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Application of Lubrication Theory and Study of Roughness Pitch During
Laminar, Transition, and Low Reynolds Number Turbulent Flow at
Microscale
Timothy P. Brackbilla; Satish G. Kandlikara
a
Rochester Institute of Technology, Rochester, New York, USA
Online publication date: 01 February 2010
To cite this Article Brackbill, Timothy P. and Kandlikar, Satish G.(2010) 'Application of Lubrication Theory and Study of
Roughness Pitch During Laminar, Transition, and Low Reynolds Number Turbulent Flow at Microscale', Heat Transfer
Engineering, 31: 8, 635 — 645
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Heat Transfer Engineering, 31(8):635–645, 2010
C Taylor and Francis Group, LLC
Copyright ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903466621
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Application of Lubrication Theory
and Study of Roughness Pitch During
Laminar, Transition, and Low
Reynolds Number Turbulent Flow
at Microscale
TIMOTHY P. BRACKBILL and SATISH G. KANDLIKAR
Rochester Institute of Technology, Rochester, New York, USA
This work aims to experimentally examine the effects of different roughness structures on internal flows in high-aspect-ratio
rectangular microchannels. Additionally, a model based on lubrication theory is compared to these results. In total, four
experiments were designed to test samples with different relative roughness and pitch placed on the opposite sides forming
the long faces of a rectangular channel. The experiments were conducted to study (i) sawtooth roughness effects in laminar
flow, (ii) uniform roughness effects in laminar flow, (iii) sawtooth roughness effects in turbulent flow, and (iv) varying-pitch
sawtooth roughness effects in laminar flow. The Reynolds number was varied from 30 to 15,000 with degassed, deionized
water as the working fluid. An estimate of the experimental uncertainty in the experimental data is 7.6% for friction factor
and 2.7% for Reynolds number. Roughness structures varied from a lapped smooth surface with 0.2 µm roughness height
to sawtooth ridges of height 117 µm. Hydraulic diameters tested varied from 198 µm to 2,349 µm. The highest relative
roughness tested was 25%. The lubrication theory predictions were good for low relative roughness values. Earlier transition
to turbulent flow was observed with roughness structures. Friction factors were predictable by the constricted flow model
for lower pitch/height ratios. Increasing this ratio systematically shifted the results from the constricted-flow models to
smooth-tube predictions. In the turbulent region, different relative roughness values converged on a single line at higher
Reynolds numbers on an f–Re plot, but the converged value was dependent on the pitch of the roughness elements.
INTRODUCTION
Literature Review
Work in the area of roughness effects on friction factors in internal flows was pioneered by Colebrook [1] and Nikuradse [2].
Their work was, however, limited to relative roughness values
of less than 5%, a value that may be exceeded in microfluidics
application where smaller hydraulic diameters are encountered.
Many previous works have been performed through the 1990s
with inconclusive and often contradictory results.
Moody [3] presented these results in a convenient graphical
form. The first area of confusion is the effect of roughness structures in laminar flow. In the initial work, Nikuradse concluded
that the laminar flow friction factors are independent of relative
roughness ε/D for surfaces with ε/D < 0.05. This has been accepted into modern engineering textbooks on this topic, as is
Address correspondence to Satish G. Kandlikar, Mechanical Engineering
Department, Rochester, NY 14623, USA. E-mail: [email protected]
evidenced through the Moody diagram. Previous work [4, 5]
has shown that the instrumentation used in Nikuradse’s experiments had unacceptably high uncertainties in the low Reynolds
numbers range. Additionally, all experimental laminar friction
factors were seen to be higher than the smooth channel theory
in Nikuradse’s study. Works beginning in the late 1980s began
to show departures from macroscale theory in terms of laminar
friction factor; however, the results were mixed and contradictory. These works are numerous, and for brevity are summarized
in Table 1. High relative roughness channels are also of interest
in this study, and ε/D values up to 25% are tested in this article.
The effect of pitch on friction factor is another important area.
Rawool et al. [6] performed a computational fluid dynamics
(CFD) study on serpentine channels with sawtooth roughness
structures of varying separation, or pitch. They showed that the
laminar friction factors are affected with varying pitch. This
effect has not been studied in the literature, and is an open area.
Several models have attempted to characterize the effect of
roughness on laminar microscale flow. Chen and Cheng [7]
635
636
T. P. BRACKBILL AND S. G. KANDLIKAR
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Table 1 Previous experimental studies
Study
Year
Roughness
Dh (mm)
Mala and Li [11]
Celata et al. [15]
1999
2002
1.75 µm
0.0265
Li et al. [20]
2000
0.1% RR to 4% RR
Kandlikar et al. [29]
2001
1.0–3.0
Bucci et al. [12]
2003
0.3% to 0.8% RR
172–520
Celata et al. [27]
2004
31–326
Peng et al. [23]
Pfund et al. [8]
Tu et al. [13]
1994
2000
2003
0.05 µm smooth, 0.2–0.8 µm
rough
—
Smooth 0.16 and 0.09, rough 1.9
Ra < 20 nm
Baviere et al. [14]
Hao et al. [21]
2004
2006
5–7 µm
Artificial 50 × 50 µm RR 19%
Shen et al. [22]
2006
4% RR
Wibel et al. [24]
Wu et al. [19]
Wu et al. [18]
Weilin et al. [26]
2006
1983
1984
2000
1.3 µm (∼0.97% RR)
0.05–0.30 height
0.01 height
2.4–3.5%
Wu et al. [17]
2003
3.26e-5 to 1.09e-2
f
50–254
130
Greater than predicted
Re < 583 classical, 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
Re < 800–1000 follows classical
79.9–449
620 and 1,067
Tentatively propose higher than
normal friction
α makes some +, some –
Higher, highest for rough
RR < 0.3%, conventional, RR =
0.35%, f is 9% higher
Increased laminar friction
Follows theory until Re = 900,
then higher, indicating transition
Higher, and Po number increases
with Re, nothing at low Re
Near classical values
Greater than predicted
Greater than predicted
Higher and larger slope for Px–Re
(18–32%)
Roughness increased it, surface
type varied it
0.133–0.343
252.8–1,900
69.5–304.7
153–191
436
∼133
45.5–83.1
134–164
51–169
∼100
created a model for pressure drop in roughened channels based
on a fractal characterization and an additional empirical modification. The additional experimental data was drawn from the
results by Pfund [8]. Bahrami et al. [9] used a Gaussian distribution of roughness in the angular and longitudinal directions
for a circular microtube. Although not presented in the work,
the average error of this model when compared to experimental
results from multiple authors appears to be about 7%, judging
from the 10% error bars presented. Zou and Peng [10] used a
constricted area model based on the height Rz of the elements.
They then applied an additional empirical correction to account
for reattachment of laminar flow past the roughness elements.
Finally, Mala and Li [11] constructed a model by modifying
the viscosity of the fluid near the roughness elements. Their
modification is based on the results of CFD studies.
A few studies have looked at turbulent flow in microchannels
in the past. Due to the high pressure drops required and difficulty in testing, very limited work is available. Some previous
researchers found that microchannel turbulent results matched
the Colebrook equation in the few tests that went into the
turbulent regime [12–14]. Another study by Celata et al. [15]
found that the Colebrook equation overpredicted the results of
experimentation.
Reo
Increases with decreasing Dh
1,881–2,479 is transition region
1,700–1,900 for rough tubes
Lowered w/ roughness
1,800–3,000, abrupt transition for
high RR
200–700
Approach 2,800 w/ larger
2,150–2,290 w/ RR < 0.3%, 1,570
for 0.35%
increased with roughness
Transition ∼900
N/A
1,800–2,300; varies with aspect ratio
1,000–3,000
N/A
N/A
parameters are illustrated graphically in Figure 1. The parameters are listed next, as well as how they are calculated. These
values are established to correct for the assumption that different
roughness profiles with equal values of Ra , average roughness,
may have different effects on flows with variations in other profile characteristics. For example, a roughness surface with twice
the pitch but the same Ra may have a different pressure drop.
•
The Mean Line is the arithmetic average of the absolute values
of all the points from the raw profile, which physically relates
to the height of each point on the surface. Note that Z is the
height of the scan at each point, i. It is calculated from the
following equation:
1
|Zi |
n
i=1
n
Mean Line =
(1)
Roughness Characterization
Recently, Kandlikar et al. [16] proposed new roughness parameters of interest to roughness effects in microfluidics. These
heat transfer engineering
Figure 1 Generic surface profile showing roughness parameters in a graphical manner.
vol. 31 no. 8 2010
T. P. BRACKBILL AND S. G. KANDLIKAR
•
Rp is the maximum peak height from the mean line, which
translates to the highest point in the profile sample minus the
mean line.
Rp = max(Zi ) − Mean Line
(2)
•
RSm is defined as the mean separation of profile irregularities,
or the distance along the surface between peaks. This is also
defined in this article as the pitch of the roughness elements.
It can be seen in Figure 1.
• Fp is defined as the floor profile. It is the arithmetic average
of all points that fall below the mean line value. As such, it is
a good descriptor of the baseline of the roughness profile.
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Let z ⊆ Z s.t. all zi = Zi iff Zi < Mean Line
Fp =
•
n
1 zi
nz
i=1
(3)
FdRa is defined as the distance of the floor profile (Fp) from
the mean line:
FdRa = Mean Line − Fp
(4)
εFP , or the value of the roughness height, is determined by
the following equation:
(5)
εFP = Rp + FdRa
Using these parameters, Kandlikar et al. [16] replotted the
Moody diagram using a constricted diameter defined as Dt = D
– 2 εFP . The resulting expression for the friction factor based on
Dt is as follows:
(Dt − 2εFP ) 5
fMoody,cf = fMoody
(6)
Dt
The constricted-diameter-based friction factor and Reynolds
number yielded a single line in the laminar region on the Moodytype plot. In the turbulent region, all values of relative roughness
between 3 and 5% plateau to a friction factor of f = 0.042 for
high Reynolds numbers.
Objectives of the Present Work
637
5. Turbulent flow—In the turbulent regime, sawtooth samples
are tested to high Reynolds numbers.
Application of Lubrication Theory
The application of the constricted parameter set is based on
theory, in addition to being a practical method for predicting
channel performance. A simple derivation from the Navier–
Stokes (NS) equation with lubrication approximations yields a
very similar concept. Originally intended for looking at hydrodynamic effects in fluid bearings, lubrication theory allows one
to account for slight wall geometry variances while keeping the
solution analytical. The structure of the problem is as follows.
A rectangular duct is formed in two dimensions using unknown
functions f(x) for the bottom face and h(x) for the top face. The
simple diagram for analysis can be seen in Figure 2.
To analyze the system, the following assumptions are made.
The separation of the system is assumed to be much smaller than
the length, and the slope of the roughness is also assumed to be
small. The gravity effects are negligible compared to pressure
drop in the x direction. The flow is assumed to be incompressible
and steady, with entry and exit regions ignored, since the analysis
is applied to the fully developed flow. It is also assumed that
there is no velocity in the y direction. Referring to Figure 2, the
following assumptions are made:
1. (h – f) << L for all x.
2. uy = 0: No flow into/out of page.
3. Lubrication approximation: Neglect uz in Navier–Stokes
equations.
4. Incompressible flow.
5. Ignore gravity; (h – f) is small for all x.
x
= 0.
6. ∂u
∂x
7. Flow does not vary in y direction.
8. Steady flow.
9. Flow is unidirectional and fully developed.
Using the assumption of incompressibility and no flow in y
direction, the continuity equation, Eq. (7), simplifies to Eq. (8):
∂uy
∂uz
∂ux
+
+
=0
∂x
∂y
∂z
(7)
∂ux
∂uz
+
=0
∂x
∂z
(8)
The objectives of the present work are summarized here:
1. Investigate the applicability of lubrication theory and examine it as a basis of constricted diameter
2. Laminar flow—Examine effects of both sawtooth and uniform roughness structures at higher values of ε/D, from
smooth to 25% relative roughness.
3. Laminar flow—Examine pitch effect on laminar flow for
sawtooth roughness using samples with the same roughness
height but varying pitches.
4. Laminar–turbulent transition—Study the effect of roughness
on the laminar–turbulent transition.
heat transfer engineering
Figure 2 Illustration of lubrication problem.
vol. 31 no. 8 2010
638
T. P. BRACKBILL AND S. G. KANDLIKAR
The Navier–Stokes equations are written and simplified in
each direction. The simplified forms are as follows:
Downloaded By: [Rochester Institute of Technology] At: 19:21 10 June 2010
∂2 ux
1 ∂P
=
x − direction
µ ∂x
∂z2
y − direction
∂P
=0
∂y
(10)
z − direction
∂P
= ρgz = 0
∂z
(11)
With the NS equations, continuity equation, and boundary conditions (BCs), we have enough information to analytically solve
this problem. First, the velocity in the x direction is found. After
integrating the x direction, two constants arise, which are found
with BCs 1 and 2. The resulting form of flow in the x direction
is given by Eq. (12):
1 ∂P
[(z − f)2 − (h − f)(z − f)]
2µ ∂x
∂ux
dz +
∂x
f
h
∂ux
dz + uz |hf = 0
∂x
This equation is integrated once to get the form shown in
Eq. (16). It can be intuitively seen that integrating x velocity
across the gap will give volumetric flow rate (Q) per width of
the channel (a). As such, the constant of integration is expressed
as Q/a:
h
ux dz = constant = Q/a
The expression derived in Eq. (16) is substituted in for ux from
Eq. (12) and then integrated. The result of this integration is:
− (h − f)3 dP
Q
=
a
12µ dx
h
ux dz =
f
dh
∂ux
df
dz + ux |h + ux |f
∂x
dx
dx
(17)
This equation is very similar to the equation encountered
when solving the simple unidirectional problem of flow through
a narrow gap, while neglecting end effects. Now to have a more
useful form of this expression, Eq. (17) is solved for the partial derivative of pressure in the x direction. In actuality, this
partial derivative is in fact a normal derivative, since the NS
equations cancel the pressure terms in the y and z directions.
Since the problem is steady, pressure is only a function of
the x direction. This allows us to integrate Eq. (18) to obtain
Eq. (17).
−12µQ
P2 − P1 =
a
L
0
1
dx
(h − f)3
(18)
(13)
From boundary conditions 1 and 2, we see that uz evaluated
at both f and h is 0, which removes that term. To integrate the
remaining term, we apply Liebnitz’s Rule to rewrite the first
term as shown in Eq. (14):
h
(16)
f
−b3eff dP
Q
=
a
12µ dx
f
d
dx
(15)
For analysis purposes, we can now define a channel height
beff that will able to predict what friction factor will be present
when two samples of known roughness profiles are placed into
the test apparatus. If we look back to Eq. (18) and use beff defined
as beff = h – f, we can rewrite it as shown in Eq. (19):
∂uz
dz = 0
∂z
f
h
ux dz = 0
(12)
Now to account for the velocity in the z direction, we integrate
the continuity equation over the gap spacing.
h
h
f
ux = uz = 0 at z = f(x).
ux = uz = 0 at z = h(x).
P = P1 at x = 0.
P = P2 at x = L.
ux =
d
dx
(9)
Next, the boundary conditions of the problem must be set. A
no-slip (NS) boundary condition is applied at both the top and
bottom surfaces, f(x) and h(x), respectively. The pressure at each
end of the channel is also defined. Since the pressure variation
in the y direction is negligible compared to the variation in the
x direction, gravity is neglected, and the form of the pressure
boundary conditions is simply defining a single static pressure of
both entrance and exit. The boundary conditions are listed here:
1.
2.
3.
4.
(13) in a form that is easy to integrate:
(14)
Integrating this function as we did before, we can obtain a
function for change in pressure using the effective height, given
in Eq. (20):
P2 − P1 =
f
At this point, we again use boundary conditions 1 and 2 to
eliminate the last two terms in Eq. (14). We can now rewrite Eq.
heat transfer engineering
(19)
−12µLQ
a (beff )3
(20)
To obtain a relationship to determine the effective height, we
can equate the right sides of Eqs. (18) and (20). When simplified,
vol. 31 no. 8 2010
T. P. BRACKBILL AND S. G. KANDLIKAR
639
we are left with the expression in Eq. (21):
⎡
⎤1/3
⎢
⎢
beff,theory = ⎢ L
⎣
0
L
1 dx
(h−f)3
⎥
⎥
⎥
⎦
(21)
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To derive an heff value from experimentation, all that is
needed is a rearrangement of Eq. (20) into the form of Eq.
(22). Since P1 ,P2 , Q, a, L, and µ are known in the experiment,
it is easy to find beff in Eq. (22).
−12µLQ 1/3
(22)
beff,exp =
a (P2 − P1 )
This theory should be able to predict the effects of small
roughness elements of low slope. Once we surpass the assumptions of this theory, that is, have roughness heights that are not
much less than the channel gap, irreversible effects will cause
the uniform flow assumption to break down. To further this theory to apply to truly two-dimensional flows, a model needs to
be added to account for these added effects on flow.
EXPERIMENTAL SETUP
Test Setup
The test setup is developed to hold the roughness samples and
vary the gap. A simple schematic of the arrangement is shown in
Figure 3. All test pieces are machined with care to provide a true
rectangular flow channel. 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 number 60 drill (diameter
of 1.016 mm) along the channel. The taps begin at the entrance
to the channel and are spaced every 6.35 mm along the 88.9 mm
length. Each tap is connected to a 0–689 kPa (0–100 psi) differential pressure sensor with 0.2% FS accuracy. For turbulent
testing, a single pressure transducer set up in differential mode
is used past the developing region of the channel. The pressure
sensor outputs are put through independent linear 100 gain amplifiers built into the NI SCXI chassis to increase the accuracy.
The separation of the samples is controlled by two Mitutoyo micrometer heads, with ±2.5 µm accuracy. There is a micrometer
head at each end of the channel to ensure parallelism.
Degassed, deionized water is delivered via one of the three
pumps, depending on the test conditions. For turbulent testing
a Micropump capable of 5.5 lpm at 8.5 bar is used. For laminar
testing, a motor drive along with two Micropump metered pump
heads are used. One pump is for low flows (0–100 ml/min) and
the other for high flows (76–4,000 ml/min). The flow rate is
verified with three flow meters, one each for 13–100 ml/min,
60–1,000 ml/min, and 500–5,000 ml/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
heat transfer engineering
Figure 3 Experimental test setup: apparatus schematic.
a known period of time. Thermocouples are mounted on the
inlet and outlet of the test section. Fluid properties are calculated
at the average temperature. All of the data is acquired and the
system is 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.
Samples
For this testing, multiple roughness structures machined into
different sets of samples are used. The two types of roughness
examined were a patterned roughness with repeating structures
and a less structured cross-hatch design. For samples with sawtooth roughness elements, a ball end mill cutter is used in a
CNC (computer numerical control) mill to make patterned cuts
across the sample at very shallow depths. The remaining protrusions from the surface form the sawtooth-shaped elements.
The second method of roughness is formed using different grits
of sandpaper. The sandpaper is manipulated in a cross-hatch
pattern on the surface of the samples. With these two methods,
various different samples were created.
To validate the setup against conventional macroscale theory,
smooth samples were made by grinding everything square and
flat and then lapping the testing surface to reduce roughness. The
roughness parameters for the surfaces studied in this work are
summarized in Table 2. Figure 4 shows high-resolution images
of some of these surfaces using an interferometer and a confocal
microscope along with the traces normal to the flow direction.
Table 2 Summary of roughness on samples
Dimensions
Sample
∈FP µm
Ra µm
Pitch µm
Fitch to
Height Ratio
Smooth (Ground)
Smooth (Lapped)
100 Grit
60 Grit
405 µm Sawtooth
815 µm Sawtooth
503 µm Sawtooth
1008 µm Sawtooth
2015 µm Sawtooth
2.01
0.20
2.30
6.09
99.71
105.55
46.41
52.51
50.11
0.31
0.06
2.64
6.09
27.43
24.19
6.89
6.36
4.62
—
—
—
—
405
815
503
1008
2015
—
—
—
—
4.06
7.72
10.84
19.2
40.22
vol. 31 no. 8 2010
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640
T. P. BRACKBILL AND S. G. KANDLIKAR
Uncertainties
RESULTS
The propagation of uncertainty to the values of friction factor
and Reynolds number is obtained using normal differentiation
methods. The uncertainties of the sensors and readings are found
from the calibration performed on each sensor. For the pressure
sensors, 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
in these 30 points is recorded. The average of these maximum
errors is used for the error of the pressure sensors. The same
procedure is performed for each of the three flow sensors. This
approach yields conservative error values of 1% for pressure
sensors and around 2.2% for the flow sensors. Using this analysis, the maximum errors occur at the smallest value of b at the
lowest flow rates encountered. These uncertainties are at worst
7.6% for friction factor and 2.7% for Reynolds number.
Smooth Channel Validation
The smooth channel friction factors are plotted against
Reynolds number over a range of hydraulic diameters tested
in Figure 5. Note that not all data points for each hydraulic diameter are shown for simplicity of the plot. To acquire this range
of hydraulic diameters, the lapped samples are held at varying
gap spacing. Laminar theoretical friction factor is plotted as a
solid black line (Eq. (23)) and is given as by Kakaç et al. [30]
in Eq. (23). The agreement is quite good as expected, within the
experimental uncertainties of 7%. Transition to turbulence is
deduced as a departure from the laminar theory line. The range
of turbulent transition Reynolds numbers is between 2,000 and
2,500, as is also expected. For accurately calculating turbulent
transition, the data points are normalized to laminar theory, and
Figure 4 High-resolution images of the tested roughness surfaces and line traces in a direction normal to the flow.
heat transfer engineering
vol. 31 no. 8 2010
T. P. BRACKBILL AND S. G. KANDLIKAR
Downloaded By: [Rochester Institute of Technology] At: 19:21 10 June 2010
Figure 5 Verification of friction factor versus Reynolds numbers for five
hydraulic diameters spanning the range used in experimentation. Amount of
data presented culled for clarity.
a 5% departure is used to determine transition.
f=
24
(1 − 1.3553α + 1.9467α 2 − 1.7012α 3
Re
+ 0.9564α − 0.2537α )
4
5
641
increases, regardless of whether the roughness structures are repeating or uniform roughness. At the highest relative roughness
of 27.6%, the data is far above the theory. These data also contradict Nikuradse’s finding that roughness less than 5% relative
roughness (RR) has no effect on laminar flow.
When the experimental data are replotted with the constricted
parameters using bcf as the gap, the experimental data fit the
theoretical curve quite well. This confirms the validity of using
the constricted flow diameter in predicting the laminar friction
factors as recommended in [16].
The other interesting feature of Figure 6 is that the transition
to turbulence decreases dramatically and systematically as the
relative roughness increases. For the 27.6% samples, transition
to turbulence can be observed at Reynolds numbers as low as
200. This can be explained by noting that adding perturbations
near the channel walls will increase chaos in the flow even before
smooth channel turbulence.
Laminar Regime—Varying Pitches
(23)
Laminar Regime—Varying Relative Roughness
Once the smooth channel results validated the setup and
testing methods, widely varied roughened samples were tested
using the same methodology.
The experimental friction factors of selected roughened sawtooth and uniform samples are plotted against Reynolds number
in Figure 6. On the left side the data are plotted using the unconstricted base parameters of the channels. The gap (b) in this
unconstricted case is defined as the distance from Fp of the
top roughness sample to Fp of the bottom roughness sample.
When plotted with the unconstricted parameters, a clear disparity is seen with respect to the theory. As the relative roughness
increases, the disparity between theory and experiment also
The preceding roughened results hold for roughness that has
structures that are close to each other, similar to the resultant
surface profiles of machined parts. It is apparent that as the
pitch of roughness elements becomes larger and larger, eventually the channel will more resemble a smooth channel with
widely spaced protrusions into the flow. At large enough separations between roughness structures, or large pitches, the use of
constricted parameters will stop providing meaningful results.
To test where this occurs, samples with pitches varying from
503 to 2,015 µm with nearly equivalent roughness heights are
tested in the laminar regime. The roughness element height in
all cases is close to 50 µm and has the same sawtooth shape. The
samples are tested at two constricted separations, 400 µm and
500 µm. The plots of friction factor versus Reynolds numbers
for both separations are shown in Figure 7 and are plotted with
constricted parameters.
Figure 6 Data plotted with (a) root parameters and (b) constricted parameters.
heat transfer engineering
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642
T. P. BRACKBILL AND S. G. KANDLIKAR
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Figure 9 Effect of increasing pitch on constricted prediction.
To examine the effect of pitch further, a parameter β, defined
by the following equation is introduced:
β=
Figure 7 Differing pitched samples at two different constrictions.
From Figure 7 we can see that as the pitch of the elements
increases, the experimental data begin departing from the constricted theory that worked well with more closely spaced elements. Not only does the friction factor depart more from the
constricted theory, but the transition Reynolds number also increases with increasing pitch. These two trends are intuitively
explained because as the pitch increases, the channel more
closely resembles a smooth channel. Additionally, the root parameters more closely predict the hydraulic performance for the
longest pitch tested. To show this, we plot the same data with
constricted and unconstricted parameters in Figure 8.
pitch
εFP
(24)
As β approaches zero, the maximum effect of having closely
spaced, high roughness structures is evident. At low values of
β, the use of constricted parameters is appropriate. As β approaches infinity, the surface appears more and more like a
smooth channel, and thus one would expect the corresponding
smooth channel results. Figure 9 shows a plot of the product f ×
Re calculated using the unconstricted parameters for each data
point versus β. The data shows a downward trend with increasing β. This shows that the effect of pitch lies in between the two
extreme limits, one with closely spaced elements represented by
the constricted flow diameter, and the other with infinite spacing
represented by the smooth channel.
Laminar–Turbulent Transition
As a result of the roughness, the transition to turbulence
occurs sooner than it would in a smooth channel. This transition
is recorded for all the tests that have been performed in this work.
Kandlikar et al. [16] characterized the results of their previous
testing on sawtooth roughness structures with the following
correlation:
0 < εDh,cf ≤ 0.08 Ret,cf = 2300 − 18,750 (εFP /Dh,cf )
0.08 < εDh,cf ≤ 0.15 Ret,cf = 800 − 3,270
× (εFP /Dh,cf − 0.08)
Figure 8
rameters.
Comparison of largest pitch results of constricted versus root pa-
heat transfer engineering
(25)
Based on additional experimental evidence, Brackbill and
Kandlikar [5] further modified this criterion to include the
smooth channel transition at a Reynolds number of ReO . The resulting correlation to determine the transition Reynolds number
vol. 31 no. 8 2010
T. P. BRACKBILL AND S. G. KANDLIKAR
643
Figure 11 f–Re characteristics for sawtooth samples.
Downloaded By: [Rochester Institute of Technology] At: 19:21 10 June 2010
Figure 10 Transition Reynolds numbers for all the tests.
is given by:
0 < εFP Dh,cf ≤ 0.08 Ret,cf = ReO −
ReO − 800
(εFP Dh,cf )
0.08
0.08 < εFP Dh,cf ≤ 0.25 Ret,cf = 800 − 3,270
× (εFP Dh,cf − 0.08)
(26)
where Re0 is the transition Reynolds number for a smooth channel with the same geometry and aspect ratio.
The transition points for each of the tests run are plotted in
Figure 10. Note that the samples with large β do not correlate
well with this criterion and are marked with red for distinction.
As β increases, the transition is delayed to higher Reynolds
numbers. In the limit, for an infinite value of β, the transition
Reynolds number will be same as the smooth channel value of
Re0 .
Additionally, with increasing relative roughness, the transition to turbulence decreases from its smooth channel transition
value of around 2700. The lowest relative roughness in Figure
10 is 1.4%, which yielded an experimental critical Reynolds
number of Recf = 2,604. When the relative roughness increases
to 4.9%, this transition occurs much lower at Recf = 1,821.
These data serve as one of the first systematic study of channels
with the exact same roughness structures and varying hydraulic
diameters.
turbulent regime. For the lower pitch of 405 µm, the data converge to a single friction factor value in the upper series of points.
The second set of data points from the 1,008 µm pitch samples
are also shown in Figure 11 in the lower series of data points.
Again, the turbulent regime appears to be converging to a single
value for friction factor, although to a lower value from the 405
µm samples. The effect of pitch is thus clearly seen. It is postulated that as β tends to infinity, the constricted-diameter-based
friction factor approaches the smooth channel values depicted
in the original Moody diagram in the fully developed turbulent
region. As β approaches zero, the constricted-diameter-based
friction factor approaches the constant value of 0.042 as depicted in the modified Moody diagram in [16]. For intermediate
values of β, the converged friction factors based on the constricted parameters lie in between these two extreme values.
This work is the first study that reports experimental data
with systematic variation of roughness height and pitch in the
turbulent region. In order to gain a complete understanding of
the effect of these parameters, further experimental study with a
wide range of β values is recommended. This work is currently
in progress in the second author’s laboratory at the Rochester
Institute of Technology.
Results of Lubrication Theory
The results from lubrication theory are applied to see which
parameter is best able to represent the laminar flow friction data.
Turbulent Regime
Additional experiments are run to look at the roughness effects past the transition region. Reynolds numbers tested in this
section range up to 15,000. Following the constricted parameter
definition roughened samples past a relative roughness of 3%
plateau to a single value of friction factor in the turbulent regime.
This results in a modified Moody diagram [16]. First, the 405
µm sawtooth results are examined. The results are shown in
Figure 11 with the constricted friction factor plotted against the
constricted Reynolds number. It can be seen that for high relative
roughness values, all of the runs converge to a single line in the
heat transfer engineering
Figure 12 Parameters for separation normalized with experimental results.
vol. 31 no. 8 2010
644
T. P. BRACKBILL AND S. G. KANDLIKAR
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Each parameter is normalized with respect to the gap beff,exp obtained from the experimental data. If the parameter is a good
fit to the experimental data, its normalized value will be close
to a value of one. The resulting plot is shown in Figure 12. At
ε/D of 1%, the use of lubrication theory, beff,theory results in 3%
error from experimental results. Below ε/D of 0.5% the theory
is applicable with minimal error. This follows because this is
where the asymptotic method used to model the non-flat wall
surfaces is valid, that is, for εFP << b. The plots shown in Figure
11 indicate that the constricted diameter yields the best result
in the entire range. Other parameters, b and mean line separation, yield significantly larger errors at higher roughness values.
From a theoretical perspective, since the lubrication theory is no
longer applicable at higher roughness values, a better method of
incorporating irreversible viscous effects is needed.
Dt
Dh
fMoody
f
Fp
FdRa
g
L
P
Q
Ra
ReO
Rp
RSm
u
Z
root diameter of the tube, m
hydraulic diameter, m
turbulent friction factor from Moody diagram, dimensionless
friction factor, dimensionless
floor profile, m
distance of the floor profile from the mean line, m
gravity, m/s2
length, m
pressure, N/m2
flow rate, m3 /s
average roughness, m
smooth channel turbulent transition, dimensionless
maximum peak height, m
mean spacing of irregularities, m
fluid velocity, m/s
height of the scanned profile, m
CONCLUSIONS
1. By comparing an idealized version of Nikuradse’s roughness
elements, εFP was shown to better characterize the roughness
elements, as compared to the commonly used Ra.
2. Contrary to other studies, and the seminal paper on roughness
by Nikuradse [2], roughness structures of less than 5% relative roughness (RR) were shown to have appreciable effects
on laminar flow.
3. Uniform roughness less than 5% RR also led to earlier transition to turbulence from the smooth channel values.
4. The use of constricted parameters was shown to work
well for roughness of two different structures, as long
as the pitch of roughness elements was not excessively
large. Both uniform roughness and sawtooth roughness elements were tested. Additionally, constricted parameters are
easy to calculate, and require no CFD results or empirical
parameters.
5. Lubrication theory is able to predict roughness with RR less
than 0.5% well. Past this point, the irreversible effects and
2-D nature of the flow around the roughness elements limit
the applicability of the lubrication theory.
6. As pitch of roughness elements increases, the friction factor and transition data approach those of a channel without
roughness elements. The ratio of roughness pitch to roughness height, defined as β, is shown to be a good parameter
to represent the pitch effects.
7. To further predict hydraulic performance with higher relative
roughness, irreversible effects need to be incorporated in the
modeling.
8. With increasing relative roughness, more abrupt transitions
to turbulence were observed.
NOMENCLATURE
a
beff
width of channel, m
distance between top and bottom surface, m
heat transfer engineering
Greek Symbols
ratio of pitch of the roughness structures to their
height, dimensionless
roughness height—new parameter, m
viscosity, Ns/m2
density, kg/m3
β
εFP
µ
ρ
Subscripts
cf
exp
theory
constructed
experimental value
theoretical values
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Timothy P. Brackbill is currently a mechanical engineering graduate student at the University of California, Berkeley. He is currently in the field of
BioMEMS. He received his master’s degree at the
Rochester Institute of Technology under Satish Kandlikar for studying the effects of surface roughness
on microscale flow.
Satish G. Kandlikar is the Gleason Professor of Mechanical Engineering at RIT. He received his Ph.D.
degree from the Indian Institute of Technology in
Bombay in 1975 and was a faculty member there
before coming to RIT in 1980. His current work focuses on the heat transfer and fluid flow phenomena
in microchannels and minichannels. He is involved in
advanced single-phase and two-phase heat exchangers incorporating smooth, rough, and enhanced microchannels. He has published more than 180 journal
and conference papers. He is a Fellow of the ASME, associate editor of a number of journals including ASME Journal of Heat Transfer, and executive editor
of Heat Exchanger Design Handbook published by Begell House. He received
the RIT’s Eisenhart Outstanding Teaching Award in 1997 and Trustees Outstanding Scholarship Award in 2006. Currently he is working on a Department
of Energy-sponsored project on fuel cell water management under freezing
conditions.
vol. 31 no. 8 2010