Ting-Yu Lin
e-mail: [email protected]
Satish G. Kandlikar1
Fellow ASME
e-mail: [email protected]
Thermal Analysis, Microfluidics,
and Fuel Cell Laboratory,
Department of Mechanical Engineering,
Rochester Institute of Technology,
76 Lomb Memorial Drive,
Rochester, NY 14623
Heat Transfer Investigation
of Air Flow in Microtubes—Part I:
Effects of Heat Loss, Viscous
Heating, and Axial Conduction
Experiments were conducted to investigate local heat transfer coefficients and flow characteristics of air flow in a 962 lm inner diameter stainless steel microtube (minichannel).
The effects of heat loss, axial heat conduction and viscous heating were systematically
analyzed. Heat losses during the experiments with gas flow in small diameter tubes vary
considerably along the flow length, causing the uncertainties to be very large in the
downstream region. Axial heat conduction was found to have a significant effect on heat
transfer at low Re. Viscous heating was negligible at low Re, but the effect was found to
be significant at higher Re. After accounting for varying heat losses, viscous heating and
axial conduction, Nu was found to agree very well with the predictions from conventional
heat transfer correlations both in laminar and turbulent flow regions. No early transition
to turbulent flow was found in the present study. [DOI: 10.1115/1.4007876]
Keywords: heat transfer coefficient, microtube, heat loss, axial heat conduction, viscous
heating, minichannel
1
Introduction
Heat transfer to gas flow in small diameter tubes is important in
a number of MEMS devices. To design these microdevices properly, a fundamental understanding of heat transfer and fluid flow
characteristics in small diameter channels is essential. Although
there is no reason for heat transfer to be affected by diameter at
these scales, large deviations from theory are reported in existing
literature on experimental data for gas flow in microtubes. The
study is aimed at providing quantitative information on the sources for such discrepancies.
Heat transfer research in microchannels was first conducted by
Tuckerman and Pease [1] in 1981. They demonstrated that the planar integrated circuit chips can be effectively cooled by a laminar
flow of water through microchannels with hydraulic diameters of
86–95 lm. Their work was followed by others who explored the
heat transfer and fluid flow characteristics in microchannels. For liquid flow in smooth microchannels, the available literature clearly
indicates that friction factors are in good agreement with the conventional (macroscale) correlations [2,3]. A number of studies investigated the heat transfer characteristics of liquid flows in
microchannels [3–10]. The earlier studies revealed that the heat
transfer coefficients can deviate significantly from the predictions of
conventional correlations [7–10] due to experimental uncertainty,
while some of the recent studies indicated that the heat transfer can
be predicted well by conventional correlations for liquid flow [3–6].
A few studies on gas flow characteristics in microchannels
found that due to roughness effects, the friction factor was higher
than in conventional correlations [11,12]. Peiyi and Little [11]
studied trapezoidal channels with a hydraulic diameter in the
range of 45–83 lm, while Tang et al. [12] investigated circular
tubes with smooth and rough surfaces. The friction factors in both
studies were in good agreement with predictions from conventional correlations for smooth tubes, but were higher than the predictions for rough tubes. Microchannels with hydraulic diameters
1
Corresponding author.
Contributed by the Heat Transfer Division of ASME for publication in the
JOURNAL OF HEAT TRANSFER. Manuscript received September 7, 2011; final
manuscript received May 25, 2012; published online February 11, 2013. Assoc.
Editor: Jose L. Lage.
Journal of Heat Transfer
on the order of 1–10 lm were investigated by a few investigators
who found that the friction factors were lower than the predictions
from conventional theory due to rarefaction effects [13–18]. Kohl
et al. [19] reported that their friction factors were in good agreement with the predictions from the conventional theory for gas
flow in 25 lm channel in laminar flow and for 100 lm channel in
turbulent flow. A delayed transition to turbulent flow was
observed with the transition occurring at a Re of 6000 for smooth
tubes. Turner et al. [20] indicated that for hydraulic diameters
higher than 4.7 lm and Kn < 0.04, the friction factors agreed well
with continuum theory. The friction factors for rough channels
with a relative roughness of up to 6%, were also found to be the
same in laminar flow.
There are very few studies reported in literature on turbulent
flow, due to difficulties encountered by the higher pressure drops
in the experimental setup design. Similarly, in the case of gas flow
in microtubes, there are very few experimental studies reported in
literature for heat transfer due to complexities in heat transfer
experiments at microscale [21]. Acosta et al. [22] applied heat and
mass transfer analogy and studied mass transfer coefficients in
narrow channels by applying the limiting current method. Choi
et al. [14] and Wu and Little [23] indicate that the heat transfer
coefficients in microchannels were significantly higher than predictions from the conventional correlations, and that the Colburn
analogy was not valid. The actual reasons for this discrepancy
will be the focus of this work.
From the reviewed literature, it can be summarized that for liquid flow in microchannels there are no differences between experimental data and the theoretical predictions. With channel
diameters greater than 15 lm, friction factors agree well with conventional correlations. Similarly, available data for channel diameters greater than 0.1 mm shows that heat transfer coefficients are
also in good agreement with predictions from conventional correlations. For gas flow in microchannels with hydraulic diameters
smaller than about 7 lm at atmospheric pressure, friction factors
are found to be lower than the conventional theory due to rarefaction effects. Yu et al. [24] reported that their friction factor data of
water and gas flows in 52–102 lm channels were similar and
lower than predictions. However, Judy et al. [2] indicated that in
this diameter range, the data for water were in good agreement
C 2013 by ASME
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with conventional correlations. Since the friction factor data of N2
gas by Yu et al. [24] were the same as that of water, microscale
effects were certainly not expected for liquid flow in microchannels within that diameter range [3,4,6,25]. Some other effects,
such as heat loss, axial conduction and viscous dissipation, may
not be important at macroscale and/or in incompressible flow.
However, these effects may become important at microscale, Heat
loss is another important consideration in heat transfer at microscale, and not accounting for it correctly is believed to be one of
the major reasons for the discrepancies found in experimental data
reported in literature. Axial conduction was indicated to have an
important effect; although it was not quantitatively assessed and
corrected in comparisons presented by previous investigators.
Regarding viscous dissipation, accurate formulations are available
for laminar flow, although there are no correlations available for
turbulent flow.
With the above shortcomings in the available literature in mind,
the present study is undertaken with the following objectives:
(i) Experimentally investigate the heat transfer coefficient of
compressible air flow in 962 lm diameter smooth tube.
(ii) Carefully evaluate the effects of heat loss, axial conduction
and viscous dissipation and compare the results with the
theoretical predictions in laminar and turbulent regions.
2
Experimental Setup
Figure 1(a) shows a schematic of the present experimental setup.
The test facility was an open system with the outlet exposed to
atmosphere. A high-pressure air bottle and a pressure regulator capable of regulating pressures up to 20 MPa were used to supply air
flow in the tube. Mass and volume flow meters with flow rate
ranges of 100, 10, and 1 standard liter per minute, and accurate to
within 0.8% reading were used to measure the flow rates. Pressure
drop of fluid flowing through the tube was measured by two different working range pressure sensors to increase the accuracy.
Pressure sensors were calibrated by a pressure calibrator with an
Fig. 2
Schematic drawing of test section assembly
accuracy of 52 Pa. The inlet air temperature was measured by resistance temperature detectors (RTD) with an accuracy of 0.1 C.
Fine wire 36 gage E type thermocouples were attached on the tube
to measure the outside wall temperature. Thermocouples were calibrated to be within an uncertainty of 0.1 C. A high current dc
power supply with 0–100 A and 0–20 V was used to supply power
to the stainless steel tube. The test section was enclosed in a vacuum chamber to reduce the natural convection heat losses. Flow
meters, thermocouples, a pressure transducer and a power supply
were interfaced with a LabView program for data acquisition. The
inlet pressure was measured by a pressure sensor mounted near the
inlet as shown in Fig. 1(b). The outlet was at atmospheric pressure
as the flow was discharged into the atmosphere after exiting the test
section.
A schematic drawing of the test tube assembly enclosed in a
vacuum chamber is shown in Fig. 2. The two ends of the test tube
were bonded to low thermal conductivity Garolite extensions with
epoxy. One end of the test section was connected to a highpressure compression tube fitting for fluid inlet while the other
side was exhausted to the atmosphere. Two copper connectors
were soldered to the two sides of the stainless steel test tube for
power supply wires following the same design as reported by
Kandlikar et al. [26]. The test section was heated directly by a dc
power supply. Three E-type thermocouples were attached on the
outer surface of the tube at uniform spacing to measure the surface
temperatures. All thermocouple wires and electric wires inside the
vacuum chamber were connected to data acquisition system and
power supply through vacuum feed-through national pipe thread
fittings. The tube cross-section image taken by an optical microscope and the internal tube surface image taken by a laser confocal scanning microscope are shown in Fig. 3. The average tube
diameter is 962.0 lm which is calculated from the tube flow area
of several individual cross-section images. The surface roughness
was determined from the images to be Ra ¼ 0.8 lm.
3
Data Reduction
In the present study, the friction factor and Nusselt number
were calculated to obtain the fluid flow and heat transfer performance data for air flow in a microtube. The pressure drop and mass
flow rate in the microtube were measured to calculate the friction
factor. Neglecting the entrance region effect as the tube diameter
to length ratio is quite large (343), the friction factor is calculated
from the following equation:
f ¼ Dpf
Dh qavg
L 2G2
(1)
where DPf is the pressure drop due to friction and qavg is the average
of inlet and outlet fluid densities, and Dh, L, and G are the hydraulic
diameter, tube length and mass flux, respectively. In compressible
flow, the measured total pressure drop DPmea is the sum of friction
pressure drop DPf and acceleration pressure drop DPa
Fig. 1 (a) Schematic of the experimental setup for heat transfer
and pressure drop investigation in a microtube and (b) 3D drawing of measurement equipment mounting
DPmea ¼ DPf þ DPa
(2)
The acceleration pressure drop DPa is given by
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while in turbulent flow with Re higher than 2300 the viscous heating is given by
2
qviscous ¼ plu2m L
7
(9)
The input heat to the fluid qin is calculated by the following
equation:
qin ¼ qmea þ qaxial þ qviscous qloss
(10)
where qmea is the measured total heat input by the DC power supply and qloss is estimated by a separate experiment as described in
the next section.
4
Fig. 3 Test tube cross-section viewed from an optical microscope and the internal surface image obtained from a laser confocal scanning microscope, Ra 5 0.8 lm
DPa ¼
G2 1
1
2 q2 q1
(3)
where q1 and q2 are the fluid densities at the inlet and outlet sections, respectively. The friction factor can also obtained from the
equation for 1D isothermal compressible flow given by [27]
Dh P21 P22
P1
(4)
2
ln
f¼
L RTG2
P2
where P1 and P2 are the inlet and outlet pressures, respectively.
The local heat transfer coefficient and Nusselt number are calculated by the following equations:
h¼
q00
Tw;x Tf;x
and Nu ¼
hDh
kf
(5)
where Tw,x is the internal tube wall temperature that can be
derived from the external wall temperature measurement using a
one-dimensional, steady-state heat conduction equation with heat
generation [26]. A detailed derivation of this equation is given in
Appendix A. Tf,x is the local fluid bulk mean temperature calculated from the energy balance for the compressible flow as shown
in Appendix B. The local input heat flux q00 is calculated as
follows:
q00 ¼
qmea qloss
Ah
Results
4.1 Heat Loss Estimation. Since the heat input needed during an actual test is quite small, on the order of a few Watts, heat
losses constitute a large portion of the heat supplied in the present
study. Therefore, it is essential to reduce the heat losses from the
system. This was accomplished by enclosing the test section in a
vacuum chamber. Heat losses are determined by heating the tube
without any fluid flowing through the test section. The tube was
heated directly by the power supply and the applied power was
quantified as the heat loss corresponding to the difference between
the local wall surface and the ambient temperature.
Figure 4 shows DT (difference between the wall and the ambient temperature) versus qloss for different levels of vacuum in the
chamber. It is seen from Fig. 4 that qloss increases with increasing
DT. As expected, qloss decreases with a decrease in the vacuum
chamber pressure. At DT ¼ 40 C, the heat loss values at pressures
of 760 Torr (atmospheric pressure) and 0.008 Torr are 1.7 W and
0.7 W, respectively. The heat loss values at pressures of
0.012 Torr and 0.008 Torr are the same, which indicates that the
heat loss at the lower pressures is due to radiation. Further reduction in pressure did not lower the heat losses.
Heat loss is a function of wall temperature as shown in Fig. 4,
while the wall temperature during a test with fluid flow is a function of the distance from the entrance. It was therefore essential to
calculate the local heat loss separately near each wall thermocouple location. To achieve this, heat loss qloss versus the temperature
difference DT at three different locations at distance Lh from the
entrance was recorded as shown in Fig. 5. The heat loss estimates
in the actual heat transfer experiments were then calculated using
this plot. As shown in the figure there is no significant difference
in the heat loss estimation at these locations. The heat loss is thus
a function of the local temperature alone for all three locations.
(6)
where qmea, qloss, and Ah are the measured power from the power
supply, the heat loss, and the heat transfer area. Axial conduction
is calculated by the following equation. Details of the axial conduction estimation are given by Lin and Kandlikar [28]
qaxial ¼ kw A
Tw;x Tw;1
Lh;x
(7)
Viscous heating is calculated using the detailed equations given in
Appendix C. In laminar flow with Re less than 2300 the viscous
heating is given by
qviscous ¼ 16plu2m L
Journal of Heat Transfer
(8)
Fig. 4 Comparison of heat losses from the test section at different vacuum levels
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Fig. 5 Heat loss versus wall to ambient temperature difference
for three test sections with different heating lengths
In an actual experiment, since the wall temperature increases
along the tube length, the heat losses also increase accordingly.
To account for the heat losses during a test at a given location,
this varying heat loss needs to be integrated along the tube length.
To accomplish this, the test section was divided into three zones
and the heat losses were calculated for each zone using the following equation:
h
i L L
h;n
h;n1 =2
(11)
qloss;n ¼ qloss;DTðn1Þ þ qloss;DTðnÞ
Lh;total
where qloss, DT(n1) is the heat loss calculated from the local temperature difference DTn1 at location n 1.
The heat input to the fluid in the zone located between Lh,n1
and Lh,n is given by
Lh;n Lh;n1
qin;n ¼ qin;n1 þ qmea
qloss;n
(12)
Lh;total
Figure 6 shows qloss/qin as a function of Re at three different locations along the test section length. It is observed that qloss/qin
increases with increasing distance Lh. This is because of the
higher local wall temperatures that result in higher heat losses at
downstream locations. The qloss/qin decreases with an increase in
Re, as the heat losses become a smaller fraction of the heat carried
away by the fluid.
Fig. 7 Comparison of local net heat flux along the test tube
with and without considering heat losses for Re 5 4700 and
Re 5 2100
Even though the test tube was uniformly heated by the power
supply, a constant heat flux assumption needs to be modified due to
the uneven heat loss in the different zones of the test tube. Figure 7
shows the calculated local net heat flux along with the heating
length with and without considering heat loss at different Re (4700
and 2100). The net heat flux along the tube is a constant value without considering the heat loss, while the local net heat flux value
decreases with increasing Lh due to the heat losses. The local heat
flux values without considering heat losses at Re ¼ 4700 and 2100
are 4.01 kW/m2 and 2.00 kW/m2, respectively, while the local net
heat flux values considering different heat losses for Lh ¼ 25 mm,
125 mm, and 229 mm at Re ¼ 4700 are 3.81 kW/m2, 3.52 kW/m2
and 3.17 kW/m2, respectively. The corresponding values at
Re ¼ 2100 are 1.77 kW/m2, 1.48 kW/m2 and 1.10 kW/m2, respectively. The difference in heat flux values with and without considering heat loss varies from about 5% to 21% at Re ¼ 4700, and 12%
to 45% at Re ¼ 2100. These differences are large enough to make
them necessary to be accounted for in the calculation for local net
wall heat flux.
4.2 Uncertainty Analysis. To assess the accuracy of the
measurement, an uncertainty analysis was performed. The calculated parameters such as Nu, Re, G and f are generally denoted as
dy. They are functions of variables x1, x2, … … xn. The uncertainty in a parameter is determined as described below
y ¼ f ðx1 ; x2 ; … … xn Þ;
"
2
2 #1=2
@y
@y
dx1 þ… … …
dxn
dy ¼
@x1
@xn
Fig. 6 Comparison of heat loss, qloss, normalized by the input
heat, qin, as a function of Re at three different heating lengths
(13)
where dx1, dx2, … … dxn are the uncertainties in the independent
variables.
The uncertainties in the experimental parameters are listed in
Table 1. The uncertainty in diameter was calculated using 14 sets
of diameter measurements taken at different locations in the tube
sample. A laser confocal scanning microscope was used to measure the internal surface roughness. From these measurements, the
average tube diameter was 962 lm and the uncertainty was calculated as 1.28 lm. For high Re conditions, flow rates, input power,
and the wall to fluid temperature difference (Tw,x – Tf,x) are high,
so the uncertainty is low for these parameters. For low Re conditions, however, the uncertainties are high.
Figure 8 shows the calculated Nu uncertainty as a function of
Re. As expected, the uncertainty in Nu increases with decreasing
Re. In this study, the local wall temperatures were maintained at
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Table 1
Uncertainties in the experimental parameters
Uncertainty
Nu (%)
Tf,x ( C)
Tw,x ( C)
Re (%)
d (lm)
G (%)
m_ (%)
DP (%)
f (%)
(Turbulent, laminar)
Lh ¼ 25 mm
Lh ¼ 125 mm
Lh ¼ 229 mm
5–8, 10–23
4–9, 12–103
6–17, 28–516
0.1–0.2, 0.2–0.8
0.1–0.4, 0.6–3.2
0.2
1–7
1.28
1–7
0.9–6
1–9
3–16
0.2–0.9, 1.4–7.8
Fig. 9 Comparison of friction factor data with theoretical
predictions
Fig. 8 The variation of uncertainty in Nu as a function of Re at
three heating lengths
similar values for each flow rate by controlling the input power.
This is done mainly to eliminate the effect of variations in heat
losses, which depend on the local wall temperatures. The heat
input and heat flux values are therefore higher at higher Re. In
laminar flow, the heat transfer coefficient is constant and independent of Re, which causes (Tw,x – Tf,x) to increase with heat
flux and Re. The low heat flux and (Tw,x – Tf,x) values in the low
Re case thus lead to higher uncertainty in Nu.
The uncertainty in Nu is also a function of heating location distance Lh and increases proportionately. For large Lh, the wall temperature and heat losses are high, which increases the uncertainty in
the derived fluid temperature and Nu. At the location Lh ¼ 25 mm,
the Nu uncertainty is less than 23%. At Lh ¼ 125 mm and
Lh ¼ 229 mm, the Nu uncertainty is quite high and the experimental
data are not reliable. However, in the turbulent flow regime, the Nu
uncertainty at the three different locations was less than 20%.
4.3 Friction Factor. Figure 9 shows the comparison of friction factor data derived from Eqs. (1) and (4). There is no significant difference observed in the data derived by the two different
equations indicating little influence from rarefaction effects. In the
laminar flow, the data are in good agreement with Hagen–
Poiseuille flow values, while in the turbulent region, the deviation
of experimental data from Blasius and Filonenco equations is less
than 10% and is considered as a good agreement with the conventional correlations. Comparing the slopes of the f-Re plots, it is concluded that for Re less than 2000 the flow is laminar and that earlier
transition from laminar to turbulent flow is not observed. In the
laminar flow, the friction factor is slightly higher than predicted at
higher values of Re. This is due to higher entrance region effects.
Journal of Heat Transfer
Fig. 10 Comparison of experimental data for Nu for 962 diameter tube with available correlations after accounting for heat
losses, viscous heating and axial conduction; inner diameter of
microtube 5 962 lm and Lh 5 25 mm
4.4 Heat Transfer Coefficient. Figure 10 shows Nu as a
function of Re for the 962 lm diameter tube at Lh ¼ 25 mm from
the entrance. It is seen that in turbulent flow, the data can be predicted very well by the Petukhov–Gnielinski correlation. In laminar flow, Nu agreeed well with the theoretical laminar fully
developed flow value of 4.36. In higher Re range of laminar flow,
Nu was slightly higher than the laminar fully developed flow
value. This was due to the flow being in the developing region.
The entrance length for the tube diameter of 962 lm corresponding to Re ¼ 2000 is about 67 mm, which is significantly higher
than the heating length Lh ¼ 25 mm. The transition flow was
observed at Re in the range 2000–3000.
It should be noted that the effects of heat loss, and axial conduction and viscous heating were accounted for in the data plotted
in Fig. 10. Without considering these factors, the data were not
well correlated with the conventional correlations.
4.5 Effect of Viscous Heating. Viscous heating effects are
usually negligible in macroscale systems or with liquid flow in
both microscale and macroscale systems due to the relatively low
velocities employed. During gas flow is microscale systems, however, the small hydraulic diameter results in a high velocity, and
consequently the thermal energy generated due to viscous heating
results in additional heat input to the fluid. Viscous heating
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Fig. 11 The comparison of viscous heating effect on Nu in
turbulent flow region using laminar flow equation, Eq. (8), and
turbulent flow equation, Eq. (9)
increases the fluid temperature and the calculated heat transfer is
altered accordingly.
To accurately evaluate the experimental heat transfer coefficient, it is essential to estimate the viscous heating effect. In laminar flow, viscous heating can be derived by considering the
laminar fully developed flow velocity profile and is also available
in literature [29]. However, there are no equations available in literature to account for viscous heating effects in turbulent flow. In
the present study, viscous heating in turbulent flow was estimated
from the proposed model using Eq. (9).
Figure 11 shows the Nu obtained by including viscous heating
from Eqs. (8) and (9) normalized by the theoretical Nusselt number, Nuth, as a function of Re for different heating lengths. Equation (8) represents the equation for viscous heating in laminar
flow, while Eq. (9) represents viscous heating in turbulent flow.
At the location Lh ¼ 25 mm, there was no significant difference
for the derived Nu by the two equations. This was due to the short
heating length and the lower magnitude of viscous heating. However, for all other lengths, using the laminar flow equation, Eq.
(8), in the turbulent flow region at higher Re results in significantly higher values of Nu due to overestimation of the viscous
effects. Using the turbulent flow equation, Eq. (9), in the turbulent
region results in Nu/Nuth to be close to 1 for all lengths.
The results shown in Fig. 11 indicate that the derived model in
the present study is applicable to estimate viscous heating in turbulent flow. Equation (9) was derived based on a 1/7 power law
profile as shown in Appendix C. Although the power exponent is
not constant but depends on the Reynolds number, the simple
power law equation is used in the present work for ease of calculations. Further refinements in this area are recommended.
4.6 Effect of Heat Loss. The heat losses constitute a large
fraction of the heat supplied and significantly affect the experimental uncertainty. Figure 12 shows the comparison of derived Nu data
with and without considering heat losses. In turbulent flow, the Nu
values evaluated with and without considering heat losses are very
close, implying that heat loss effects are negligible. However, in
laminar flow, the data without considering heat losses are significantly different from the predictions from the conventional theory.
The heat loss effect becomes more prominent in the laminar flow
region. Without considering heat losses, the heat input to the fluid
was overestimated, along with the heat flux, as illustrated in Fig. 7.
Similarly, the local fluid temperature was also overestimated, which
caused (Tw,xTf,x) to be lower. From Eq. (5) it is seen that heat
transfer coefficient is significantly overestimated at higher q00 and
Tf,x values.
Fig. 12 Comparison of Nu/Nuth with different heat loss correction equations, Eq. (8) based on laminar flow, and Eq. (9) based
on turbulent flow
Fig. 13 Effect of axial heat conduction in laminar and turbulent
regions and comparison of experimental data with Lin and
Kandlikar [28] model
4.7 Effect of Axial Conduction. The axial conduction effect
is not negligible in gas flow due to the low heat input and low
fluid heat capacity. This effect becomes more significant for
smaller diameter tubes, higher thermal conductivity tube materials, and lower RePr product according to the model presented by
Lin and Kandlikar [28]. Figure 13 compares the data without correcting for the axial heat conduction effects, and by including
them as described by model [28] and Eq. (14).
Nuk0
¼
Nuth
1þ4
1
kw Ah;s Nuth
(14)
kf Af ðRePrÞ2
where Nuk0 is Nusselt number neglecting the effect of axial heat
conduction, Nuth is the theoretical Nusselt number and Ah,s and Af
are the cross-sectional areas of the channel wall and the crosssectional area of fluid flow in the channel. It is seen from Fig. 13
that the axial conduction effect was negligible for Re higher than
1000 in the present study. For low Re, the Nuk0 is considerably
lower than the conventional theory and the value decreases at
lower Re.
By comparing the data in Fig. 13 to that in Fig. 10, it is seen
that the effects of axial heat conduction are small in the turbulent
flow region. In laminar flow, however, the effects of axial conduction are quite large, and the Nu decreases at lower Re if these
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Table 2
Re
16968
13297
9652
6788
4561
3081
Wall and fluid temperature variation along test section
Tw
Lh ¼ 25 mm
Tw
Lh ¼ 125 mm
Tw
Lh ¼ 229 mm
Tf
Lh ¼ 25 mm
Tf
Lh ¼ 125 mm
Tf
Lh ¼ 229 mm
30.3
32.1
34.8
36.2
36.6
36.8
38.2
42.3
48.7
52.7
54.1
54.4
45.1
51.3
61.1
67.4
69.6
69.8
26.3
26.9
27.8
28.4
28.7
30.1
34.2
37.0
41.5
44.7
46.0
46.2
42.1
47.1
55.0
60.6
62.7
62.6
effects are not accounted properly. For low Re, the Nu is significantly lower than the conventional laminar fully developed flow
theory. It is however, in good agreement with the prediction proposed by the Lin and Kandlikar [28] model. The temperature variation of the microtube at different conditions are listed in Table 2.
It shows the magnitude of temperatures gradients along the length,
as well as the higher temperatures encountered in the downstream
region for different flow conditions. The effect of axial heat conduction in the diameter range of the present study is therefore not
negligible, especially for low Re.
5
Conclusions
Heat transfer and friction factor of air flow in a smooth 304SS
962 lm inner diameter tube was investigated in this study. The
effects of heat loss, axial conduction and viscous heating were analyzed in detail. An uncertainty analysis was performed to evaluate the uncertainty in the data as a function measurement location
along the flow length and flow rate. Based on the experimental
data the following conclusions were drawn:
(1) Heat losses are a function of local wall temperature. Since
the wall temperature variation along the tube length
increases at lower Re for a given power input, the heat
losses are higher in the downstream section.
(2) For laminar flow at low Re, the wall temperatures in the
downstream region are higher resulting in large heat losses
and larger uncertainty, making data in this region unreliable. The experimental Nu uncertainty increases with
increasing distance (downstream), and with decreasing Re.
(3) Viscous heating effects increase with Re and are not negligible for high Re. A new equation is developed for viscous
heating in the turbulent region.
(4) The effects of axial heat conduction at low Re are not negligible. The model of Lin and Kandlikar [28] is able to predict this effect well.
(5) The heat transfer coefficient and friction factor for air flow
in a 962 lm tube can be predicted very well by conventional correlations after accounting for the variable heat
losses along the length of the test section, axial conduction
and viscous heating effects.
Acknowledgment
This material is based upon work supported by the National
Science Foundation under Award No. CBET-0829038. Any opinion, findings, and conclusions or recommendation expressed in
this material are those of the authors and do not necessarily reflect
the views of the National Science Foundation.
Nomenclature
A¼
Af ¼
Ah,s ¼
Dh ¼
f¼
G¼
area, m2
cross-sectional area of fluid flow in channel, m2
cross-sectional area of the channel wall, m2
hydraulic diameter, m
fanning friction factor, dimensionless
mass flux, kg/m2 s
Journal of Heat Transfer
h¼
kf ¼
kw ¼
Kn ¼
L¼
Lh,x ¼
Nu ¼
Nuth ¼
P1 ¼
P2 ¼
DP ¼
DPa ¼
DPf ¼
DPmea ¼
qaxial ¼
qin ¼
qloss ¼
qmea ¼
qviscous ¼
q00 ¼
Re ¼
Tw,x ¼
Tf,x ¼
Tf,i ¼
um ¼
heat transfer coefficient, W/m2 C
thermal conductivity of liquid, W/m C
thermal conductivity of tube, W/m C
Knudsen number, dimensionless
tube length, m
local heating length, m
Nusselt number, dimensionless
theoretical Nusselt number, dimensionless
pressure in the inlet region, Pa
pressure in the outlet region, Pa
pressure drop, Pa
acceleration pressure drop, Pa
friction pressure drop, Pa
measured pressure drop, Pa
axial conduction heat, W
input heat to fluid, W
heat loss, W
measured heat from power supply, W
viscous heat, W
heat flux, W/m2
Reynolds number, dimensionless
local wall temperature, C
local fluid temperature, C
inlet fluid temperature, C
mean fluid velocity, m/s
Greek Symbols
l¼
q1 ¼
q2 ¼
qavg ¼
viscosity, Ns/m2
fluid density in tube inlet region, kg/m3
fluid density in tube outlet region, kg/m3
averaged fluid density, kg/m3
Appendix A: Internal Tube Wall Temperature
Calculation
From energy equation, for one-dimensional heat conduction
with heat generation
q
1d
dT
r
þ
¼0
r dr
dr
kw
(A1)
where kw is the conductivity of micro tube and q is the internal
heat source generated inside the micro tube by the electric power
defined as
q¼
q
Lh p ro2 ri2
(A2)
With the adiabatic condition at the outer wall, the inside temperature is given by
q 2
q 2
ro
2
ro ri ro Ln
Tw ¼ Tw;o þ
4kw
2kw
ri
(A3)
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Appendix B: Calculation of Local Fluid Temperature
U¼
The inlet velocity u1 can be determined from the correlation as
follows:
4m_
u1 ¼
q1 pd2
(B1)
_ q1 are mass flow rate and inlet fluid density, p
respecwhere m,
ffiffiffiffiffiffiffiffiffiffi
tively. The inlet speed of sound, a1 is expressed as a1 ¼ cRT 1 ,
where c, R, and T1 are ratio of specific heats, gas constant and
inlet fluid temperature. Hence, inlet flow Mach number can be
determined M1 ¼ ðu1 =a1 Þ and
T01
c1 2
M1
¼1þ
2
T1
(B2)
where T01 is the temperature which the fluid element achieves
when u ¼ 0, defined as total pressure. Hence, T01 is obtained as
c1 2
(B3)
M1
T01 ¼ T1 1 þ
2
From the energy balance
T02 ¼
q_
þ T01
cp
(B4)
_
where q_ is heat added per unit mass defined as q_ ¼ ðqin =mÞ.
Therefore
T01
ðc þ 1ÞM12
¼
½2 þ ðc 1ÞM12 T0
ð1 þ cM12 Þ2
(B5)
T02 T02 T01
¼
T0
T01 T0
(B6)
From the compressible flow table, local Ma M2 can be determined.
The local fluid temperature T2 can be determined from the following equation where T =T1 can be determined from M1:
T2
1þc 2
2
¼ M2
T
1 þ cM22
T2 ¼
T2 T T1
T T1
(B7)
(B8)
Appendix C: Viscous Dissipation
(
2 2
2 1 @vh
@w
1 @vh 1 @w 2
þ
þ
þ vr
þ
r @h
@z
2 @z r @h
2 )
2
1 @w @vr
1 1 @vr
@ vh
þ
þ
þr
(C1)
þ
2 @r
2 r @h
@r r
@z
U ¼ 2l
@vr
@r
Assume steady and
@w 0;ð@=@zÞ 0
uniform
flow,
vh 0; ð@=@hÞ 0;
2
@vr
U ¼ 2l
@r
(C2)
r 2 uðrÞ ¼ 2u 1 R
(C3)
@u
r
¼ 4u 2
@r
R
(C4)
In laminar flow
ðL ðR
0
0
r2
2l 16u2 4 2prdrdx
R
(C5)
U ¼ 16plu2 L
(C6)
u r 1=7
¼
um
R
(C7)
In turbulent flow
@u 1 1 r 6=7
¼ um
@r 7 R R
ðL ðR 1 2 1 r 12=7
2prdrdx
um 2
2l
U¼
49 R R
0 0
2
U ¼ plu2m L
7
(C8)
(C9)
(C10)
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