Proceedings of the Fifth International Conference on Nanochannels, Microchannels and Minichannels
Proceedings of ASME ICNMM2007
ICNMM2007
th
5 International Conference on Nanochannels, Microchannels
and Puebla,
Minichannels
June 18-20, 2007,
Mexico
June 18-20, 2007, Puebla, Mexico
ICNMM2007-30031
Paper No. ICNMM2007- 30031
EFFECTS OF LOW UNIFORM RELATIVE ROUGHNESS ON SINGLE-PHASE FRICTION FACTORS IN
MICROCHANNELS AND MINICHANNELS
Timothy P. Brackbill
Rochester Institute of Technology
[email protected]
Satish G. Kandlikar
Rochester Institute of Technology
[email protected]
Q
Ra
Re
Rec
Recf
Rec.cf
Rp
Reo
Rv
Sm
v
µ
x
z
ABSTRACT
Nikuradse’s [1] work on friction factors focused on the
turbulent flow regime in addition to being performed in large
diameter pipes. Laminar data was collected by Nikuradse,
however only low relative roughness values were examined. A
recent review by Kandlikar [2] showed that the uncertainties in
the laminar region of Nikuradse’s experiments were very high,
and his conclusion regarding no roughness effects in the laminar
region is open to question. In order to conclusively resolve this
discrepancy, we have experimentally determined the effects of
relative roughness ranging from 0-5.18% in micro and
minichannels on friction factor and critical Reynolds numbers.
Reynolds numbers were varied from 30 to 7000 and hydraulic
diameters ranged from 198µm to 1084µm. There is indeed a
roughness effect seen in the laminar region, contrary to what is
reported by Nikuradse. The resulting friction factors are well
predicted using a set of constricted flow parameters. In addition
to higher friction factors, transition to turbulence was observed
at decreasing Reynolds numbers as relative roughness increased.
NOMENCLATURE
α
Aspect ratio
a
Channel base
b
Chanel height
Constricted channel height
bcf
ε
Roughness element height
εFP
ε based on proposed parameters
Dh
Hydraulic diameter
Dh,cf
Constricted hydraulic diameter
f
Friction factor
fcf
Constricted friction factor
FdRa Floor profile line to average line
Fp
Floor profile line
L
Length along channel
Mass flow rate
&
m
p
Pitch of roughness elements
ρ
Density of the fluid
P
Pressure or perimeter, apparent from eqn.
Volumetric flow rate
Average roughness
Reynolds number
Critical Reynolds number
Constricted Reynolds number
Critical constricted Reynolds number
Maximum peak height
Critical Reynolds number for ε/Dh,cf =0
Roughness – valley method
Mean spacing of roughness irregularities
Flow velocity
Dynamic viscosity
Distance from channel beginning
Profilometer scan heights
m2/s
m
m
m
m
m/s
Ns/m2
m
m
INTRODUCTION
Literature Review
Nikuradse [1] conducted pioneering work on examining the
effect of roughness on friction factors. Using sand grain
roughness, he showed friction factor deviation from smooth tube
theory in roughened tubes. His tubes ranged from 2.42cm to
9.92cm in diameter. The conclusion he reached was that
roughness has no effect on laminar friction factor. However
when one looks at their plot of friction factor versus Reynolds
number all laminar points lay above the theoretical friction
factor prediction.
Kandlikar [2] later showed that the manometers that were
used to determine pressure drop had unacceptably high
uncertainty in the laminar regime, based on the small pressure
drops encountered. The uncertainty observed by Nikuradse was
3-5% in the turbulent region, but due to smaller pressure drops
in the laminar region, it is expected that the uncertainty there is
much higher. Looking at other literature from the late 80’s and
early 90’s, Kandlikar also found that inaccurate measurements
of geometrical dimensions and pressures led many researchers
to erroneously believe a departure from continuum theory
occurred in microchannels.
m
m
m
m
m
m
m
m
m
kg/s
m
kg/m3
Pa or m
1
Copyright © 2007 by ASME
Mala and Li
[6]
Celata et al.
[7]
Li et al. [8]
Kandlikar et
al. [9]
Bucci et al.
[10]
Celata et al.
[11]
Peng et al.
[12]
Pfund et al.
[13]
Tu and
Hrnjak [14]
Baviere et al.
[15]
Hao et al.
[16]
Shen et al.
[17]
Wibel and
Ehrhard [18]
Wu and
Little [3]
Wu and
Little [19]
Weilin et al.
[20]
Wu and
Cheng [21]
Year
Roughness
Shape
Dh (µm)
h/w
Re
1999
1.75µm
Capillaries
50-254
~
<2100
2000
0.0265
Capillaries
130
~
1008000
2000
0.1%RR to
4%RR
Capillaries
79.9-449
~
<2400
Capillaries
620 and
1067
~
<2300
2006000
2001
1.0-3.0
laminar f
greater than predicted, increases
w/ decreasing Dh
Re<583 classical, f is greater with
higher Re numbers
Smooth Tubes follow macroscale,
Rough have 15-37% higher f
no effect on Dh 1067, highest f
and Nu from roughest 620
2003
0.3% to 0.8% RR
Capillaries
172-520
~
2004
.05µm smooth,
.2-.8µm rough
Capillaries
31-326
~
1994
~
Rectangular
.133-.343
.333-1
2000
.16 and .09µ,
rough 1.9µ
Rectangular
252.81900
69.5304.7
.0128.105
.09.24
153-191
0.390.55
<2400
Follows theory until Re=900, then
higher, indicating trans.
Higher, and Po number increases
with Re, nothing at low Re
2003
Ra < 20nm
Rectangular
2004
5-7µm
Rectangular
2006
Artificial 50x50µm
RR 19%
Rectangular
Re<800-1000 follows classical
504000
<3600
1129180
.018000
tentatively propose higher than
normal friction
Aspect ratio makes some f higher,
some f lower
Higher f than theory, highest for
rough
RR<0.3%, conventional,
RR=0.35% f is 9% higher
Increased laminar friction factor
2006
4% RR
Rectangular
436
0.375
1621257
2006
1.3µm
(~0.97%RR)
Rectangular
~133
0.2-1
<4000
near classical values
1983
0.05-0.30 height
Trapezoidal
45.5-83.1
~
~
greater than predicted
1984
0.01 height
Trapezoidal
134-164
~
~
greater than predicted
2000
2.4%-3.5%
Trapezoidal
51-169
~
<1500
2003
3.26e-5-1.09e-2
Trapezoidal
~100
.0382.3573
141100
Higher and larger slope for Px-Re
(18-32%)
roughness increased it, surface
type varied it
Rec
1881-2479 is transition
region
1700~1900 for rough
tubes
Error
9.2%f
3%Re
9%f 5%Re
20%Nu
Lowered w/ roughness
1800-3000, abrupt
transition at high RR
200-700
approach 2800 w/ larger
2150-2290 w/ RR<0.3%
1570 for 0.35%
increased with
roughness
8.36%f
1.8%Re
10%f,
8%Re
11.1%f
3.4% Re
6.3%f
2%Re
Transition ~900
N/A
7.1%f
6.95%Re
1800-2300 - varies with
aspect ratio
1000-3000
N/A
N/A
7.6%f
4.6%Re
10.3%fappR
e 7.8%Nu
Table 1 – Summary of experimental works
0 < ε/Dh,cf < 0.08
0.08 < ε/Dh,cf < 0.15
Other studies have observed friction factors higher than
theory would predict. Wu and Little [3] first noticed friction
factors higher than theory would predict in microchannels. This
prompted more studies in this area. The rest of the studies have
varied results but all report higher friction factors than theory
would predict, along with other roughness influenced effects.
The results of these studies are summarized in Table 1.
Past work by the authors [4, 5] has shown the applicability
of considering only the unobstructed flow cross section in
calculating the hydraulic diameter of channels. Using
constricted parameters the Moody diagram was redrawn as
shown in Kandlikar [5].
Using these constricted flow
parameters, friction factor reaches a plateau at 0.042 for all
values of relative roughness up to 5% in the turbulent flow
regime.
Critical Reynolds number, noted here as the point at which
the friction factor departs laminar correlations, is also observed
to vary with roughness present in the channels. Increasing
relative roughness was observed to lower this critical transition
Reynolds number. The lowest value reported by other studies
shows a transition value of Rec of 900 by Hao et al. [16] with
artificially created roughness elements with a relative roughness
of 19%.
Past work by the authors [4] determined that with high
relative roughness, transition Reynolds number could be
lowered to as low as 210 for a relative roughness of 27.6%. The
correlation proposed by Kandlikar et al. [5] for critical Reynolds
number is given by Eq. 1.
Rec,cf = 2300 – 18,750(ε/Dh,cf)
Rec,cf = 800 – 3,270(ε/Dh,cf – 0.08)
(1)
One problem encountered when comparing experimental
studies in literature is the method of reporting the roughness
value ε. Often times, the value for a roughness height will be
given, without defining which parameter was calculated to
determine the element height that is reported. The methodology
for determining the element height has not been standardized for
this type of application either. In different studies, the surface
element height could be measured by average roughness (Ra),
ten-point roughness (Rz), or other similar parameters. However,
in Nikuradse’s experiments, which form the basis of the modern
Moody diagram, the diameter of sand grains lacquered to the
inside of pipes was used as the value of roughness height. The
proposed method for determining the roughness parameter εFP in
this paper aims to accurately represent any surface and is given
later in the discussion.
OBJECTIVES OF CURRENT WORK
In the present work, experiments are conducted to provide
data with low uncertainty using carefully selected instruments
for measuring pressure drop, mass flow rate and channel
geometric parameters. Relative roughness values used range
from 0 to 5.18%. Reynolds numbers range from 30 to 7000.
A new parameter, εFP is used to determine the height of the
roughness elements. The effects of roughness on laminar flow
are examined in depth, including laminar-turbulent transition
and friction factor effects.
2
Copyright © 2007 by ASME
THEORETICAL CONSIDERATIONS
Roughness height determination
A brief overview of the method of calculation of the
proposed peak to floor roughness height, εFP, is presented here.
The full context can be found in Kandlikar et al. [5] along with
Taylor et al. [22]. The following calculations are performed
after the profile of a surface is obtained using a stylus
profilometer or other similar instrument. All parameters are
illustrated in Fig. 1.
The Mean Line is the arithmetic average of all the points
from the raw profile, which physically relates to the height of
each point on the surface. It is calculated as shown in Eq. 2.
Sm1
Rp1
Sm2
Rp2
Sm3
Rp3
Main Profile
Mean Line
FdRa
Figure 1 – Illustration of Roughness parameters
Floor
Profile
Let z ⊆ Z s.t. all zi = Zi iff Zi < Mean Line
1 nz
Fp = ∑ z i
n z i =1
1 n
MeanLine = ∑ Z i
n i =1
(2)
Rp is the maximum peak height from the mean line, which
translates to the highest point in the profile sample minus the
mean line. It is calculated by Eq. 3.
Rp = max(Zi ) − MeanLine
(3)
(4)
FdRa is defined as the distance of the floor profile (Fp)
from the mean line. It is found with Eq. 5.
FdRa = MeanLine − Fp
(5)
εFP, or the value of the proposed roughness height, is
determined by Eq. 6. It is the distance between the floor profile
and the peak of the roughness.
(6)
ε FP = Rp + FdRa
The reason for the conception of this parameter will
become more clear in the discussion on Nikuradse’s method of
determining roughness. It is proposed because the current
parameter widely used, Ra, does not accurately predict
roughness height.
Sm is defined as the mean separation of profile
irregularities, or the distance along the surface between peaks.
As it doesn’t play a role in the current work, its definition is not
included.
Fp is defined as the floor profile. It is the arithmetic
average of all the points that fall below the mean line value. As
such, it is a good descriptor of the baseline of the roughness
profile. It can be calculated from Eq. 4.
Evaluation of Nikuradse’s Tube Surface Roughness Profiles
A brief look will be taken at the method used by Nikuradse
[1] to generate the original data on roughness in 1933.
Nikuradse collected pressure drop data on roughened pipes,
using sand grains as roughness elements in commercially
available smooth tubes. He defined roughness height to be the
diameter of the grains of sand coating the walls of the pipe.
Unfortunately, relating a real life roughness profile to a sand
grain diameter for calculations is nearly impossible. Therefore,
a parameter to relate roughness encountered in microchannels
and minichannels to the data collected by Nikuradse is
determined. To begin, the method Nikuradse used to create his
roughness must be examined.
Nikuradse [1] first sifted grains from ordinary building sand
to obtain grains that were 800µm in diameter using an 820µm
and a 780µm sieve. These grains were then put under a
micrometer to verify the desired size. Pipes used in the
experiments were filled with a lacquer, drained, and allowed to
become tacky. When the lacquer became tacky, the pipes were
filled with the sieved sand, and then emptied again. He then
again filled the tubes with lacquer and emptied them, in the
interest of achieving better adhesion of sand grains to the walls
of the tubes. To allow the lacquer to properly dry, heat lamps
were applied over an extended amount of time. These pipes
were then tested in the experimental apparatus. The resulting
surface of the pipes looks something like that give in Fig. 2 (b).
Figure 2 – Idealized profile of Nikuradse’s work
3
Copyright © 2007 by ASME
A question this process brings to mind is whether a coating
of lacquer over the sand grains would appreciably change their
diameter. To alleviate this worry, Nikuradse applied this same
procedure to a flat plate, and measured the height of the
resulting roughness formations post-lacquering with a
micrometer. Confirmation of the height of roughness was
expressed in his work. A microscope picture of these grains
was taken, and it showed that small gaps were present in
between the sand grains, however this was included only to
show that the hydrodynamic influence of these grains was
indeed the diameter. Thus the gap size between the grains is of
definite variability, but from the microscope photo shown in
Nikuradse’s work it appears to be at most 400µm and at least
100µm as a rough estimate.
A model is made, assuming the sand grains are perfect
spheres, of an ideal representation what a 2D stylus profilometer
scan of this surface would look like. This model is implemented
in a spreadsheet. Using this example profile, just as one would
use the results of a stylus profilometer, parameters of the surface
can be determined. Since the average gap distance between
grains is not noted in the paper, the exercise was performed for
many gap sizes. With a reasonable assumption that the average
distance between grains is taken to be 300µm; the resulting Ra
is 291µm and the proposed parameter εFP is 756µm. The value
that Nikuradse would have used for this profile is 800µm, and
thus that is what the currently compiled body of data for friction
factor relies on. This alone shows the inadequacy of using the
Ra parameter as roughness height. Both the profile of roughness
(at 300µm gaps between sand grains) and the plot of the
parameters versus the gap size can be seen in Fig. 2. In addition,
Fig. 2(a) shows clearly that using εFP the roughness height
asymptotically approaches 800µm, which is the value that the
modern Moody Diagram is based on. Using Ra as the roughness
parameter will give a smaller value of roughness than intended,
and will introduce large errors in the Moody diagram
representation.
Note that with increasing gap size, εFP approaches the actual
sand grain size. One could observe that use of the peak-to-valley
roughness parameter, which is simply the maximum point in the
profile subtracted by the minimum point, would yield the
correct size of 800µm. However practical applications with nonideal roughness require a parameter that is easy to calculate with
simple algorithms, and peak-to-valley would be wrong in any
case where a profile contained errant peaks or valleys. This
simple exercise shows that εFP can characterize roughness height
well on a theoretical level.
Definition of Constricted Parameters
The derivation of constricted parameters is paramount to
determining important predictors for the friction factor in high
roughness channels. First, we have to define the constricted
channel height. An ordinary channel has a cross section of
height b, and width a. However, with roughness on 2 sides of
the channel the parameter bcf represents the new constricted
channel height. These parameters are illustrated with generic
ribbed roughness in Fig. 3. Note that the pitch, p, of the
roughness elements does not play a large role in the uniform
roughness used in this current experimentation.
Now, to recalculate new constricted parameters, we will use
bcf and Acf, defined as follows
(13)
b cf = b − 2ε FP
A = ab
A cf = ab cf
(14)
Perimeter and constricted perimeter follow, substituting the
bcf value.
P = 2a + 2b
(15)
Pcf = 2a + 2b cf
(16)
Hydraulic diameter is calculated using and constricted
hydraulic diameter are found with the following
4A
P
4A cf
=
Pcf
Dh =
D h,cf
(17)
(18)
Using these constricted parameters, we can now find the
theoretical friction factors. In the laminar regime, friction factor
for rectangular channels is predicted by Kakac, et al [23] by Eq.
19. The aspect ratio α is defined by Eq. 20. Again, the
constricted aspect ratio, αcf, is defined with the constricted
channel height in Eq. 21.
f =
24
1 − 1.3553α + 1.9467α 2 − 1.7012α 3 + .9564α 4 − .2537α 5
Re
(19)
(
)
b
a
(20)
b cf
α cf =
a
(21)
The theoretical turbulent friction factor is calculated using
the following equation from Colebrook [24].
 ε

 Dh
1
2.51 
(22)
=
−
2
log
+

0.5 
3
.
7
f 0.5
Re
f




To relate the turbulent friction factor we have to look at the
governing equation determining friction factor. From the
pressure drop equation, we solve for f and group terms in Eq.
23.
α=
Figure 3 – Side view of channel with parameters marked
4
Copyright © 2007 by ASME
∆P =
2fLρv 2
Dh
 ∆P 1
f = 
2
 ∆x 2ρQ
88.9mm length. Each tap is connected to a 0-689kPa (0-100psi)
differential pressure sensor with 0.2% FS accuracy. The
pressure sensor outputs are put through independent linear 100
gain amplifiers built into the NI SCXI chassis to increase
accuracy. The separation of the samples is controlled by two
Mitutoyo micrometer heads, with ±2.54µm accuracy. There is a
micrometer head at each end of the channel to ensure
parallelism.
Water is delivered via a Micropump motor drive along with
two Micropump metered pump heads, one for low flows (0100mL/min) and one for high flows (76-4000mL/min). The
flow rate is verified with three flow meters, one each for 13100mL/min, 60-1000mL/min, and 500-5000mL/min. Each flow
meter is accurate to better than 1% FS. Furthermore, each flow
meter was calibrated by measuring the weight of water collected
over a period of time. Thermocouples are mounted on the inlet
and outlet of the test apparatus. Fluid properties are calculated
at the average temperature. All of the data is acquired and the
system controlled by a LabVIEW equipped computer with an
SCXI-1000 chassis.
Testing equipment allows for fully
automated acquisition of data at set intervals of Reynolds
number. A test setup schematic can be found in Fig. 4. All of
the circles with P’s are pressure sensors.
(23)

D h A 2

To compare the friction factor of the constricted channel,
we will substitute Dh,cf for Dh and Acf for A. Then, Eq. 23 for f
will be divided by Eq. 23 for fcf to obtain a correlation for the
constricted friction factor. This is given in Eq. 24.
 ∆P 1 

D h, cf A cf 2
f cf  ∆x 2ρQ 2 
=
f
 ∆P 1 
(24)

D A 2
2  h
∆
x
2
ρ
Q


f cf = f
D h, cf A cf
2
DhA2
Now we calculate the Reynolds number. It is given by Eq.
25. The constricted Reynolds number is given in Eq. 26.
&
4m
Re =
µP
(25)
Re cf =
&
4m
µPcf
Testing
One of the two samples of each roughness set is secured in
place to remain immobile, and the other sample is placed in the
movable ground steel holder opposite to the first. The samples
are brought together with the set screws onto two precision gage
blocks of the same known width and the micrometer heads
zeroed on them. One gage block is placed at each end of the
channel, to ensure that the channel is parallel. This gage block
width is then known to be the bcf value of the channel. The
separation of the samples is then set to the desired bcf value with
the micrometer heads and held in place with set screws. The
channel is then sealed. The flow is started and controlled by the
computer while acquiring data. While the experiment is
running, the pressure taps are monitored to make sure they are
outside the developing region and operating properly.
(26)
VERIFICATION
Experimental Setup
The experimental setup was developed based on a previous
variable hydraulic diameter setup from Brackbill and Kandlikar
[4]. Improvements were made in many areas to accommodate
easier testing. Any metallic surface critical to accurate
measurements was ground smooth, planar, and square in a
precision surface grinder. The channel is sealed with sheet
silicone gaskets around the outside of the samples to prevent
leaks. The base block acts as a fluid delivery system and also
houses 15 pressure taps, each drilled with a #60 drill (diameter
of 1.016mm) along the channel. The taps begin at the entrance
to the channel and are spaced every 6.35mm along the channel’s
Samples
The sample blanks are machined to near-dimensions, and
then are precision ground to exact dimensions. The parallelism
and flatness is then verified to ensure proper channel geometry.
The smooth channel samples were then lapped first with 5µm
lapping compound and then 1µm lapping compound to create a
mirrored smooth surface.
Smooth
100 Grit
60 Grit
Ra
µm
0.06
2.64
6.09
Rp
µm
0.15
6.87
17.09
Rv
µm
0.08
7.66
20.81
FdRa
µm
0.05
2.30
6.09
εFP
µm
0.20
9.17
23.19
Table 2 – Roughness summary
Figure 4 – Test Section Diagram
5
Copyright © 2007 by ASME
The first roughened channel set is then formed by using 100
grit sandpaper in a perpendicular crosshatch pattern. The
uniform roughness is formed sanding 45 degrees in both
directions from the axis along the length of the channel. The
profile of this surface is then taken with a stylus profilometer,
and parameters are determined using both εFP and the
conventional Ra. The results of 8 tests are averaged and a value
εFP = 9.17µm is found for the roughness element height. The
procedure is repeated on another sample set, except 60 grit
sandpaper is used instead of 100 grit sandpaper. This yielded a
much higher value of εFP = 23.19µm for the roughness element
height. An example profile of each sample used in testing can
be seen in Fig. 5, along with a 3D digital microscope scan of the
surfaces. Note that the charts given in Fig. 5 represent the
results of one profilometer scan, and thus the values for each of
these calculated parameters may vary slightly from the
presented average of all 8 scans.
The roughness parameter Ra has often been used in studies
to represent the height of the roughness elements. For
comparison, this parameter was calculated by the profilometer
used in this study. Both parameters for all samples used in
testing can be seen in table 2. For a more in depth look at the
parameterization of different machined surfaces using this
method, refer to Young et al [25].
Uncertainty
The uncertainty in the measure of friction factor is now to
be determined. First the equation for the friction factor is given
in Eq. 7.
f=
P2 − P1 ρD h A 2
&2
x
2m
(7)
To find the error, the propagation of errors in f (δf) by the
changes in each of the variables was found by the differentiation
given in Eq. 8.
∂f
∂f
∂f
∂f
∂f
∂f
& (8)
δf =
δP1 +
δP2 + δx +
δD h +
δA +
δm
&
∂P1
∂P2
∂x
∂D h
∂A
∂m
Figure 5 – Roughness Profiles given by 3D profile and 2D profilometer data
6
Copyright © 2007 by ASME
Now, the uncertainty in each variable, depicted here as an
arbitrary y1, is then defined as is shown in Eq. 9.
δy
uy = 1
y1
(9)
Rearranging Eq. 7 using the definition of Eq. 8, dividing
through by f, and using a RMS approach yields an uncertainty in
f of given in Eq. 10.
1
1
2
2
2
 P ∂f
2
  P ∂f
  x ∂f

 1

u P1  +  2
u P2  + 
ux 
 f ∂P1

  f ∂P2
  f ∂x 
u f = ±

2
2
2
  A ∂f
& ∂f
  D h ∂f
 m
 
u& 
+  f ∂D u D h  +  f ∂A u A  +  f ∂m
& m  
 
h
 
 
(10)
Defining each partial derivative by its respective equation
yields uncertainty in friction in Eq. 11.
1
 P  ρD A 2   2  P  ρD A 2
h
h
 1  −
u P  +  2 
& 2  1   f  2xm
&2
 f  2xm

2
 x  (P − P )ρD A 2  
1
h


  − 2
u
 x +
& 2x2
2m
 f 
 

2
2
 D h  (P2 − P1 )ρA 





u f = ±

u D h 

& 2x
2m

 f 


2
  A  (P2 − P1 )ρD h A  


+
u


A
 f
 +
& 2x
m
 
  
2

&  (P2 − P1 )ρD h A 2  
 m

−
u m&
 
 f 
& 3x
m
 


u P
 2

2
 2
 +
 

















Figure 6 – Plot of all smooth data points
(11)
The uncertainties of the other variables involved in this
equation are calculated in the same manner. The procedure is
repeated in uncertainty of the Reynolds number, and gives an
uncertainty equation given in Eq. 12.
1
u Re
2
2
 m
&
&
 2
  a
4
− 8m


u m&  + 
u
a  
2
 Re µ(2a + 2b)
  Re µ(2a + 2b)
 
= ±

2
&

− 8m
  b



+  Re µ(2a + 2b) 2 u b 


 

Figure 7 – Plot of Constricted and unconstructed data
maximum errors occur at the smallest value of b at the lowest
flow rates encountered. These uncertainties are 7.58% for
friction factor and 2.67% for Reynolds number.
(12)
RESULTS
Verification was first performed on the smooth samples.
Excellent agreement was observed with macroscale theory.
Verification was performed on hydraulic diameters from 198µm
to 961µm. Transition to turbulence began at Reynolds numbers
ranging from 2449 to 2613. The difference from the accepted
value of 2300 can be attributed to the fine finish of the samples
and rectangular channel cross-section. All of the smooth channel
data can be seen in Fig. 6. To calculate the transition to
turbulence, a linear regression was performed on the linear
portion of the f vs Re plot. When the friction factor consistently
deviated more than 1% from this value, transition was assumed
to have begun.
This analysis is based on being able to find the uncertainty
of each measurement in the experiment. To do this, the
calibration performed on each sensor is used. The points used
for the linear calibration are used to find the error between
measured and the calibration value. For each sensor, 30 points
are checked, and the maximum value of error of the 30 is
recorded. The average of these maximum errors is used for the
error of the pressure sensors. The same is performed for each of
the three flow sensors. This approach yields extremely
conservative error values, of 0.998% for pressure sensors and
around 2.2% for the flow sensors. Using this analysis, the
7
Copyright © 2007 by ASME
Figure 8 – Plots of a representative sample of data sets plotted as (a) unconstructed and (b) as constricted
With the verification completed, testing on the 100 grit
samples in the range of 0.97% to 2.69% relative roughness is
performed. The test at a relative roughness of 1.42% is shown
in Fig. 7. When the data is plotted with the non-constricted
parameters (plus data set), error can clearly be seen in the results
as friction factor is higher than predictions. When the
constricted parameters are used (circle data set), the
experimental data correlates well with that which is calculated
using Dh,cf and fcf. This plot clearly shows a laminar effect of
roughness, contrary to what is commonly accepted in this type
of work. The critical Reynolds number for these samples ranges
from 2124 to 2604. As the relative roughness increased, the
critical Reynolds number was lowered.
Next, the 60 grit samples were run in the range of relative
roughness from 2.70% to 5.18%. Again, use of constricted
parameters accurately predicted the friction factors in the
channels, which also means that a laminar roughness effect was
observed throughout testing. The critical Reynolds numbers
varied from 1282 to 2032.
A representative sample of the rough data was plotted using
the unconstructed parameter which is plotting the data using the
b parameter. This is plotted in Fig. 8(a). Not only is there a
laminar effect from the roughness, but the effect is magnified by
increasing relative roughness. The lowest relative roughness
plotted is 1.42%, which shows the smallest deviation from
laminar theory. As the relative roughness increases, increasing
error is present between experimental and theoretical values.
Note that many intermediate relative roughness values were
tested, but were omitted from the plots for clarity.
A representative sample of the constricted data from the
roughened samples was then plotted together in Fig. 8(b). When
plotted with the constricted parameters, it can be seen that the
observed laminar effect is predicted by theory. In addition, a
trend can be seen between critical Reynolds number and relative
roughness. The 1.42% samples transitions around 2604, then
each higher relative roughness data set has sequentially
decreasing transitions, a trend which is clear on the plot. To
further investigate this relation, the critical constricted Reynolds
number is then plotted against the relative roughness of the
sample in Fig. 9. Also in Fig. 9 is data from Brackbill and
Kandlikar. [4], labeled with its year.
After plotting the critical transition data, the Kandlikar
correlation introduced was modified slightly. Rather than
beginning the correlation at 2300 for ε/Dh,cf = 0, which is what
was given by Kandlikar in literature, a starting value of 2500
was used to plot the critical Reynolds number, with the
Figure 9 – Critical Reynolds numbers and Modified Kandlikar Correlation
8
Copyright © 2007 by ASME
equations given in Eq. 27. It is recommended that use of this
correlation modify the Rec,cf value at ε/Dh,cf = 0, or Reo, to the
transition point of a smooth channel with similar geometric
dimensions.
0 < ε/Dh,cf < 0.08
Microminiature Joule-Thomson Refrigerators”, Cryogenics,
Vol. 23, pp 273-277, 1983.
[4] T.P. Brackbill and S.G. Kandlikar, “Effect of Triangular
Roughness Elements on Pressure Drop and LaminarTurbulent Transition in microchannels and minichannels”,
Proceedings of the International Conference on
Nanochannels,
Microchannels,
and
Minichannels
ICNMM2006-96062, 2006.
[5] S.G. Kandlikar, D. Schmidt, A.L. Carrano, and J.B. Taylor.
“Characterization of Surface Roughness Effects on Pressure
Drop in Single-Phase Flow in Minichannels and
Microchannels.” Phys. Fluids 17, Paper No. 100606, 2005.
[6] G. M. Mala and D. Li, “Flow characteristics of water in
microtubes”, International Journal of Heat and Fluid Flow
20, pp. 142-148, 1999.
[7] G. P. Celata, M. Cumo, M. Guglielmi and G. Zummo,
“Experimental Investigation of Hydraulic and Single Phase
Heat Transfer in 0.130mm Capillary Tube”, Proceedings of
the International Conference on Heat Transfer and
Transport Phenomena in Microscale, pp. 108-113, 2000.
[8] Z. Li, D. Du and Z. Guo, “Experimental Study on Flow
Characteristics of Liquid in Circular Microtubes”,
Proceedings of the International Conference on Heat
Transfer and Transport Phenomena in Microscale, pp. 162167, 2000.
[9] S.G. Kandlikar, S. Joshi and S. Tian, “Effect of Channel
Roughness on Heat Transfer and Fluid Flow Characteristics
at Low Reynolds Numbers in Small Diameter Tubes”,
National Heat Transfer Conference NHTC01-12134, 2001.
[10] A. Bucci, G.P. Celata, M. Cumo, E. Serra and G. Zummo,
“Water Single-Phase Fluid Flow and Heat Transfer in
Capillary
Tubes”,
International
Conference
on
Microchannels and Minichannels ICMM2003-1037, 2003.
[11] G. P. Celata, M. Cumo, S. McPhail and G. Zummo,
“Hydrodynamic Behaviour and Influence of Channel Wall
Roughness and Hydrophobicity in Microchannels”,
International Conference on Microchannels and
Minichannels ICMM2004, 2004.
[12] X.F. Peng, G.P. Peterson and B.X. Wang, “Frictional Flow
Characteristics of Water Flowing Through Rectangular
Microchannels”, Experimental Heat Transfer 7, pp. 249264, 1994.
[13] D. Pfund, D. Rector, A. Shekarriz, A. Popescu and J.
Welty, “Pressure Drop Measurements in a Microchannel”,
AlChE Journal, Vol. 46, No. 8, August 2000.
[14] X. Tu, P. Hrnjak, “Experimental Investigation of SinglePhase Flow Pressure Dop Through Rectangular
Microchannels”,
International
Conference
on
Microchannels and Minichannels ICMM2003-1028, 2003.
[15] R. Baviere, F. Ayela, S. Le Person and M. Favre-Marinet,
“An Experimental Study of Water Flow in Smooth and
Rough Rectangular Micro-channels”, International
Conference on Microchannels and Minichannels
ICNMM2004, 2004.
[16] P. Hao, Z. Yao, F. He, and K. Zhu, “Experimental
investigation of water flow in smooth and rough silicon
Re o − 800
(ε / Dh,cf )
0.08
= 800 − 3270(ε / Dh ,cf − 0.08)
Re c ,cf = Re o −
0.08 < ε/Dh,cf < 0.15 Re c ,cf
where:
Reo = transition for ε/Dh,cf = 0
Reo = 2500, for this work
(27)
Average absolute error for Eq. 27 is 13%.
CONCLUSIONS
An experimental study of friction factor in rectangular
channels with variable channel height was performed.
Hydraulic diameters ranged from 198µm to 1084µm, Re varied
from 30 to 7000, and relative roughness varied from 0 to 5.18%.
Conclusions reached are presented in the following enumerated
parts.
1) Roughness was shown to affect the friction factor in
the laminar flow regime, contrary to Nikuradse’s
results for relative roughness from 0-5%. Friction
factors were observed to be higher than laminar theory
would predict using the unconstricted channel
dimensions as is customary in fluidics applications.
2) The use of constricted flow parameters accurately
predicts the friction factors that were encountered in
laminar, low relative roughness experiments in this
study
3) Use of parameter εFP, a measure from profile peak to
the floor profile, as roughness element height was
shown to be a good predictor of sand grain roughness
with an ideal model. εFP was shown to better predict
roughness height than the currently accepted Ra value.
4) Increasing relative roughness yields a lower critical
Reynolds number, which can be predicted by a
modified correlation based on the critical transition of a
smooth channel of same geometry. This correlation is
given in Eq. 27.
REFERENCES
[1] J. Nikuradse, “Forschung auf dem Gebiete des
Ingenieurwesens”, Verein Deutsche Ingenieure, Vol. 4, pp.
361, 1933.
[2] S. G. Kandlikar, “Roughness effects at microscale –
reassessing Nikuradse’s experiments on liquid flow in
rough tubes”, Bulletin of the Polish Academy of Sciences,
Vol 53, No. 4, 2005.
[3] P. Wu and W.A. Little, “Measurement of Friction Factors
for the Flow of Gases in Very Fine Channels used for
9
Copyright © 2007 by ASME
microchannels”,
J.
of
Micromechanics
and
Microengineering 16, pp 1397-1402, 2006.
[17] S. Shen, J. L. Xu, J. J. Zhou, and Y. Chen, “Flow and heat
transfer in microchannels with rough wall surface”, Energy
Conversion and Management 47, pp. 1311-1325, 2006.
[18] W. Wibel and P. Ehrhard, “Experiments on Liquid
Pressure-Drop in Rectangular Microchannels, Subject to
Non-Unity Aspect Ratio and Finite Roughness”,
International Conference on Nanochannels, Microchannels,
and Minichannels ICNMM2006-96116, 2006.
[19] P. Wu and W.A. Little, “Measurement of Heat Transfer
Characteristics in the Fine Channel Heat Exchangers used
for Microminiature Refrigerators”, Cryogenics, Vol. 24, pp
415-420, 1984.
[20] Q. Weilin, G. M. Mala and L. Dongquing, “Pressure-driven
water flows in trapezoidal silicon microchannels”, Int. J. of
Heat Mass Transfer 43, pp. 353-364, 2000.
[21] H.Y. Wu and P. Cheng, “An experimental study of
convective heat transfer in silicon microchannels with
different surface conditions”, International Journal of Heat
and Mass Transfer 46, pp. 2547-2556, 2003.
[22] J.B. Taylor, A.L. Carrano, and S.G. Kandlikar,
“Characterization of the effect of surface roughness and
surface texture on fluid flow: past, present, and future”,
International Journal of Thermal Sciences. Volume 45.
Issue 10. pp 962-968, 2006.
[23] S. Kakac, R.K. Shah and W. Aung, Handbook of SinglePhase Convective Heat Transfer. John Wiley and Sons, pp
3-122, 1987.
[24] F. C. Colebrook, “Turbulent flow in pipes, with particular
reference to the transition region between the smooth and
rough pipe laws,” J. Inst. Civ. Eng., Lond., 11, pp. 133-156,
1939.
[25] P.L. Young, T.P. Brackbill, and S.G. Kandlikar,
"Estimating Roughness Parameters Resulting from Various
Machining Techniques for Fluid Flow Applications",
Proceedings of the International Conference on
Nanochannels,
Microchannels,
and
Minichannels
ICNMM2007-30033, 2007.
10
Copyright © 2007 by ASME