of heat transfer coefficients in small tubes
compared to conventional correlations was explained with the
increased influence of surface roughness.
Lin and Yang [13] proposed a non-contacted Liquid Crystal Thermography (LCT) method to measure the surface temperature of
microtubes. It is successfully avoid the thermal shunt and contact
p
q
q00
R
Ra
Red
T
Ti
Tx
Twx
TLC
x
l
q
Dp
pressure (Pa)
heat transfer rate (W)
heat flux (W/m2)
specific gas constant (J/kg K)
average roughness (m)
reynolds number, dimensionless
temperature (°C)
inlet water temperature (°C)
local water temperature (°C)
local tube inside wall temperature (°C)
thermochromic liquid crystal
axial position of tubes (m)
viscosity (N/m2 s)
density (kg/m3)
pressure drop (Pa)
problem caused by using thermocouple. Yang and Lin [14] used this
method to measure the heat transfer performance of water flow in
microtubes with inside diameters from 123 to 962 lm. The test results showed that the conventional heat transfer correlations for
laminar and turbulent flow can be well applied for predicting the
fully developed heat transfer performance in microtubes. The transition from laminar to turbulent flow occurs at Reynolds number
from 2300 to 3000. This is also the same range as that for conventional tubes. There is no significant size effect for water flow in tubes
within this diameter range.
Celata et al. [15] presented the work deals with the compressible flow of nitrogen gas inside microtubes ranging from 30 to
500 lm and with different values of the surface roughness (<1%),
for different flow regimes. Their results showed that classic correlations can predict friction factor in laminar flow without revealing
any evident influence of the surface roughness. The laminar–
turbulent transition starts for Reynolds number not lower than
2000 for smooth pipes. In the fully developed turbulent regime,
an agreement between experimental data and the Blasius correlation has been verified for smooth pipes.
In summarizing the above literature review, we may find that
most of the early studies showed significant discrepancy between
the experimental results and convention correlations prediction values. However, in the recent years, the friction factors test results for
both liquid and gas in microtubes can be adequately predicted by the
conventional correlations. The heat transfer test results for liquid
can also be well predicted by the traditional forced convection heat
transfer correlations. But owing to the measurement difficulties,
none of the heat transfer test results for gas flow in microtubes have
been published in the literature. The conventional heat transfer
correlations have not been verified to be applied for flow in microtubes. This study provides an experimental investigation on laminar
and turbulent forced convective heat transfer characteristics of air
flow in microtubes. The LCT method proposed by Lin and Yang
[13] was used in this study to measure the surface temperature of
microtubes.
2. Experimental method
2.1. Tubes size measurement and experiment system setup
Three steel tubes with inside diameter of 920.1, 308.4 and
85.6 lm were tested in the present study. The tubes inner diameters
were measured from the enlarged photographs taken by scanning
electron microscope (SEM) for tube with inner diameter of 85.6
14
C.-Y. Yang et al. / Experimental Thermal and Fluid Science 37 (2012) 12–18
920 µm
308 µm
86 µm
Fig. 1. Enlarged photographs of the microtubes.
Table 1
Detail dimensions and surface roughness of the tubes tested.
Tube
notation
Tube length,
L (mm)
Average outside
diameter, do (lm)
Average inside
diameter, di (lm)
Standard
deviation (lm)
Surface
roughness, Ra
(lm)
Heating length,
Lh (mm)
Temperature measuring
position, Lm (mm)
920
308
86
181.5
179.3
96.3
1260
550
270
920.1
308.4
85.6
3.02
2.74
1.28
0.704
0.685
0.135
78.14
82.67
32.55
28.6
19.5
9.2
and 308.4 lm, and optical microscope (OM) for tube with inner
diameter of 920.1 lm. Fig. 1 shows the sample enlarged photographs of the cross-section view of the tubes. For reducing the measurement uncertainties, seven tubes were bundled together, cut and
ground to have smooth cross section surface. Each tube diameter
was measured and all values were averaged to obtain the average
tube diameter. The tubes inside surface roughness were measured
by atomic force microscope (AFM) for 85.6 lm tube and by surface
texture measuring instrument for 308.4 lm and 920.1 lm tubes.
Table 1 gives the detail dimensions and surface roughness of these
tubes. The Detail drawing of the corresponding tube dimensions is
shown in Fig. 2.
The schematic diagram of the test facilities is shown in Fig. 3.
High pressure air flows from a storage tank through a regulator
to the test section. The inlet air temperature was measured by a
resistance temperature detector (RTD). A differential pressure
transducer was installed on both ends of the test tube to measure
the flow pressure drop. A mass flow meter was connected after the
test tube to measure the flow rate of the working fluid. DC power
was clapped on both ends of the test tube to heat the tube wall. The
DC voltage and current were measured by connecting an ampere
and a volt meter to the electrodes directly. The power input was
calculated by the product of measured current and voltage.
Tube surface temperature was measured by the LCT method
that proposed by Lin and Yang [13] for avoiding the thermocouple
wire thermal shunt effect and will be described in the next section.
The heating length and temperature measuring positions are
shown in Fig. 2 for each tube and there values are also listed in Table 1. The measuring position was designed to be longer than the
maximum theoretical laminar flow entrance length. But because
of the experimental space limitation, the length for 920 lm tube
is slightly shorter than its theoretical entrance length. The experimental apparatus and derived parameters uncertainties are listed
in Table 2.
Since the heat transfer coefficient of gaseous flow is low, the
heat loss by natural convection from outside of the test section
may be important in the heat transfer measurement. For minimizing the heat loss, the test section was enclosed in a vacuum chamber. The chamber was evacuated by a vacuum pump before test to
maintain the inside pressure below 13 mTorr. The heat loss was
calibrated for each tube before its heat transfer performance test.
The test tube was heated inside the vacuum chamber without
working fluid through it. It was maintained at the temperature that
same as it was expected for the heat transfer performance test by
adjusting the power input. The power input thus can be treated as
the heat loss that would be resulted in the heat transfer performance test.
2.2. LCT temperature measurements
Fig. 2. Detail drawing of the test tube.
The LCT method that proposed by Lin and Yang [13] was used in
this study to measure the surface temperature of microtubes. For
C.-Y. Yang et al. / Experimental Thermal and Fluid Science 37 (2012) 12–18
15
Fig. 3. Schematic diagrams of the test facilities.
h¼
Table 2
Uncertainties of the experimental apparatus and derived parameters.
Apparatus
Uncertainties
Calibration range
RTD (°C)
T type thermocouple (°C)
Differential pressure
transducers
Pressure transducer
Mass flow meter
±0.1
±0.2
±0.075%
Twx (LCT) (°C)
0.5
0–100
0–100
0–10 kPa, 0–500 kPa and 0–
9 MPa
0–2 MPa
0–100 SCCM, 0–5 SLM, 0–50
SLM
28–43
±0.4%
±0.6%
q
AðT wx T x Þ
where A is the heat transfer area, A = pdiLh, di is the tube inside
diameter. Twx is the local inside tube surface temperature that can
be derived from the LCT measured outside surface temperature by
the method of one-dimensional heat conduction analysis. The temperature difference between the inside and the outside wall was
calculated as less than 0.03 °C which is among the experimental
uncertainty range. The Reynolds number and Nusselt number are
defined as the following:
Derived parameters
86 lm
308 lm
920 lm
Friction coefficient (f) (%)
Nusselt number (Nud) (%)
Reynolds number (Red) (%)
2.0–8.0
7.3–27.0
0.7–4.0
0.9–6.1
6.4–26.5
0.3–3.0
0.8–10.0
6.0–14.4
0.3–4.8
increasing the accuracy of temperature measurement, two thermochromic liquid crystals (TLCs) with 5 °C band width from 28–33 °C
and 38–43 °C were used. The diameters of the encapsulated TLCs
are from 5 to 15 microns. The TLCs was painted on the tested surface with thickness of approximately 30 lm. A black paint was also
painted under the TLCs as the background for improving the color
resolution by absorbing un-reflected light.
The relation between the hue value and temperature was calibrated in a constant temperature box. Electrical heating wires were
attached on inside surfaces of the box to maintain the entire box
space at the designated temperatures. Seven T-type thermocouples
were evenly placed near the test tube in the box to measure its
temperature distribution. The Liquid Crystal Thermograph and
temperature measured by thermocouples were recorded simultaneously. The temperature uniformity in the constant box at different temperature can be maintained within ±0.2 °C. The detail
process and uncertainty of the LCT temperature measurement
was described in Lin and Yang [13]. The standard deviation for
the calibrated temperature-hue curve was evaluated within
±0.5 °C.
Red ¼
The heat transfer rate q, was measured from the DC power input
deducted by the corresponding heat loss calibrated. It equals to the
increased enthalpy of air flow. Since the electrical power was
added uniformly on the tube surface, the local air temperature,
Tx, at the position x from the heating entrance can be estimated by:
x
_ p ðT x T i Þ
¼ mc
Lh
ð1Þ
_ is the air flow rate, Lh is the tube heating length and Ti is
where m
the air inlet temperature. From the Newton’s Law of cooling,
q00 ¼
q
¼ hðT wx T x Þ
A
The local heat transfer coefficient h can be derived as:
ð2Þ
Gdi
and Nud ¼
l
hdi
kf
ð4Þ
_ c , Ac is the tube cross-section
where G is the air mass flux, G ¼ m=A
area.
Since the tubes are small, the tube wall thickness is comparable
with the inside diameter, the heat conduction in the wall along axis
direction may be important. This axial conduction was estimated
by the method of Maranzana et al. [16]. The results show that
the ratio of axial conduction to the tube inside convection is less
than 0.02 for all tubes and thus can be neglected.
3. Results and discussions
3.1. Friction coefficients
The total flow pressure drop, Dpt was measured at the condition
of no heating power added. The frictional pressure drop Dpf was
evaluated by deducted the inlet (Dpi), exit (Dpe) and acceleration
(Dpa) terms from the measured total pressure drop (Dpt).
Dpf ¼ Dpt Dpi Dpe Dpa
ð5Þ
Dpi, Dpe and Dpa were calculated by following those suggested by
Kays and London [16]:
Dpi ¼
G2
ð1 r2 þ K c Þ
2q i
ð6Þ
Dpe ¼
G2
ð1 r2 K e Þ
2qe
ð7Þ
2.3. Data reduction
q
ð3Þ
Dpa ¼
G2
qi
qi
1
qo
ð8Þ
where G is the air mass velocity, r is the test section connectors’
contraction ratio and Kc and Ke are the entrance and exit loss coefficients which can also be obtained from Kays and London [17].
Since the Kn numbers range in the present study, 2.8 105 to
2.28 106, is far below the slip-continuum flow boundary (103)
that suggested by Beskok and Karniadakis [18], continuum flow
condition was considered in the present study. The friction coefficients can be derived directly from the Darcy’s equation listed
below.
16
C.-Y. Yang et al. / Experimental Thermal and Fluid Science 37 (2012) 12–18
heat loss by natural convection from outside of the tube for long
tubes with L di, the flow can be considered as on the isothermal
condition. Shapiro [19] proposed a theoretical equation for calculating friction coefficient that includes the effect of flow compressibility as:
1
Laminar (16/Red)
Turbulent (Blasius)
86 µm
308 µm
920 µm
0.1
f ¼
f
1000
10000
3.2. Effect of heat loss
Red
Fig. 4. Variation of friction coefficients versus Reynolds number.
f ¼
ð10Þ
The friction coefficients evaluated by the above equation are
shown in Fig. 5. It shows that the friction coefficients in turbulent
flow regime for all tubes agree reasonably with those predicted by
the Blasius equation. The above comparison shows that the acceleration (Dpa) terms from Kays and London [17] (Eq. (8)) over estimated the effect of momentum change for a large pressure
variation flow condition. The frictional coefficient of gas flow in
microtube is the same as that in the conventional larger tubes if
the compressibility effect is evaluated by the method that proposed by Shapiro [19].
0.01
0.001
100
Dh p2i p2o
pi
2
ln
4L G2 RT
po
Dpf 2q di
G2 4L
ð9Þ
Fig. 4 shows the variation of friction coefficients versus Reynolds
number for each tube. It clearly shows that the laminar to turbulent
transition Reynolds number is around of 2200 for all tubes. This is
the same as that for conventional larger tubes. In laminar flow regime, the friction coefficients can be well predicted by Poiseuille
theory (f = 16/Red). In turbulent flow regime, the friction coefficients for 920 lm tube still agree well with those predicted by
the Blasius equation. However, for 308 lm tube, the friction factor
departed from the Blasius prediction values while Reynolds higher
than 10,000. The discrepancy increases with increasing Reynolds
number. For the smallest tube, 86 lm, the friction factors are significantly lower than those predicted by Blasius equation. These
results are the same as those tested by Yang et al. [9].
Since the pressure drop was tested under no heating condition,
and the viscous shear heating was much lower than the possible
Since the heat transfer coefficient of air flow is low, the heat loss
by natural convection from outside of the test tube may not be neglected in the heat transfer measurement. The ratio of heat loss to
the input heating power at various flow rates for each tube has
been measured and shown in Fig. 6. The ratios varied from 0.76%
to 46.6% depending on the air flow rate and the tube size. From
the fundamentals of convective heat transfer, the in-tube forced
convection heat transfer coefficient increases with increasing Reynolds number but with decreasing tube diameter. The outside natural convection heat transfer coefficient is almost independent of
tube diameter. Therefore, the heat loss ratio is smaller for smaller
tube and decreases with increasing Reynolds number as that
shown in Fig. 6.
3.3. Heat transfer coefficients
The heat transfer rate was measured from the DC power input
deducted by the corresponding heat loss calibrated. The derived
Nusselt numbers for each tube at various Reynolds number are
1.0
1
920 µm
308 µm
86 µm
Laminar (16/Red)
Turbulent (Blasius)
86 µm
308 µm
920 µm
0.1
f
qloss /q
0.1
0.01
0.0
0.001
100
1000
10000
Red
Fig. 5. Friction factors in microtubes evaluated by the Shapiro [19] equation.
100
1000
10000
Red
Fig. 6. Ratio of heat loss to total heat transfer.
C.-Y. Yang et al. / Experimental Thermal and Fluid Science 37 (2012) 12–18
100
17
taken consideration. The conventional heat transfer correlation
for laminar and turbulent flow can be well applied for predicting
the fully developed air flow heat transfer performance in microtubes. If we combine the present results and the results by Yang
and Lin [14] for water flow, we may conclude that the conventional
friction and heat transfer correlations can be well applied for both
gas and liquid flow in microtubes ranged from 86 to 920 lm. There
is no significant size effect for air flow in tubes within this diameter
range.
Furthermore, since the heat transfer coefficient of gaseous flow
is low, the heat loss by natural convection from outside of the test
tube is not negligible in the heat transfer measurement. The heat
loss percentage is smaller for smaller tube and decreases with
increasing Reynolds number.
Laminar (Nud = 4.36)
Turbulent (Gnielinski [1976])
Turbulent (Gnielinski [1995])
Nud
86 µm
308 µm
920 µm
10
Acknowledgment
1
100
1000
10000
Red
The study was financially supported by the National Science
Council under Grant No. NSC 98-2221-E-008-088-MY3.
References
Fig. 7. Nusselt number versus Reynolds number for all tubes.
shown in Fig. 7. Morini et al. [20] suggested that for flow in microchannels, the low values of the inner diameter limit the significance of the Grashof number (which depends on the third power
of the inner diameter) and, hence, of mixed convection. Gnielinski,
[21] proposed a series of correlations for the prediction of the Nusselt number for pure forced convection. The prediction values by
Gnielinski [21] and the original Gnielinski correlation [22] for conventional sized tubes were also plotted in Fig. 7 for comparison.
The result shows that in the turbulent regime, the conventional
Gnielinski [22] correlation is able to well predict the present test
results for the 920 lm tube. However, for the 86 lm tube, the
Gnielinski [21] correlation for pure force convection provides a
better prediction. This agrees with that suggested by Morini et al.
[20].
For the flow in laminar regime, Nusselt numbers agree well
with the theoretical constant heat flux value, 4.36. For Reynolds
numbers greater than 1000, the heat transfer coefficients increase
with increasing Reynolds numbers. This shows that the tubes
length is not long enough for fully developed flow at high Reynolds
number conditions and the flow is still in the developing regime.
The thermal entrance length is longer than that estimated by the
correlation from Incropera et al. [23]. This is in agreement with
those tested by Yang and Lin [14] for water flow.
The above test results show that the conventional heat transfer
correlation for large tubes can be well applied for predicting the
heat transfer performance of air flow in microtubes in both laminar
and turbulent flow regime. Furthermore, if we combine the test
results by Yang and Lin [14] for water flow, we may conclude that
the conventional heat transfer correlation can be well applied for
predicting the heat transfer performance of both air and water flow
in microtubes ranged from 86 to 920 lm. There is no significant
size effect for air flow in tubes within this diameter range.
4. Conclusions
This study provides an experimental investigation on the pressure drop and heat transfer performance of air flow through microtubes with inside diameter of 86–920 lm. The experimental
results show that the frictional coefficient of gas flow in microtube
is the same as that in the conventional larger tubes if the effect of
gaseous flow compressibility that proposed by Shapiro [19] was
[1] P. Wu, W.A. Little, Measurement of friction factor for the flow of gases in very
fine channels used for microminiature Joule–Thomson refrigerators,
Cryogenics May (1983) 272–277.
[2] P. Wu, W.A. Little, Measurement of the heat transfer characteristics of gas flow
in fine channel heat exchangers used for microminiature refrigerators,
Cryogenics 24 (1984) 415–420.
[3] B. Choi, R.F. Barron, R.O. Warrington, Fluid flow and heat transfer in
microtubes, Micromechanical Sensors, Actuators, and Systems ASME 32
(1991) 123–134.
[4] D. Yu, R. Warrington, R. Barron, T. Ameel, An experimental and theoretical
investigation of fluid flow and heat transfer in microtubes, ASME/JSME
Thermal Engineering Conference 1 (1995) 523-530.
[5] T.M. Adams, S.I. Abdel-Khalik, S.M. Jeter, Z.H. Qureshi, An experimental
investigation of single-phase forced convection in microchannels,
International Journal of Heat and Mass Transfer 41 (1998) 851–857.
[6] Gh.M. Mala, D.Q. Li, Flow characteristics in microtubes, International Journal of
Heat and Fluid Flow 20 (1999) 142–148.
[7] G.P. Celata, M. Cumo, M. Guglielmi, G. Zummo, Experimental investigation of
hydraulic and single phase heat transfer in 0.130 mm capillary tube,
Microscale Thermophysical Engineering 6 (2002) 85–97.
[8] Z.X. Li, D.X. Du, Z.Y. Guo, Experimental study on flow characteristics of liquid in
circular microtubes, Microscale Thermophysical Engineering 7 (2003) 253–
265.
[9] C.-Y. Yang, J.-C. Wu, H.-T. Chien, S.-R. Lu, Friction characteristics of water, R134a, and air in small tubes, Microscale Thermophysical Engineering 7 (2003)
335–348.
[10] T.-H. Yen, N. Kasagi, Y. Suzuki, Forced convective boiling heat transfer in
microtubes at low mass and heat fluxes, International Journal of Multiphase
Flow 29 (2003) 1771–1792.
[11] D. Lelea, S. Nishio, K. Yakano, The experimental research on microtube heat
transfer and fluid flow of distilled water, International Journal of Heat and
Mass Transfer 46 (2004) 149–159.
[12] S. Grohmann, Measurement and modeling of single-phase and flow-boiling
heat transfer in microtubes, International Journal of Heat and Mass Transfer 48
(2005) 4073–4089.
[13] T.-Y. Lin, C.-Y. Yang, An experimental investigation on forced convection heat
transfer performance in micro tubes by the method of liquid crystal
thermography, International Journal of Heat and Mass Transfer 50 (2007)
4736–4742.
[14] C.-Y. Yang, T.-Y. Lin, Heat transfer characteristics of water flow in micro tubes,
Journal of Experimental Thermal and Fluid Science 32 (2007) 432–439.
[15] G.P. Celata, M. Lorenzini, G.L. Morini, G. Zummo, Friction factor in micropipe
gas flow under laminar, transition and turbulent flow regime, International
Journal of Heat and Fluid Flow 30 (2009) 814–822.
[16] G. Maranzana, I. Perry, D. Maillet, Mini- and micro-channels: influence of axial
conduction in the walls, International Journal of Heat and Mass Transfer 47
(2004) 3993–4004.
[17] W.M. Kays, A.L. London, Compact Heat Exchangers, third ed., McGraw-Hill,
Singapore, 1984. pp. 36 and pp. 111.
[18] S. Beskok, G.E. Karniadakis, A model for flows in channels, pipes, and ducts at
micro and nanoscales, Microscal Thermophysical Engineering 3 (1) (1999) 43–
77.
[19] A.K. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow,
vol. 1–2, John Wiley, 1953.
[20] G.L. Morini, M. Lorenzini, S. Salvigni, G.P. Celata, Experimental analysis of
microconvective heat transfer in the laminar and transitional regions,
Experimental Heat Transfer 23 (2010) 73–93.
18
C.-Y. Yang et al. / Experimental Thermal and Fluid Science 37 (2012) 12–18
[21] V. Gnielinski, Ein Neues Berechnungsverfahren fur die Warmeubertragung im
Ubergangsbereich Zwischen Laminaren und Turbulenter Rohstromung,
Forschung im Ingenieurwesen–Engineering Research 61 (1995) 240–248
(quote in Mirini et al. [19]).
[22] V. Gnielinski, New equation for heat and mass transfer in turbulent pipe and
channel flow, International Journal of Chemical Engineering 16 (1976) 359–368.
[23] F.P. Incropera, D.P. Dewitt, T.L. Bergman, A.S. Lavine, Fundamentals of Heat and
Mass Transfer, John Wiley & Sons, New York, 2007. pp. 492.