Proceedingsofofthe
theASME
Seventh
International
ASMEConference
Conferenceon
onNanochannels,
Nanochannels,Microchannels
Microchannelsand
andMinichannels
Minichannels
Proceedings
2009
7th International
ICNMM2009
ICNMM2009
June22-24,
22-24,2009,
2009,Pohang,
Pohang,South
SouthKorea
Korea
June
ICNMM2009-82255
NUMERICAL SIMULATION OF SINGLE PHASE LIQUID FLOW IN NARROW RECTANGULAR
CHANNELS WITH STRUCTURED ROUGHNESS WALLS
Rishabh R. Srivastava*, Nicholas M. Schneider, Satish G. Kandlikar
Mechanical Engineering Department
Kate Gleason College of Engineering
Rochester Institute of Technology, Rochester, NY, USA
*[email protected]
ABSTRACT
Laminar flow in rough channels is receiving considerable
attention recently due to its application in microfluidic devices.
Developing a proper understanding of the fundamental effects
of roughness in narrow channels is essential. In the present
study, single-phase laminar flow of an incompressible fluid
through narrow rectangular channels with structured roughness
walls is investigated numerically using CFD software,
FLUENT. The results for smooth channel geometries are first
compared with the previously validated experimental data of
Brackbill [1]. Triangular ribbed roughness structures, defined
by pitch and height, are incorporated in the CFD model and the
fully developed region is analyzed. Pressure drop results from
CFD simulations are presented and the friction factor, f , is
calculated for comparison with published experimental results
of Brackbill and Kandlikar [2-4] and Brackbill [1]. The
validated numerical scheme will be used in future work for the
evaluation of the effect of different roughness features on
laminar flow.
INTRODUCTION
An extensive experimental study into the effects of surface
roughness on pressure drop was conducted by Nikuradse [5] in
1933. Known roughness was achieved by depositing sifted sand
grains on the inner walls of tubes with diameters ranging from
2.42 cm to 9.92 cm. In his work, Nikuradse concluded that
roughness had no effect on laminar flow. However, the
experimental friction factor data were found to lie above the
predicted theoretical values.
The manometers that were used by Nikuradse were later
shown by Kandlikar [6] to have unacceptably large uncertainty
for the small pressure drops found in the laminar regime. The
uncertainty in Nikuradse’s turbulent results was much smaller
than in the laminar regime due to the much large pressure drops
in turbulent flow. Kandlikar also found that large inaccuracies
in measurements such as pressure drop and surface geometry
were the reasons behind erroneous conclusions made by
researchers about roughness effects in the 80’s and 90’s.
In the early 80’s, Wu and Little [7] found an early
transition to turbulent flow in microminiature refrigerators.
Channels with Dh varying from 45 µm to 83 µm were etched in
glass and silicon and tested with H2, N2, and Ar. The authors
then performed a similar study [8] with Dh around 150 µm. In
this study the channels were prepared with the same methods as
their previous work. The results yielded friction factors greater
than conventional theory predictions. The authors noted that the
values were higher, but had the same slope as presented in the
conventional Moody Diagram for friction factors. The
experiments showed a critical Reynolds number as low as
around 400.
More recently, Kandlikar et al. [9] found the occurrence of
early transition to turbulence in rectangular channels of Dh
ranging from 325 µm to 1819 µm. The recorded pressure drop
data were noted to deviate from the conventional values for air
and water as the working fluids. The critical Reynolds number
was correlated with the relative roughness (ε/Dh,cf), and the
friction factor could be calculated well with the use of the
constricted hydraulic diameter (Dh,cf). In a later work, Kandlikar
et al. [10] presented the relationship for critical Reynolds
number shown in Eq. (1). The work shows that increasing
relative roughness causes a reduction in the critical Reynolds
number. In their work, the authors also present a modified
Moody Diagram using the constricted diameter over the entire
range of Reynolds numbers.
1
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0 < ε/Dh,cf ≤ 0.08
0.08 < ε/Dh,cf ≤ 0.15
Ret,cf = 2300 - 18,750(ε/Dh,cf)
Ret,cf = 800 - 3,270(ε/Dh,cf - 0.08)
(1)
In representing the roughness effects on microscale,
Kandlikar et al. [9] proposed a new set of roughness
parameters. Figure 1 shows the new set of parameters. Rp is the
maximum height from the mean line along the profile. Next, Sm
is defined as the mean separation of profile irregularities, which
correspond to the pitch of roughness elements in this work.
Lastly, FdRa is the distance of the floor profile (Fp) which lies
below the mean line. These values are established to replace the
assumption that a surface needs only be defined by the average
roughness, Ra. The roughness height, ε, is the sum of FdRa and
Rp. These parameters detail the surface profile in a more indepth fashion compared to Ra.
Figure 1. Schematic diagram of roughness parameters [9]
Rawool et al. [11] performed CFD analysis of flow through
microchannels having roughness in the form of obstructions
along the channel walls. They studied the effect of roughness
height and geometry on friction factor and concluded that
pressure drop decreases with increase in roughness pitch. Croce
et al. [12] investigated roughness effects on pressure drop and
heat transfer rate using finite element CFD code. They modeled
roughness through a set of random generated peaks and
concluded that increase in Poiseuille number is a function of
Reynolds number. Croce et al. [13] also modeled roughness as
three dimensional conical peaks and showed that surface
roughness has a significant effect on pressure drop in
microchannels.
Experimental Data on Roughness Effects
In recent years, Brackbill and Kandlikar [2-4], and Brackill
[1] have generated a considerable collection of data on the
effects of surface roughness on fluid flow and friction factor.
Their work experimentally investigated an array of surface
geometries and parameters. Variations included saw-tooth
roughness, triangular elements, and uniform surface finishes
with relative roughness ranging from 0 to 24.8%. The authors
found early transition to turbulence as the relative roughness
increased, and demonstrated that the use of the constricted
hydraulic diameter would cause the data to collapse on to the
conventional theory line for laminar flow.
In the present study the data generated in the previous
works of Brackbill and Kandlikar [2-4] and Brackbill [1] are
utilized for the purpose of comparison and validation of the
proposed numerical scheme. In their experimental works, the
authors developed a variable separation test section designed to
accommodate test pieces with known surface parameters. Two
identical profile test pieces of known profiles were assembled
in the test section creating a narrow rectangular cross-section
channel. Once assembled, a positive displacement pump was
used to control flow rate of water through the test section.
Pressure was measured at the inlet, outlet, and along the
channel length. Experimental results were acquired via
LABVIEW and were processed to calculate the friction factors
for a given set of parameters.
OBJECTIVE
The current work is performed in order to extend our
understanding of the effect of roughness structures on fluid
flow characteristics. First, smooth channel geometries, utilized
in the previous experimental works, were modeled and
numerically simulated using commercial CFD software
FLUENT. These geometries accounted for entrance region
effects. Fully developed region was analyzed and the results
were compared and validated with published data. Numerical
scheme was then extended to two roughness geometries which
had also been used in prior experimental works. Results of
simulation are presented in the form of constricted flow friction
factor, fcf..
NOMENCLATURE
A
Area
a
Channel width
b
Channel height
Dh
Hydraulic diameter
Fp
Floor profile line
FdRa Distance from Fp to average, µm
Friction factor
f
Mass flow rate
P
Perimeter
Ra
Average roughness, µm
Rp
Maximum peak height, µm
Re
Reynolds number
RR
Relative roughness
Mean spacing of irregularities, µm
Sm
V
Volumetric flow rate
Subscripts
cf
Constricted flow parameter
exp
Experimental based calculation
fluent Fluent based calculation
Greek
α
αcf
ε
λ
Aspect ratio
Constricted aspect ratio
Roughness element height
Pitch
THEORETICAL CONSIDERATIONS
The derivation of the constricted parameters is useful in
accurately calculating friction factor in high roughness channels
[2]. This work utilized the constricted parameter scheme of
Kandlikar et al. [9] and Brackbill and Kandlikar [2-4]. In the
smooth wall case, a channel has a cross-section of height b, and
width a. For roughness on two sides of the channel, the
2
Copyright © 2009 by ASME
parameter bcf represents the constricted channel height. Figure 2
shows a generic representation of the constricted
icted parameters
parameters.
RRcf =
ε
Dh,cf
(12)
Now Reynolds number and constricted
c
Reynolds number
can be calculated in the following way:
•
•
m =V ρ
(13)
•
4m
Re =
µP
Figure 2. Generic constricted parameters – Side vview [1]
•
Re cf =
In order to recalculate the constricted parameters, bcf and
Acf can be defined as follows:
(2)
bcf = b - 2ε
A = ab
Acf = abcf
(3)
Dh =
Dh,cf =
4A
4ab
=
P 2(a + b)
4 Acf
4abcf
=
Pcf
2(a + bcf )
(6)
(7)
Theoretical friction factors are calculated usi
using the
constricted parameters. For laminar flows in rectangular
channels, the theoretical friction factor is predicted by Kakac et
al. [14], by Eq. (8). Constricted aspect ratio
tio given in Eq. (10) is
derived from the conventional aspect ratio defined in Eq. (9)
with the new parameters.
f =
24
1 − 1.3553α + 1.9467α 2 − 1.7012α 3 + 0.9564α 4 − 0.2537α 5
Re
(
α=
b
a
α cf =
)
(8)
(9)
bcf
2
dP ρDh A
•
dx
2 m2
2
dP ρDh,cf Acf
=
•
dx
2 m2
f exp =
(4)
(5)
Hydraulic diameter and constricted hydraulic
ydraulic diameter can
be found through a similar process:
4m
µPcf
(15)
The theoretical pressure drop can be calculated for
comparison with the simulation results using Eq. (16). The
pressure drop can also be found based on constricted
parameters in Eq. (17).
Constricted perimeter follows by substituting bcf into
perimeter as follows:
P = 2a +2b
Pcf = 2a +2bcf
(14)
f exp,cf
(16)
(17)
CFD MODELING
The first step in the CFD modeling is to compare smooth
channel results from FLUENT to both conventional theory and
previously published experimental results.
results After validation with
smooth channel results, CFD modeling is extended to channels
with designed roughness structures.
A schematic diagram of the general channel outline is
presented in Fig. 3. The depth of the channel perpendicular to
the plane of the paper is 12.7 mm. The length of the actual
channel is 88.9 mm with an inlet and outlet, yielding a total
length of 114.3 mm. Two
wo inner surfaces noted in the Fig. 3
compose the walls of the tested channel. They are also the
location where designed surfaces are machined. The separation
is varied to represent the experimentally tested conditions in the
previously published work [1-4].
]. Experimentally tested flow
rates are selected and used to calculate the flow velocity, which
is assumed
med to be uniform at the inlet. Geometries are meshed
and solved individually to accurately represent the realistic
experimental conditions.
(10)
a
Relative roughness and constricted relative
elative roughness can
be calculated using Eq. (11) and Eq. (12)
12) respectively.
RR =
ε
Dh
Figure 3.. General channel outline
o
(11)
Geometry 1, Smooth Channel
The smooth channel geometry was modeled to replicate the
ideal case of a perfectly smooth channel. The walls were
3
Copyright © 2009 by ASME
modeled to the general case dimensions shown in Fig. 3 with a
depth of 12.7 mm into the page, and flow being defined from
left to right. Three separations of 200 µm, 300 µm, and 500 µm
were simulated with constant mass flow rates of 0.0106 kg/sec,
0.0112 kg/sec, and 0.0110 kg/sec being defined at the inlets on
the left, respectively. Inlet temperatures were taken as 27.7°C,
26.6°C, and 27.1 °C respectively to match the experimental
conditions.
The commercial software package FLUENT was used to
numerically simulate the flow through the channel. The presolver software GAMBIT was utilized for mesh generation.
The meshed geometry is shown in Fig. 4. Tet/hybrid T grid
scheme was used to mesh the geometries. In order to ensure
grid independence, a study was carried out by varying the mesh
spacing. The number of elements required for analysis for each
separation is detailed in Table 1. It was found that a grid having
the reported number of elements (Table 1) is appropriate for
each analysis, as any further reduction in the mesh spacing did
not produce any change in the pressure profiles at various
cross-sections. The model was simulated using pressure based
solver with a convergence criterion of E-6.
geometries. Finer mesh was created near the roughness
elements to more accurately simulate the flow in this complex
region. The model parameters of inlet flow rate and number of
mesh elements are summarized in Table 1. Each simulation was
carried out using a pressure-based solver for a convergence
criterion of E-6.
Pitch
0.127mm between all
13.08mm
To end
Height
1 Segment
Figure 5. Roughness element parameters
Figure 6. Isometric view- Rough channel
Figure 4. Geometry 1, smooth channel mesh
Rough Channels
Two channel configurations containing structured
roughness were studied. The roughness geometries were taken
to replicate the test sections of Brackbill and Kandlikar [2].
The walls were designed to consist of uniform, triangular
roughness elements defined by the pitch and height parameters.
A generic element sketch can be found in Fig. 5 with the
necessary parameters for fabrication of the designer surfaces.
Figure 5 is drawn to show how the surface cross-section would
look utilizing this machining process and what was modeled for
simulation purposes. The rough channels follow the overall
channel scheme mentioned earlier, with the roughness elements
being placed on the noted surfaces in Fig. 1.
FLUENT and the provided pre-solver software GAMBIT
were utilized for the roughness effect study. Figure 6 shows an
isometric view of the rough channel modeled in GAMBIT.
Each roughness geometry was meshed for two separations of
400µm, and 500µm. Geometry 2, rough channel (λ = 508 µm)
can be found in Fig. 7 and a detailed meshed view is shown in
Fig. 8. Likewise, details of geometry 3, rough channel (λ =
1016 µm) and its mesh views are shown in Fig. 9 and Fig. 10
respectively. Tet/hybrid T grid scheme was used to mesh the
The geometry 2 was designed with 174 segments with each
segment having a total length of 508µm. Each segment
consisted of three arcs of a circular cut with diameter of 381µm
and centers spaced by 127 µm longitudinally. Each of the cuts
had a depth of 51µm relative to the original, smooth plane. This
process is mirrored for the opposing channel wall. These
dimensions result in a uniform set of roughness elements
defined by a pitch of 508 µm and height of 51µm. The
geometry was set to have a separation of 400µm, and 500µm.
Mass flow rates were taken to be 0.0046 kg/sec and 0.0052
kg/sec respectively. Inlet velocity was assumed to be uniform at
the inlet and was taken for water at 26.7 °C. These parameters
are summarized in Table 1.
Figure 7. Geometry 2, rough channel—Detail
4
Copyright © 2009 by ASME
Table 1. Model Parameters for the Three Geometries Tested
Mass flow rate [kg/s]
Pitch, λ [μm] Height, ε [μm] Dh [μm] D h,cf [μm]
α
Geometry 1, Smooth
Separation 500 [μm]
Separation 300 [μm]
Separation 200 [μm]
Geometry 2, Rough
Separation 500 [μm]
Separation 400 [μm]
Geometry 3, Rough
Separation 500 [μm]
Separation 400 [μm]
Mesh Elements
N/A
0
0
0
394
586
775
394
586
775
0.016
0.025
0.033
0.0061
0.0112
0.0106
2248568
2177144
2085196
508
508
51
51
586
961
389
768
0.033
0.049
0.00461
0.00528
3265616
2945244
1016
1016
51
51
586
961
389
768
0.033
0.049
0.00315
0.00530
3187120
2885549
Figure 8. Geometry 2, rough channel mesh
The geometry 3 was designed with 87 segments with each
segment having a total length of 1016 µm. Each segment
consisted of seven circular cuts with diameter of 381 µm and
centers spaced by 127 µm longitudinally. Each cut has a depth
of 51 µm relative to the original, smooth plane. This process is
mirrored for the opposing channel wall. These dimensions
result in a uniform set of roughness elements defined by a pitch
of 1016 µm and height of 51 µm. The geometry was set to have
a separation of 400 µm and 500 µm. Mass flow rates were
taken to be 0.0053 kg/sec and 0.0031 kg/sec for the two
separations respectively. The velocity at the inlet was assumed
to be a uniform profile for water at 26.7 °C. These parameters
are also summarized in Table 1.
Figure 9. Geometry 3, rough channel – Detail
Figure 10. Geometry 3, rough channel mesh
RESULTS AND DISCUSSIONS
Meshed geometries were simulated using pressure based
solver under the prescribed conditions in the CFD software
FLUENT. The resulting pressure contours were examined and
pressure drop data were extracted. The fully developed region
was determined to be the region where pressure decreased
linearly with respect to distance along the flow direction.
Pressure data was used to calculate the friction factor under the
prescribed conditions for each geometry and separation. The
published experimental results of Brackbill [1] and Brackbill
and Kandlikar [2-4] were used as a basis for evaluation of the
numerical scheme.
The controlling parameters that defined a given geometry
were channel separation, surface geometry, and mass flow rate.
Hydraulic diameters ranged from Dh = 389 µm to Dh = 961 µm
and Reynolds number was chosen below the reported critical
Reynolds number [1]. Three geometries were simulated in the
numerical scheme: smooth, 508 µm pitch triangular roughness,
and 1016 µm pitch triangular roughness. Constricted
parameters were calculated to more accurately describe the
flow caused by roughened channel geometries. The simulated
fcf results for the chosen geometries were compared with the
experimental fcf to determine the accuracy of the numerical
scheme.
Geometry 1, Smooth Channel Results
The smooth channel geometry was simulated for three
separation values (200 µm, 300 µm, and 500 µm). Pressure
5
Copyright © 2009 by ASME
drop along the length of the channel is plotted for both
experimental and numerical results in Fig. 11 for the 300 µm
separation. Pressure drop data obtained from CFD correlate
well with experimental results,, thereby proving that the
numerical scheme can accurately predict the real physical
setting for smooth channel flow. Similar results were found for
the 200 µm and 500 µm separations.
Geometry 3, Rough Channel Results
The roughness geometry (λλ = 1016 µm) was numerically
simulated using FLUENT for the two separations of 400 µm
and 500 µm. f fluent,cf , calculated using simulated results and
Eq. (17), was compared to experimental results as summarized
in Table 4. The agreement between simulated and experimental
exp
results was found to be good,, and errors calculated were less
than 5.4%.
€
Table 4. Fully Developed, Laminar Friction Factor for
Geometry 3, Rough Channel (λ = 1016 µm)
bcf [µm]
500
400
Figure 11. Pressure drop along channel length ccomparison of
geometry 1, smooth channel - 300 µm
m separation
Using Eq. (17) and numerical results, ffluent,cf is calculated and
compared to experimental results in Table 2. Percent
Percentage error
calculated between the numerical and experimental results are
all less than 0.8%. The experimental values of fcf for smooth
channels obtained by Brackbill [1] are within an average of
7.6% error as compared to theory. The theoretical values of f
were calculated using Eq. (8), which is an approximation for
the exact expression for the fully developed f [14].
Table 2. Fully Developed, Laminar Friction Factor for Geometry
1, Smooth Channel
bcf
[µm]
200
300
500
Re
ftheory
fexp,cf
ffluent
%Errorfluent-exp
2006
2063
1108
0.0117
0.0113
0.0205
0.0115
0.0120
0.0219
0.0116
0.0119
0.0220
0.8
0.8
0.4
Geometry 2, Rough Channel Results
The roughness geometry (λ = 508 µm) was numerically
simulated using FLUENT for the two separations of 400 µm
and 500 µm. The results of the simulations and Eq. (17) were
used to calculate f fluent,cf . Table 3 outlines numerical results
and percent error in
f fluent,cf
for the two separation heights.
Simulated results correspond well with experimental data and
were found to have error less than 4.9% for both cases.
€
Table 3. Fully Developed, Laminar Friction
ction Factor for
€Geometry 2, Rough Channel (λ = 508 µm)
bcf [µm]
500
400
Re
837
989
fexp,cf
0.0280
0.0203
ffluent,cf
0.0289
0.0213
%Errorfluent-exp
3.2
4.9
Re
586
990
fexp,cf
0.0295
0.0174
ffluent,cf
0.0311
0.0182
%Errorfluent-exp
5.4
4.6
CONCLUSIONS
A numerical simulation of three-dimensional,
three
narrow,
rectangular channels was performed using
us
the commercial CFD
software FLUENT. Three different geometries,
geometries one smooth and
two rough, were studied for varying separations. Entrance
region effects were accounted for in the model and fully
developed laminar flow regions were analyzed with Dh ranging
from 389 µm to 961 µm. For each of the simulated geometries,
the constricted flow model was used
ed to calculate friction factor,
fcf. The ideal, smooth channel case showed the validity of the
numerical scheme wherein all the generated results were found
to be in very good agreement with experimental results
(absolute error < 0.8%). The two roughness geometries (λ =
508 µm and λ = 1016 µm)) selected for simulation also showed
similar agreement. Errors for
or geometries with roughness were
found to range from 3.2% to 5.4%.
The numerical scheme implemented with FLUENT
software has been found to be an accurate tool for simulating
simul
experimental conditions and can be used for research in
microchannels and minichannels. It will help in better
understanding of transport phenomena at the microscale level
and will aid researchers in the design of appropriate
experimental setups.
ACKNOWLEDGMENTS
The authors would like to thank the members of the RIT
Thermal Analysis, Microfluidics, and Fuel Cell Laboratory for
their continued support.
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