Proceedings of the Fifth
International Conference on Nanochannels, Microchannels and Minichannels
th
Proceedings of the 5 International Conference on Nanochannels, Microchannels and Minichannels
ICNMM2007
ICNMM2007
June 18-20, 2007, Puebla,
Mexico
June 18-20, 2007, Puebla, Mexico
ICNMM2007-30033
ICNMM2007-30033
ESTIMATING ROUGHNESS PARAMETERS RESULTING FROM VARIOUS MACHINING
TECHNIQUES FOR FLUID FLOW APPLICATIONS
Perry L. Young
Rochester Institute of Technology
[email protected]
Timothy P. Brackbill
Rochester Institute of Technology
[email protected]
ABSTRACT
Recently, a set of new roughness parameters was proposed
by Kandlikar et al. [1] and Taylor et al. [2] for reporting
surface roughness as related to fluid flow. The average
roughness Ra parameter is often used in microfluidic
applications, but this parameter alone is insufficient for
describing surface roughness; a specimen with deep grooves
and sharp obstructions can share the same average roughness
value as a relatively smooth surface with low uniform surface
roughness. Since the average roughness parameter is broad, it is
difficult to access the surface topography features that result
from different machining processes or etches. A profilometer
and a digital microscope are used to examine the surface
roughness profiles of various materials submitted to different
machining techniques. The materials studied will be similar to
those used for microchannels including aluminum, stainless
steel, copper, and silicon. Depending on the material, these
samples are submitted to several machining processes including
milling, grinding, fly cutting, and microfabrication techniques.
These machining processes and microfabrication techniques are
of practical interest in microfluidics applications. After
studying the surface roughness patterns exhibited in these
samples, the roughness parameters employed in some of the
recent roughness models are evaluated. This study is expected
to provide more understanding of assorted surface roughness.
INTRODUCTION
The correlation between surface roughness and fluid flow
was initially drawn by Darcy, among others, in the early
nineteenth century [3]. He conducted careful pressure drop
experiments and concluded that the surface roughness was an
important factor in fluid flow. Fanning later performed more
experiments and proposed correlations between surface
roughness and pressure drop [4]. More experiments were done
Satish G. Kandlikar
Rochester Institute of Technology
[email protected]
by Nikuradse [5], who sifted sand into groups of known
diameters. He then lined pipes of 250, 500, and 1000 mm
diameters with a combination of the sand grains and Japanese
lacquer. These pipes were then studied to determine the effect
of the surface roughness on the pressure drop of the fluid flow
through the pipe [5]. The Reynolds numbers studied in his
experiments ranged from 600 to 106.
Colebrook later
conducted studies on sixteen spun concrete pipes, including the
Ontario Tunnel [6]. These concrete pipes ranged in diameters
of 101.6 mm to 5486 mm. He developed the well known
Colebrook equation through these experiments. Moody then
used the Colebrook equations, as well as the experimental data
from Nikuradse’s work, to develop the Moody diagram for
pipes with relatively low values of surface roughness in the
range of 0-5% relative roughness [7].
There are various ways to measure the surface roughness
of a material. Many studies have used a profilometer to obtain
a surface profile. However, this process can cause localized
damage to the surface being examined. Sundararajan et al. [8]
used profilometry and Atomic Force Microscopy (AFM) to
investigate the effects of various etching processes on silicon
surfaces. The two etches the authors focused on were
potassium hydroxide (KOH) and tetramethyl ammonium
hydroxide (TMAH). They also studied the effects of adding
isopropyl alcohol (IPA) additive to these etches. They found
that the TMAH etch produced rougher surfaces than the KOH
etch at larger scan sizes than 5 µm. Other researchers employ
optical or scanning electron microscopy to evaluate surface
roughness. Chen et al. [9] have proposed using atomic force
microscopy with a bent tapered fiber probe to measure surface
roughness and profiles. Using Digital Holographic Microscopy
(DHM) has also been investigated by Montfort et al. [10] as
another approach to measuring the topography of a surface.
Kandlikar et al. [1] and Taylor et al. [2] proposed six new
roughness parameters in recent work in order to better depict
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Copyright © 2007 by ASME
surface roughness characterization for single-phase fluid flow
applications. Of the six parameters, three correspond to surface
roughness and three correspond to localized hydraulic diameter
variations. The three roughness characterization parameters
that were proposed are Rp (maximum profile peak height), Rsm
(mean spacing of profile irregularities), and FdRa (floor
distance to mean line). A new roughness parameter, εFp, was
proposed in place of average roughness (Ra). These parameters
will be discussed in more depth later.
NOMENCLATURE
AFM
atomic force microscope
DHM
digital holographic microscope
DRIE
deep reactive ion etching
εFp
roughness
FdRa
floor distance to mean line
IPA
isopropyl alcohol
KOH
potassium hydroxide
Ra
average roughness
RP
maximum profile peak height
Rv
maximum profile valley
Rsm
mean spacing between profile irregularities
TMAH
tetramethyl ammonium hydroxide
XeF2
xenon diflouride
SURFACE ROUGHNESS PARAMETERS
There are many surface roughness parameters that can be
used to analyze a surface. The most common surface
roughness parameter used in industry is the average roughness.
The average roughness can be calculated using equation 1:
Ra =
1 n
∑ Yi
n i =1
(1)
In this equation, n is the number of sample points evaluated,
and Yi represents the absolute value of the profile deviation
from the mean line. This roughness parameter is well known
in industry, and is very easy to calculate and use.
Manufacturers often specify an average roughness value and
are confident they will receive a surface that fits their needs.
However, the average roughness parameter fails to accurately
represent the surface topography. To make up for this
shortcoming other surface roughness parameters are often used
in addition to the average roughness.
Kandlikar et al. [1] proposed the use of three surface
roughness parameters, as well as a new roughness parameter,
εFp, to be used to represent the roughness in replacement of
average roughness for fluid flow. A diagram of these surface
roughness parameters is shown in Fig. 1. These surface
roughness parameters include Rsm, Rp and Rv. Rsm represents
the mean spacing of profile irregularities. It is calculated using
equation 2:
Rsm =
1 n
∑ Smi
n i =1
(2)
Figure 1: Diagram of Surface Roughness Parameters, from
Taylor et al. [8]
To calculate this, two lines are drawn near the mean line.
These two lines could be a percentage value away from the
mean line or represent a standard height of the profile. The
area between these two lines represents the dead zone of the
profile. Smi represents the peak and valley pairs of the profile;
the distance between when the data drops below the dead zone
to where it rises above the dead zone. This parameter
represents the pitch of the surface profile.
The maximum profile peak height, Rp, represents the
distance between the mean line and the highest point of the
profile. The maximum profile valley, Rv, is a similar
parameter and represents the distance between the mean line
and the lowest point of the profile. These two parameters can
be calculated using equations 3 and 4:
RP = max(Zi) - mean(Zi)
(3)
RV = min(Zi) – mean(Zi)
(4)
where Zi represents all the points of the surface profile.
Another useful surface parameter is FdRa, which
represents the floor distance to mean line. FdRa can be
calculated by finding the average of all profile data points that
fall below the mean line, as shown in equation 4.
Let z⊆Z such that all zi = Zi iff Zi < Mean Line
FdRa =
1
nz
nz
∑z
i
(5)
i =1
The roughness parameter, εFp, that was proposed to be used
rather than the average roughness has also been reviewed in
this paper. This roughness parameter is equal to the sum of the
roughness parameters Rp and FdRa:
εFp = Rp + FdRa
(6)
This distance, shown in Fig. 1, is represented as the distance
from the average of the peak values to the averages of the floor
values of the surface roughness profile. Kandlikar et al. [1] can
be referred to for more information.
OBJECTIVES OF THE PRESENT WORK
• Collect and examine a variety of surface roughness
samples of practical interest in microfluidics
applications
• Examine the surfaces using a Mitutoyo SJ-401 Stylus
Profilometer with a 5 micron diamond tip, and a
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Copyright © 2007 by ASME
•
Keyence VHX-500K Digital Microscope with a VHX515 profile measurement unit
Calculate Ra and εFp for all surfaces, and discuss the
suitability of the roughness parameters in representing
different roughness features
ANALYSIS
The following surface roughness samples were collected
for analysis:
• Copper, aluminum, and stainless steel pieces were
milled with a 12.7 mm (½”) diameter 2 flute high
speed end mill
• Samples of copper, aluminum, and stainless steel were
flycut with a single point, carbide tipped tool bit with a
1 mm (0.040”) radius
• Samples of copper, aluminum, and stainless steel were
surface ground using a 60 grit wheel
• Copper, brass, and stainless steel capillary tubes were
milled open to examine the surface roughness inside
• Copper microchannels, machined with a 200 µm thick
roller
• A pure nickel microfinish surface comparator, having
22 surfaces from a range of machining techniques
including
o grinding
o blanchard grinding
o shape turning
o lapping
o profiling
o milling
• Three silicon wafers with
etched
microchannels
were also analyzed. The
etching
processes
included
o potassium
hydroxide
(KOH)
o xenon diflouride
(XeF2)
o deep
reactive
ion
etching
(DRIE)
For this review, two methods
were used to measure the surface
roughness of all the specimens
gathered; a Mitutoyo SJ-401
Stylus Profilometer with a 5
micron diamond tip, and a
Keyence VHX-500K Digital
Microscope with a VHX-515
profile measurement unit. Various
roughness
parameters
were
accessed for all the surface
roughness specimens, including
Ra, Rp, and Rv. The parameters
proposed by Kandlikar et al. [1]
were also examined; FdRa and the
roughness, εFp. Mitutoyo Surfpak software was used with the
profilometer to collect surface profile data and the average
roughness values. Microsoft Excel was used to calculate the
Rp, Rv, and FdRa parameter values using the profilometer
profile data. The profilometer was run to collect data eight
times for each surface examined. The measurement length of
each test was 4 mm, and out of the eight surface profiles
generated using the profilometer, the top six profiles were
chosen for calculating the surface roughness parameters in
order to reduce faulty data.
The surface roughness parameters Ra, Rv, Rp, FdRa, and
εFp were calculated for every surface examined using equations
1-6, and the results are tabulated and shown in Table 1. In the
following sections, detailed results for the surfaces examined
are presented.
The roughness parameter, εFp, is being
considered for replacing the currently accepted average
roughness value, Ra, and the validity of this roughness
parameter will be assessed.
FLYCUT SAMPLES
The samples of copper, aluminum, and stainless steel that
had been flycut were examined first. Fly cutting is a process
where a single point cutting bit similar to that of a lathe is
mounted in an end mill using a special holder. This process can
be used for many machining techniques, including cutting,
drilling, grinding, and sanding.
Figure 2 shows a three dimensional photograph of the
copper flycut surface as well as a surface roughness profile
obtained from the stylus profilometer. Color was added to the
Table 1: Surface Roughness Parameters
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Copyright © 2007 by ASME
Figure 2: Copper Flycut Surface, magnified 450x, and Surface Profile [Ra=0.31 µm, εFp =1.04 µm]
three-dimensional surface picture to make the height
differences more visible. The stainless steel and the aluminum
flycut surfaces looked very similar to the copper flycut surface.
However, the stainless steel flycut surface appeared slightly
rougher, and the aluminum surface appeared smoother than the
copper flycut surface.
As seen from Table 1, the average roughness of the
aluminum, copper, and stainless steel flycut surfaces were 0.22
µm, 0.31 µm and 0.29 µm, respectively. These average
roughness values portray that the aluminum has the smoothest
of all three flycut surfaces and that the copper surface is the
roughest. The roughness parameter, εFp, shows a slightly
different picture, with aluminum having the smallest εFp of 0.10
µm, copper having a εFp of 1.04 µm, and stainless steel having
the highest value of 1.06 µm. These values are shown in table
1, as well as the calculated Rp, Rv, and FdRa values.
The photographs of each flycut surface obtained from the
microscope correspond better to the roughness value, εFp than to
the average roughness values. The aluminum surface looked
the smoothest, and had the lowest εFp value. The stainless steel
flycut sample appeared to be the roughest, and had the highest
εFp value of all three materials. The aluminum flycut surface
looked to be overall more smooth but with areas where the
surface roughness was more prominent than others. The copper
and the stainless steel flycut surfaces appeared to be more
uniformly rough.
SURFACE GROUND SAMPLES
Surface grinding was performed on samples of copper,
aluminum, and stainless steel in a 60 grit wheel. Grit is a unit
of measurement used with sandpaper and grinding wheels; it
refers to the number of abrasive particles within a square inch
of the grinding wheel. The higher the grit, the smoother the
Figure 3: Stainless Steel 60 grit Surface Ground, 3-D photograph magnified 450x,
and Surface Profile [Ra=0.37 µm, εFp =1.33 µm]
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Copyright © 2007 by ASME
Figure 4: Aluminum Milled Surface, 3-D photo magnified 450x, and Surface Profile [Ra=0.64 µm, εFp=2.12 µm]
ground surface will be. Used mainly for heavy sanding and
stripping purposes, 60 grit is considered coarse.
The three samples that had been surface ground with a 60
grit grinding wheel were examined with the profilometer. The
Ra values that were calculated for these three materials reported
that the aluminum was the smoothest surface of the three, with
Ra equal to 0.25 µm. The highest Ra value was calculated to
be 0.37 µm for the stainless steel sample. The copper sample
fell in between these two, with an average roughness of 0.29
µm.
Figure 3 shows a three-dimensional picture of the stainless
steel surface ground sample.
In the three-dimensional
photographs, the aluminum surface ground sample looks the
smoothest of the three, but also looks to be somewhat wavy.
The roughness value εFp for the aluminum sample is the lowest
of the three at 1.12 µm. The highest value of εFp is for the
copper sample, at 1.34 µm. The stainless steel falls in between
these two values at 1.33 µm. The copper surface ground
sample seemed to have more peaks and valleys than the
aluminum and stainless steel samples, which appeared of more
uniform roughness. The tabulated values for the surface
roughness parameters can be found in Table 1.
END MILLED SAMPLES
The next set of samples analyzed were the aluminum,
copper, and stainless steel samples that had been milled in a
12.7 mm (½”) diameter 2 flute high speed end mill. A threedimensional photograph of the aluminum milled surface, and a
surface profile obtained from the profilometer is shown in Fig.
4. In this photograph, the path taken by the end mill across the
surface is evident. While this path is visible on all three
surfaces, it is most prominently viewed on the aluminum milled
surface. Though the aluminum surface shows the machining
processes the most vividly, it had the lowest calculated Ra
value of 0.64 µm, and appears to be the smoothest of the three
surfaces. The stainless steel milled surface appears to be much
rougher than the aluminum or copper milled samples, and had
the highest Ra value of 1.2 µm. The εFp values followed a
different pattern than the average roughness, with the copper
taking the smallest value of 0.90 µm. The stainless steel had a
much higher roughness value of 4.27 µm. These surface
roughness parameters, and others, are shown in Table 1. The
milled surfaces were more noticeably rough than the flycut or
surface ground samples. The εFp values that were calculated
seemed to be a better correlation for discussing surface
roughness in terms of fluid flow than the average roughness
values; the average roughness values have been consistently
lower than the εFp values for all surface roughness samples
examined. Since the εFp roughness value better represents the
height of the profile peaks than the average roughness value, it
more accurately represents the obstructions the fluid flow
observes.
NICKEL SURFACE COMPARATOR - LAPPED
The lapped surfaces on the nickel surface comparator were
the smoothest samples that were investigated. The 2L surface
represents that the surface has been lapped with a roughness
height of 2 microinches (0.05 µm). Only two lapped surfaces
were available for analysis. The 2L surface had an average
roughness value of 0.04 µm and a roughness value, εFp, of 0.15
µm. The second lapped surface had a roughness height of 4
microinches, and was represented with the characterization 4L.
This lapped surface brought in a higher average roughness
value of 0.08 µm and a roughness value εFp of 0.24 µm. It is
interesting to note that as the roughness height value doubles
from two to four, the roughness value, εFp, increases by a factor
of approximately 1.65.
NICKEL SURFACE COMPARATOR - GROUND
The nickel microscale surface comparator had six different
ground surfaces. Four were surface ground with the periphery
of a grinding wheel, and the last two were blanchard ground,
which is where the flat of the grinding wheel is used to grind
the sample smooth. The four samples that were ground with
the periphery of the wheel had roughness heights of 0.2, 0.41,
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Copyright © 2007 by ASME
0.81, and 1.6 µm (8, 16, 32, and 63 microinches). The
blanchard ground surfaces had roughness heights of 0.41 and
0.81 µm (16 and 32 microinches). The tabulated values for the
surface roughness parameters can be found in Table 1.
Figure 5 displays the three-dimensional photographs of the
16G and 16BL surfaces and their corresponding surface
profiles. It was assumed that the average roughness and εFp
value would increase as the roughness heights of the samples
increased, which proved to be a valid assumption.
For the ground surfaces, the average roughness was the
lowest for the 8G surface, at 0.13 µm. This surface also had the
lowest roughness value, with εFp of 0.58 µm. The 16G surface
had the next lowest Ra value of the six ground surfaces, with
Ra equal to 0.389 µm. The εFp value of the 16G surface was
also the second lowest, at 1.63 µm. The blanchard ground
16BL surface was just slightly rougher than the 16G surface,
with Ra equal to 0.45 µm and εFp equal to 1.76 µm. Table 1
provides are more in-depth summary of the surface roughness
parameters calculated.
As would be expected, the 32G and 32BG surfaces were
rougher than the ground surfaces with roughness heights of
0.41 µm (16 microinches). The average roughness value for the
32G surface was 1.008 µm and εFp was equal to 3.6 µm.
Interestingly enough, the blanchard ground surface was now
smoother than the ground surface. The 32BG surface had an
average roughness of 0.65 µm and εFp was equal to 2.48 µm.
There is no identifiable relationship between the blanchard
ground surface and the ground surface roughness values; one
type of grinding is not consistently rougher than the other.
The next ground roughness height on the surface
comparator was 1.6 µm (63 microinches). The average
roughness of the 63G surface was 1.44 µm while the value for
εFp was equal to 6.85 µm. As expected, both the average
roughness value and the value εFp of the samples increased as
the roughness height increased.
Figure 5: Nickel Surface Comparator, 16BL 3-D Drawing and Surface Profile [Ra=0.45 µm, εFp=1.77 µm] (Top)
and 16G 3-D Drawing and Surface Profile [Ra=0.39 µm, εFp=1.64 µm] (Bottom)
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Copyright © 2007 by ASME
Figure 6: Silicon Wafer with DRIE etch (left) and Silicon Wafer with XeF2 etch (right)
NICKEL SURFACE COMPARATOR - MILLED
The next type of surface machining that was available on
the nickel surface comparator was milling. The smallest height
of a milled surface available was 1.6 µm. As shown in Table 1,
the average roughness value for this sample was equal to 1.22
µm. This value was lower than the average roughness of the
ground surface with a 1.6 µm roughness height. For this milled
surface, the εFp value equaled 5.05 µm, which was also smaller
than that of the ground surface with a roughness height of 1.6
µm.
surface etched with KOH.
Three dimensional photographs were obtained for the
silicon wafers etched with XeF2 and DRIE, and are shown in
Fig. 6.
CAPILLARY TUBES
The three capillary tubes that were milled open were
examined using the VHX-500 digital microscope. The brass
and copper capillary tubes had a larger diameter of 1.067 mm
SILICON MICROCHANNELS
While it was possible to use the profilometer to obtain a
surface profile and roughness parameters for the silicon wafer
that had been etched with KOH, it was not possible to use the
profilometer on the silicon wafers etched with XeF2 or deep
reactive ion etching. The silicon wafer etched with XeF2 had
tiny serpentine microchannels that were not straight or long
enough to examine with the profilometer. The wafer that had
been etched with DRIE couldn’t be examined with the
profilometer due to the localized damage that would result.
The KOH wafer was examined, and it was found that the Ra
value was 0.13 µm and the εFp value was 0.34 µm. Table 1
depicts the surface roughness parameters calculated for the
Figure 8: Copper Capillary Tube, magnified 450x
with added color and height
Figure 7: Copper Capillary Tube, magnified 450x
Figure 9: Brass Capillary Tube, magnified 450x
with added color and height
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Copyright © 2007 by ASME
and the stainless steel capillary tube had a diameter of 0.62 mm.
These samples had not been milled deeply enough to allow for
examination with the profilometer, yet despite the lack of a
surface profile to perform roughness parameter calculations, the
capillary tubes were examined under the microscope. The
stainless steel and copper capillary tubes looked to have similar
surface roughness, but the brass capillary tube had more
evident roughness peaks. Figure 7 displays the interior of the
copper capillary tube as photographed with the digital
microscope and Fig. 8 displays the same photograph but with
added color and height to enhance visualization of the surface
roughness. Figure 9 shows a photograph acquired from the
microscope of the brass capillary tube, with enhanced color and
height to show the difference in the surface roughness. From
these photographs, it is evident that the brass capillary has more
prominent peaks than the other two capillary tubes.
SURFACE ROUGHNESS IN FLUID FLOW
A companion paper by Brackbill and Kandlikar [11]
examines and further confirms the effect of surface roughness
parameters on the friction factor for fluid flow. In this
experiment, three surfaces were used for analysis. The first
surface was a smooth lapped surface. The two other samples
were surface ground, one with a 60 grit grinding wheel and the
second with a 100 grit grinding wheel. This paper also
discusses the new roughness value proposed by Kandlikar et al.
[1]. Brackbill and Kandlikar [11] noticed that the parameter εFp
was more representative of the sand grain roughness in
Nikuradse’s experiments than the average roughness parameter.
DISCUSSION
The roughness parameter values calculated are shown in
Table 1 for all the surface roughness samples investigated. It is
interesting to note that surfaces resulting from different
machining processes or etches could have a similar average
roughness value but have values that do not match for εFp. For
example, the average roughness value for the silicon wafer
etched with KOH and the 8G sample of the nickel surface
comparator are the same, at 0.13 µm. Though the two surfaces
share an average roughness value, the values of εFp for the two
surfaces differ. For the silicon wafer etched with KOH, the εFp
value is 0.34 µm, and for the 8G nickel surface comparator
surface εFp is equal to 0.58 µm. The two surfaces have the
same average roughness values, yet the difference in the values
of εFp is 0.24 µm. For fluid flow applications, the εFp value is
more representative of the effect of the surface features on the
fluid flow, and provides a more accurate model of the surface
Figure 10: Relationships between Ra and εFp values for surface roughness samples
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Copyright © 2007 by ASME
topography than the currently accepted average roughness
parameter. Brackbill and Kandlikar [11] confirm this in their
paper. Using a new constricted diameter calculated using the
proposed roughness parameter εFp, they calculated the friction
factor of microscale flows. It was found that these theoretical
results more accurately predicted the friction factors
experimentally determined than if a non-constricted diameter
had been used.
A chart of the values of Ra and εFp calculated for all the
surfaces examined is shown in Fig. 10. From looking at this
chart, another example of two surfaces that share a common
average roughness value but have unique values for εFp is
evident between the 63M surface of the nickel surface
comparator and the stainless steel milled surface. The average
roughness of the 63M nickel comparator surface is 1.21 µm and
for the stainless steel milled surface, it is 1.20 µm. The εFp
values for these two surfaces shows that the two surfaces aren’t
as alike as the average roughness value implies; the εFp for the
63M nickel comparator surface is 5.05 µm and for the stainless
steel milled sample it is only 4.27 µm.
Another example where the average roughness parameter
portrays an inaccurate representation of surface roughness is
evident from a comparison between the copper ground surface
and the stainless steel flycut surface. The copper ground
surface has an average roughness equal to 0.31 µm, and the
stainless steel flycut surface has a similar Ra value of 0.3 µm.
However, the εFp value for the copper ground surface is higher
than that of the stainless steel. The εFp value for the stainless
steel flycut surface is 1.07 µm while that of the copper surface
ground sample is 1.34 µm. This trend repeats itself with other
surfaces that have been examined, further proving that the εFp
value would provide better correlations between surface
roughness and fluid flow than the currently accepted surface
roughness value.
CONCLUSIONS
The average roughness value, while commonly used in
fluid flow applications, is not adequate for characterizing
surface roughness. A surface that has many deep pits but is
otherwise smooth could have a similar average roughness value
as a surface that has a low uniform roughness, such as
sandpaper. Kandlikar et al. [1] propose the use of a new
surface roughness parameter, εFp. This value is equal to the
distance between the floor mean line and the peak value of the
surface profile. The samples examined were of practical
interest for microfluidics applications; the materials,
microfabrication techniques, and machining processes
employed to create the surface roughness samples examined are
often used to fabricate minichannels, microchannels, and
capillary tubes. When both the average roughness and εFp
parameters were calculated for various surfaces examined in
this paper, it was evident that the value of εFp is often greater
than that of the average roughness. It was also noticed that
many scenarios arise where the average roughness of two
surfaces can be similar while the εFp values may differ greatly.
Since the εFp parameter is more indicative of the height of the
surface roughness, it better models the obstructions the fluid
flow encounters. Thus, the εFp parameter would provide a more
accurate parameter for modeling the effects of surface
roughness on fluid flow.
REFERENCES
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J.B., “Characterization of Surface Roughness Effects on
Pressure Drop in Single-Phase Flow in Minichannels and
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Minichannels, ICMM2005, Jun 13-15 2005, Anonymous
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