Heat transfer engineering an international journal, tập 31, số 8, 2010

Heat Transfer Engineering, 31(8):627, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903463388

editorial

Selected Papers from the Sixth
International Conference on
Nanochannels, Microchannels,
and Minichannels
SATISH G. KANDLIKAR
Mechanical Engineering Department, Rochester Institute of Technology, Rochester, New York, USA

I am very pleased to present this special issue highlighting
some of the papers presented at the Sixth International Conference on Nanochannels, Microchannels, and Minichannels, held
in the newly built and environmentally friendly modern conference center, Wissenschafts- und Kongresszentrum in Darmstadt,
Germany, June 23–25, 2008. The conference was co-hosted by
Professor Peter Stephan, Dean of Engineering at the Technische
Universitaet of Darmstadt.
With the conference located in the center of Europe, the participation in the conference set an all-time record with more
than 250 papers presented in the three days. The conference
theme of interdisciplinary research was once again showcased
with researchers working in diverse areas such as traditional
heat and mass transfer, lab-on-chips, sensors, biomedical applications, micromixers, fuel cells, and microdevices, to name a
few. Selected papers in the field of heat transfer and fluid flow
are included in this special volume.
There are nine papers included in this special volume. The
topics covered are basic fluid flow in plain and rough channels,
application of lubrication theory for periodic roughness structures, laminar, transition, and turbulent region friction factors,

converging–diverging microchannels, axial conduction effects,
slip flow condition for gas flow, refrigerant distribution, and
finally gas transport and chemical reaction in microchannels.
These papers represent the latest developments in our understanding of some of the new areas in microscale transport that
are being pursued worldwide.
Address correspondence to Professor Satish G. Kandlikar, Mechanical Engineering Department, Rochester Institute of Technology, James E. Gleason
Building, 76 Lomb Memorial Drive, Rochester, NY 14623-5603, USA. E-mail:


The conference organizers are thankful to all authors for
participating enthusiastically in this conference series. Special
thanks are due to the authors of the papers in this special issue. The authors have worked diligently in meeting the review
schedule and responding to the reviewers’ comments. The reviewers have played a great role in improving the quality of
the papers. The help provided by Enrica Manos in the ME Department at RIT in organizing this special issue is gratefully
acknowledged.
I would like to thank Professor Afshin Ghajar for his dedication to this field and his willingness to publish this special
issue highlighting the current research going on worldwide. He
has been a major supporter of this conference series, and I am
indebted to him for this collaborative effort.

Satish Kandlikar is the Gleason Professor of Mechanical Engineering at Rochester Institute of Technology (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 singlephase 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 and Heat in History Editor for Heat Transfer Engineering. He received RIT’s Eisenhart Outstanding Teaching Award in 1997 and
its 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.

627


Heat Transfer Engineering, 31(8):628–634, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903463404

Laminar Fully Developed Flow in
Periodically Converging–Diverging
Microtubes
MOHSEN AKBARI,1 DAVID SINTON,2 and MAJID BAHRAMI1
1
Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey,
British Columbia, Canada
2
Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia, Canada

Laminar fully developed flow and pressure drop in linearly varying cross-sectional converging–diverging microtubes have
been investigated in this work. These microtubes are formed from a series of converging–diverging modules. An analytical
model is developed for frictional flow resistance assuming parabolic axial velocity profile in the diverging and converging
sections. The flow resistance is found to be only a function of geometrical parameters. To validate the model, a numerical
study is conducted for the Reynolds number ranging from 0.01 to 100, for various taper angles, from 2 to 15 degrees, and for
maximum–minimum radius ratios ranging from 0.5 to 1. Comparisons between the model and the numerical results show
that the proposed model predicts the axial velocity and the flow resistance accurately. As expected, the flow resistance is

found to be effectively independent of the Reynolds number from the numerical results. Parametric study shows that the effect
of radius ratio is more significant than the taper angle. It is also observed that for small taper angles, flow resistance can be
determined accurately by applying the locally Poiseuille flow approximation.

INTRODUCTION
There are numerous instances of channels that have
streamwise-periodic cross sections. It has been experimentally
and numerically observed that the entrance lengths of fluid flow
and heat transfer for such streamwise-periodic ducts are much
shorter than those of plain ducts, and quite often, three to five
cycles can make both the flow and heat transfer fully developed
[1]. In engineering practice the streamwise length of such ducts
is usually much longer than several cycles; therefore, theoretical works for such ducts often focus on the periodically fully
developed fluid flow and heat transfer. Rough tubes or channels
with ribs on their surfaces are examples of streamwise-periodic
ducts that are widely used in the cooling of electronic equipment and gas turbine blades, as well as in high-performance
heat exchangers.
The authors are grateful for the financial support of the Natural Sciences and
Engineering Research Council (NSERC) of Canada and the Canada Research
Chairs Program.
Address correspondence to Mohsen Akbari, Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey, BC,
V3T 0A3, Canada. E-mail:

Many researchers have conducted experimental or numerical investigations on the flow and heat transfer in streamwiseperiodic wavy channels. Most of these works are based on numerical methods. Sparrow and Prata [1] performed a numerical
and experimental investigation for laminar flow and heat transfer in a periodically converging–diverging conical section for
the Reynolds number range from 100 to 1000. They showed
that the pressure drop for the periodic converging–diverging
tube is considerably greater than for the straight tube, while
Nusselt number depends on the Prandtl number. For Pr < 1,
the periodic tube Nu is generally lower than the straight tube,

but for Pr > 1, Nu is slightly greater than for a straight tube.
Wang and Vanka [2] used a numerical scheme to study the flow
and heat transfer in periodic sinusoidal passages. Their results
revealed that for steady laminar flow, pressure drop increases
more significantly than heat transfer. The same result is reported
in Niceno and Nobile [3] and Wang and Chen [4] numerical
works for the Reynolds number range from 50 to 500. Hydrodynamic and thermal characteristics of a pipe with periodically
converging–diverging cross section were investigated by Mahmud et al. [5], using a finite-volume method. A correlation was
proposed for calculating the friction factor, in sinusoidal wavy
tubes for Reynolds number ranging from 50 to 2,000. Stalio

628


M. AKBARI ET AL.

and Piller [6], Bahaidarah [7], and Naphon [8] also studied
the flow and heat transfer of periodically varying cross-section
channels. An experimental investigation on the laminar flow
and mass transfer characteristics in an axisymmetric sinusoidal
wavy-walled tube was carried out by Nishimura et al. [9]. They
focused on the transitional flow at moderate Reynolds numbers
(50 to 1,000). Russ and Beer [10] also studied heat transfer
and flow in a pipe with sinusoidal wavy surface. They used
both numerical and experimental methods in their work for the
Reynolds number range of 400 to 2,000, where the flow regime is
turbulent.
For low Reynolds numbers, Re ∼ 0(1), some analytical and
approximation methods have been carried out in the case of
gradually varying cross section. In particular, Burns and Parkes

[11] developed a perturbation solution for the flow of viscous
fluid through axially symmetric pipes and symmetrical channels
with sinusoidal walls. They assumed that the Reynolds number
is small enough for the Stokes flow approximation to be made
and found stream functions in the form of Fourier series. Manton
[12] proposed the same method for arbitrary shapes. Langlois
[13] analyzed creeping viscous flow through a circular tube
of arbitrary varying cross section. Three approximate methods
were developed with no constriction on the variation of the wall.
MacDonald [14] and more recently Brod [15] have also studied
the flow and heat transfer through tubes of nonuniform cross
section.
The low Reynolds number flow regime is the characteristic of
flows in microchannels [16]. Microchannels with converging–
diverging sections maybe fabricated to influence cross-stream
mixing [17–20] or result from fabrication processes such as
micromachining or soft lithography [21].
Existing analytical models provide solutions in a complex
format, generally in a form of series, and are not amicable to engineering or design. Also, existing model studies did not include
direct comparison with numerical or experimental data. In this
study, an approximate analytical solution has been developed
for velocity profile and pressure drop of laminar, fully developed, periodic flow in a converging–diverging microtube, and
results of the model are compared with those of an independent
numerical method. Results of this work can be then applied to
more complex wall geometries.

629

Figure 1 Geometry of slowly varying cross-section microtube.

The governing equations for this two-dimensional (2-D) flow
are:
1 ∂
∂u
(rv) +
=0
(1)
r ∂r
∂z
ρ v

∂u
∂u
+u
∂r
∂z

=−

∂P
∂u
∂z u
1 ∂
r
+ 2
+µ
∂z
r ∂r
∂r
∂z

∂v
∂v
+u
∂r
∂z

=−

∂
∂P
+µ
∂r
∂r

ρ v

Consider an incompressible, constant property, Newtonian
fluid which flows in steady, fully developed, pressure-driven
laminar regime in a fixed cross section tube of radius a0 . At the
origin of the axial coordinate, z = 0, the fluid has reached a fully
developed Poiseuille velocity profile, u(r) = 2um,0 [1 − ( ar0 )2 ],
where um,0 is the average velocity. The cross-sectional area for
flow varies linearly with the distance z in the direction of flow,
but retains axisymmetric about the z-axis. Figure 1 illustrates
the geometry and the coordinates for a converging tube; one
may similarly envision a diverging tube.
heat transfer engineering

1 ∂

∂2 v
(rv) + 2 (3)
r ∂r
∂z

with boundary conditions
u(r, z) = 0, v(r, z) = 0;

r = a(z)

r
a0

u(r, 0) = 2um,0 1 −

(4)

2

z=0

;

P (r, 0) = P0
In this work we seek an approximate method to solve this
problem.
MODEL DEVELOPMENT
The premise of the present model is that the variation of the
duct cross section with the distance along the direction of the
flow is sufficiently gradual that the axial component of the velocity profile u(r, z) remains parabolic. To satisfy the requirements

of the continuity equation, the magnitude of the axial velocity
must change, i.e.,
u(r, z) = 2um (z) 1 −

PROBLEM STATEMENT

(2)

r
a(z)

2

(5)

where um (z) is the mean velocity at the axial location z and can
be related to the mean velocity um,0 at the origin z = 0, and
using conservation of mass as
um (z) =

a0
a(z)

2

(6)

um,0

Then the axial velocity profile u(r, z) becomes

u(r, z) = 2um,0
vol. 31 no. 8 2010

a0
a(z)

2

1−

r
a (z)

2

(7)


630

M. AKBARI ET AL.

Substituting Eq. (7) into the continuity equation, Eq. (1), and
integrating leads to
v(r, z) = 2mηum,0
where m =

da(z)
dz

a0
a(z)

2

1−

is the wall slope and η =

r
a (z)

2

(8)

r
.
a(z)

Figure 2 Schematic of the periodic converging–diverging microtube.

PRESSURE DROP AND FLOW RESISTANCE
Comparing Eqs. (7) and (8) reveals that uv = mη; thus, one
can conclude that if m is small enough, v will be small and the
pressure gradient in the r direction can be neglected with respect
to pressure gradient in the z direction.
, pressure drop in
Knowing both velocities and neglecting ∂P
∂r

a converging–diverging module can be obtained by integrating
Eq. (2). The final result after simplification is
P =

16µum,0 L ε2 + ε + 1 m2 (1 + ε)
+
3ε2
2ε5
a02

(9)

where P is the difference of average pressure at the module
inlet and outlet, a0 and a1 are the maximum and minimum
radiuses of the tube, respectively, m = tan φ is the slope of the
tube wall, and ε = aa10 is the minimum–maximum radius ratio.
Defining flow resistance with an electrical network analogy
in mind [22],
Rf =

P
Q

(10)

where Q = πa02 um,0 , the flow resistance of a converging–
diverging module becoms
Rf =

16µL ε2 + ε + 1 m2 (1 + ε)

+
3ε2
2ε5
πa04

(11)

At the limit when m = 0, Eq. (11) recovers the flow resistance
of a fixed-cross-section tube of radius a0 , i.e.
16µL
πa04

(12)

ε2 + ε + 1 m2 (1 + ε)
+
3ε2
2ε5

(13)

Rf,0 =
In dimensionless form,
Rf∗ =

The concept of flow resistance, Eq. (10), can be applied
to complex geometries by constructing resistance networks to
analyze the pressure drop.
For small taper angles (φ ≤ 10◦ ), the term containing m2
becomes small, and thus Eq. (13) reduces to

Rf∗ =

ε2 + ε + 1
3ε2

(14)

The maximum difference between the dimensionless flow
resistance, Rf∗ , obtained from Eq. (13) and that from Eq. (14)
heat transfer engineering

is 6% for φ = 1◦ . Equation (14) can also be derived from the
locally Poiseuille approximation. With this approximation, the
frictional resistance of an infinitesimal element in a gradually
varying cross-section microtube is assumed to be equal to the
flow resistance of that element with a straight wall. Equation
(14) is used for comparisons with numerical data.

NUMERICAL ANALYSIS
To validate the present analytical model, 15 modules
of converging–diverging tubes in a series were created
in a finite-element-based commercial code, COMSOL 3.2
(www.comsol.com). Figure 2 shows the schematic of the modules considered in the numerical study. Two geometrical parameters, taper angle, φ, and minimum–maximum radius ratio,
ε = aa10 , were varied from 0 to 15◦ and 0.5 to 1, respectively.
The working fluid was considered to be Newtonian with constant fluid properties. A Reynolds number range from 0.01 to
100 was considered. Despite the model is developed based on
the low Reynolds numbers, higher Reynolds numbers (Re ∼
100) were also investigated to evaluate the limitations of the
model with respect to the flow condition. A structured, mapped
mesh was used to discretize the numerical domain. Equations

(1)–(3) were solved as the governing equations for the flow for
steady-state condition. A uniform velocity boundary condition
was applied to the flow inlet. Since the flow reaches streamwise
fully developed condition in a small distance from the inlet, the
same boundary conditions as Eq. (4) can be found at each module inlet. A fully developed boundary condition was assumed
∂
= 0. A grid refinement study was conducted
for the outlet, ∂z
to ensure accuracy of the numerical results. Calculations were
performed with grids of 3 × 6, 6 × 12, 12 × 24, and 24 × 48
for each module for various Reynolds numbers and geometrical
configurations. The value of dimensionless flow resistance, Rf∗ ,
was monitored since the velocity profile in any cross section
remained almost unchanged with the mesh refinement. Figure 3
shows the effect of mesh resolution on Rf∗ for φ = 10◦ ,
ε = aa10 = 0.95, and Re = 10. As can be seen, the value of
Rf∗ changes slower when the mesh resolution increases. The
fourth mesh, i.e., 24 × 48, was considered in this study for
all calculations to optimize computation cost and the solution
accuracy.
The effect of the streamwise length on the flow has been
shown in Figures 4 and 5. Dimensionless velocity profile, u∗ =
a0
u
, is plotted at β = a(z)
= 1.025 for the second to fifth
umax (z)
vol. 31 no. 8 2010

M. AKBARI ET AL.

631

Figure 3 Mesh independency analysis.
Figure 5 Effect of module number on the dimensionless flow resistance.

modules as well as the dimensionless flow resistance, Rf∗

for the
second to seventh modules for the typical values of φ = 10◦ , ε =
a1
= 0.95, and Re = 10. Both velocity profile and dimensionless
a0
flow resistance do not change after the forth module, which
indicates that the flow after the fourth module is fully developed.
The same behavior was observed for the geometrical parameters
and Reynolds numbers considered in this work. Values of the
modules in the fully developed region were used in this work.
Good agreement between the numerical and analytical model
can be seen in Figure 6, where the dimensionless frictional flow
resistance, Rf∗ , is plotted over a wide range of the Reynolds
number, Re = 2ρuµm,0 a0 . The upper and lower dashed lines represent the bounds of nondimensional flow resistance for the
∗
investigated microtube. Rf,0
is the flow resistance of a uniform

Figure 4 Effect of the streamwise length.

heat transfer engineering

cross-sectional tube with the radius of a0 , and as expected its
∗
value is unity. Rf,1
stands for the flow resistance of a tube with
the radius of a1 . Since the average velocity is higher for the tube
∗
∗
of radius a1 , the value of Rf,1
is higher than the value of Rf,0
.
Both numerical and analytical results show the flow resistance
to be effectively independent of Reynolds number, in keeping
with low Reynolds number theory. For low Reynolds numbers,
in the absence of instabilities, flow resistance is independent of
the Reynolds number.
Table 1 lists the comparison between the present model, Eq.
(14), and the numerical results over the wide range of minimum–
maximum radius ratio, 0.5 ≤ ε ≤ 1, three typical Reynolds
numbers of Re = 1, 10, and 100, and taper angles of φ = 2.7◦
and 15◦ . The model is originally developed for small wall taper
angles, φ ≤ 10, and low Reynolds numbers, Re ∼ 0(1); however,

Figure 6 Variation of Rf∗ with the Reynolds number, φ = 10, and ε = 0.95.

vol. 31 no. 8 2010


632

M. AKBARI ET AL.
Table 1 Comparison of the proposed model and the numerical results
φ= 2
Re = 1

Re = 10

Re = 100

Model

Numerical

Error (%)

Numerical

Error (%)

Numerical

Error (%)

0.5
0.6
0.7
0.8
0.9
1

4.67
3.02
2.13
1.59
1.24
1

4.59
2.97
2.09
1.56
1.21
1

–1.7
–1.7
–1.7
–1.8
–2.2
0.0

4.59
2.98
2.09
1.56
1.22
1

–1.7
–1.7

–1.7
–1.8
–1.6
0.0

4.96
3.14
2.17
1.59
1.23
1

+6.2
+3.7
+1.9
+0.0
–0.8
0.0

0.5
0.6
0.7
0.8
0.9
1

4.67
3.02
2.13
1.59

1.24
1

4.67
3.02
2.13
1.59
1.24
1

0.0
0.0
–0.2
–0.6
0.0
1.0

4.72
3.05
2.14
1.60
1.24
1

+1.2
+0.9
+0.7
+0.7
0.0
0.0

6.00
3.55
2.33
1.66
1.25
1

+28.6
+17.2
+9.7
+4.5
+0.6
0.0

0.5
0.6
0.7
0.8
0.9
1

4.67
3.02
2.13
1.59
1.24
1

5.01

3.24
2.26
1.67
1.28
1

+7.3
+7.3
+6.2
+5.4
+3.4
0.0

+12.4
+10.0
+8.9
+6.7
+3.7
0.0

7.33
4.11
2.57
1.76
1.32
1

+57.0
+36.0
+20.7

+10.7
+6.8
0.0

ε

φ= 7

φ = 15

Re =

2ρum,0 a0
Error%
µ

=

5.25
3.33
2.32
1.70
1.29
1

∗
∗
Rf,
model −Rf, numerical
∗

Rf,
model

as can be seen in Table 1, the proposed model can be used for wall
taper angles up to 15◦ , when Re < 10, with acceptable accuracy.
Note that the model shows good agreement with the numerical
data for higher Reynolds numbers, up to 100, when ε > 0.8.
Instabilities in the laminar flow due to high Reynolds numbers
and/or large variations in the microchannel cross section result
in the deviations of the analytical model from the numerical
data.

studied when taper angle, φ = 7, was kept constant. As shown
in Figure 7, both numerical and analytical results indicate that
the frictional flow resistance, Rf , decreases by increasing of
the minimum–maximum radius ratio, ε. For a constant taper
angle, increase of ε = aa10 increases the module length as well
as the average fluid velocity. Hence, higher flow resistance can
be observed in Figure 7 for smaller values of ε. For better physical interpretation, flow resistances of two straight microtubes

PARAMETRIC STUDIES
Effects of two geometrical parameters—minimum–
maximum radius ratio, ε, and taper angle, φ—are investigated
and shown in Figures 7 and 8. Input parameters of two typical
converging–diverging microtube modules are shown in Table 2.
In the first case, the effect of ε = aa10 on the flow resistance was
Table 2 Input parameters for two typical microtubes
Parameter

Value

500 µm
10

a0
Re
Case 1
φ= 7
0.5 < ε < 1
Case 2
ε = 0.8
2 ≤ φ ≤ 15

Figure 7 Effect of ε on the flow resistance, φ = 7, and Re = 10.

heat transfer engineering

vol. 31 no. 8 2010


M. AKBARI ET AL.

633

angle. Both Rf,0 and Rf,1 increase inversely with the taper angle
φ in a similar manner.
The effect of the module length can be eliminated by nondimensionalizing the module flow resistance with respect to the
flow resistance of a straight microtube. Dimensionless flow resistance with the definition of Eq. (12) was used in Figure 9.
As can be seen, the taper angle φ effect is negligible while the
controlling parameter is the minimum–maximum radius ratio, ε.
SUMMARY AND CONCLUSIONS

Figure 8 Effect of φ on the flow resistance, ε = 0.8, and Re = 10.

with the maximum and minimum module radiuses are plotted in
Figure 7. Since the total length of the module increases inversely
with ε, a slight increase in Rf,0 can be observed. On the other
hand, the flow resistance of the microtube with the minimum
radius of the module Rf,1 increases sharply when ε becomes
smaller. Keeping in mind that the flow resistance is inversely related to the fourth power of the radius, Eq. (12), and a1 changes
with ε, sharp variation of Rf,1 can be observed in Figure 7.
Variation of the flow resistance with respect to the taper angle
when the minimum–maximum radius ratio, ε = aa10 , was kept
constant is plotted in Figure 8. Since a1 remains constant in this
case, the only parameter that has an effect on the flow resistance
is the variation of the module length with respect to the taper

Laminar fully developed flow and pressure drop in gradually
varying cross-sectional converging–diverging microtubes have
been investigated in this work. A compact analytical model
has been developed by assuming that the axial velocity profile remains parabolic in the diverging and converging sections.
To validate the model, a numerical study has been performed.
For the range of Reynolds number and geometrical parameters
considered in this work, numerical observations show that the
parabolic assumption of the axial velocity is valid. The following results are also found through analysis:
For small taper angles (φ ≤ 10), effect of the taper angle on
the dimensionless flow resistance, Rf∗ can be neglected with
less than 6% error and the local Poiseuille approximation can
be used to predict the flow resistance.
• It has been observed through the numerical analysis that the
flow becomes fully developed after less than five modules of

length.
• Comparing the present analytical model with the numerical
data shows good accuracy of the model to predict the flow
resistance for Re < 10, φ ≤ 10, and 0.5 ≤ ε ≤ 1. See Table
1 for more details.
• The effect of minimum–maximum radius ratio, ε, is found to
be more significant than taper angle, φ on the frictional flow
resistance.
•

As an extension of this work, an experimental investigation to
validate the present model and numerical analysis is in progress.
NOMENCLATURE

Figure 9 Effect of φ and ε on Rf∗ , Re = 1.

heat transfer engineering

a(z)
a0
a1
L
m
Q
r, z
Re
Rf
Rf∗
um
u, v

=
=
=
=
=
=
=
=
=
=
=
=

radius of tube, m
maximum radius of tube, m
minimum radius of tube, m
half of module length, m
slope of tube wall, [—]
volumetric flow rate, m3 /s
cylindrical coordinate, m
Reynolds number, 2ρuµm,0 a0
frictional resistance, pa ms3
Rf
normalized flow resistance, Rf,0
mean fluid axial velocity, m/s
velocity in z and r directions, m/s

vol. 31 no. 8 2010

634

M. AKBARI ET AL.

Greek Symbols
β
η
ε
ρ
µ
φ
P

=
=
=
=
=
=
=

a0
a(z)
r
a(z)
a1
a0

fluid density, kg/m3

fluid viscosity, kg/m-s
angle of tube wall, [—]
pressure drop, Pa

REFERENCES
[1] Sparrow, E. M., and Prata, A. T., Numerical Solutions for Laminar
Flow and Heat Transfer in a Periodically Converging–Diverging
Tube With Experimental Confirmation, Numerical Heat Transfer,
vol. 6, pp. 441–461, 1983.
[2] Wang, G., and Vanka, S. P., Convective Heat Transfer in Periodic
Wavy Passages, International Journal of Heat and Mass Transfer,
vol. 38, no. 17, pp. 3219–3230, 1995.
[3] Niceno, B., and Nobile, E., Numerical Analysis of Fluid Flow and
Heat Transfer in Periodic Wavy Channels, International Journal
of Heat and Fluid Flow, vol. 22, pp. 156–167, 2001.
[4] Wang, C. C., and Chen, C. K., Forced Convection in a Wavy Wall
Channel, International Journal of Heat and Mass Transfer, vol.
45, pp. 2587–2595, 2002.
[5] Mahmud, S., Sadrul Islam, A. K. M., and Feroz, C. M., Flow
and Heat Transfer Characteristics Inside a Wavy Tube, Journal of
Heat and Mass Transfer, vol. 39, pp. 387–393, 2003.
[6] Stalio, E., and Piller, M., Direct Numerical Simulation of Heat
Transfer in Converging–Diverging Wavy Channels, ASME Journal of Heat Transfer, vol. 129, pp. 769–777, 2007.
[7] Bahaidarah, M. S. H., A Numerical Study of Fluid Flow and Heat
Transfer Characteristics in Channels With Staggered Wavy Walls,
Journal of Numerical Heat Transfer, vol. 51, pp. 877–898, 2007.
[8] Naphon, P., Laminar Convective Heat Transfer And Pressure Drop
in the Corrugated Channels, International Communications in
Heat and Mass Transfer, vol. 34, pp. 62–71, 2007.
[9] Nishimura, T., Bian, Y. N., Matsumoto, Y., and Kunitsugu, K.,

Fluid Flow and Mass Transfer Characteristics in a Sinusoidal
Wavy-Walled Tube at Moderate Reynolds Numbers for Steady
Flow, Journal of Heat and Mass Transfer, vol. 39, pp. 239–248,
2003.
[10] Russ, G., and Beer, H., Heat Transfer and Flow Field in A Pipe
With Sinusoidal Wavy Surface—Ii: Experimental Investigation,
International Journal of Heat and Mass Transfer, vol. 40, no. 5,
pp. 1071–1081, 1997.
[11] Burns, J. C., and Parkes, T., Peristaltic Motion, Journal of Fluid
Mechanics, vol. 29, pp. 731–743, 1967.
[12] Manton, M. J., Low Reynolds Number Flow in Slowly Varying
Axisymmetric Tubes, Journal of Fluid Mechanics, vol. 49, pp.
451–459, 1971.
[13] Langlois, W. E., Creeping Viscous Flow Through a Circular Tube
of Non-Uniform Cross- Section, ASME Journal of Applied Mechanics, vol. 39, pp. 657–660, 1972.
[14] MacDonald, D. A., Steady Flow in Tubes of Slowly Varying
Cross-Section, ASME Journal of Applied Mechanics, vol. 45, pp.
475–480, 1978.

heat transfer engineering

[15] Brod, H., Invariance Relations for Laminar Forced Convection In
Ducts With Slowly Varying Cross Section, International Journal
of Heat and Mass Transfer, vol. 44, pp. 977–987, 2001.
[16] Squires, T. M., and Quake, S. R., Microfluidics: Fluid Physics at
Nano-Liter Scale, Review of Modern Physics, vol. 77, pp. 977–
1026, 2005.
[17] Lee, S. H., Yandong, H., and Li, D., Electrokinetic Concentration
Gradient Generation Using a Converging–Diverging Microchannel, Analytica Chimica Acta, vol. 543, pp. 99–108, 2005.
[18] Hung, C. I., Wang, K., and Chyou, C., Design and Flow Simulation

of a New Micromixer, JSME International Journal, vol. 48, no.
1, pp. 17–24, 2005.
[19] Hardt, S., Drese, K. S., Hessel, V., and Schonfeld, F., Passive
Micromixers for Applications in the Microreactor and µ-TAS
Fields, Microfluids and Nanofluids, vol. 1, no. 2, pp. 108–118,
2005.
[20] Chung, C. K., and Shih, T. R., Effect of Geometry on Fluid Mixing
of the Rhombic Micromixers, Microfluids and Nanofluids, vol. 4,
pp. 419–425, 2008.
[21] McDonald, J. C., Duffy, D. C., Anderson, J. R., Chiu, D. T., Wu,
H., Schueller, O. J., and Whiteside, G. M., Fabrication of Microfluidic Systems in Poly(demethylsiloxane), Electrophoresis,
vol. 21, pp. 27–40, 2000.
[22] Bahrami, M., Yovanovich, M. M., and Culham, J. R., Pressure
Drop of Fully-Developed, Laminar Flow in Rough Microtubes,
ASME Journal of Fluids Engineering, vol. 128, pp. 632–637,
2006.

Mohsen Akbari is a Ph.D. student at Mechatronic
System Engineering, School of Engineering Science,
Simon Fraser University, Canada. He received his
bachelor’s and master’s degrees from Sharif University of Technology, Iran, in 2002 and 2005. Currently,
he is working on transport phenomena at micro and
nano scales with applications in biomedical diagnosis
and energy systems.

Majid Bahrami is an assistant professor with the
School of Engineering at the Simon Fraser University, British Columbia, Canada. Research interests
include modeling and characterization of transport
phenomena in microchannels and metalfoams, contacting surfaces and thermal interfaces, development
of compact analytical and empirical models at micro and nano scales, and microelectronics cooling.

He has numerous publications in refereed journals
and conferences. He is a member of ASME, AIAA,
and CSME.

David Sinton received the B.Sc. degree from the University of Toronto, Toronto, Ontario, Canada, in 1998,
the M.Sc. degree from McGill University, Montreal,
Quebec, Canada, in 2000, and the Ph.D. degree from
the University of Toronto in 2003, all in mechanical engineering. He is currently an associate professor in the Department of Mechanical Engineering,
University of Victoria, Victoria, British Columbia,
Canada. His research interests are in microfluidics
and nanofluidics and their application in biomedical
diagnostics and energy systems.

vol. 31 no. 8 2010


Heat Transfer Engineering, 31(8):635–645, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903466621

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:

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

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:
Mean Line =

1
n

n

|Zi |

(1)

i=1

Roughness Characterization
Recently, Kandlikar et al. [16] proposed new roughness parameters of interest to roughness effects in microfluidics. These
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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.
Let z ⊆ Z s.t. all zi = Zi iff Zi < Mean Line
Fp =

•

1
nz

n

zi

(3)

i=1

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.
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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:
∂2 ux
1 ∂P
=
x − direction
µ ∂x
∂z2

h

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

h

∂ux
dz +

∂x
f

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

f
h

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

0

∂ux
dh
df
dz + ux |h + ux |f
∂x

dx
dx

f

1
dx
(h − f)3

−b3eff dP
Q
=
a
12µ dx

(18)

(14)

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)

Integrating this function as we did before, we can obtain a

function for change in pressure using the effective height, given
in Eq. (20):

h

ux dz =

L

−12µQ
P2 − P1 =
a

(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

(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).

f

d
dx

(16)

f

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

∂ux
dz + uz |hf = 0
∂x

(15)

(12)

Now to account for the velocity in the z direction, we integrate
the continuity equation over the gap spacing.
h

ux dz = 0
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:

−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
L
0

1 dx
(h−f)3

⎥
⎥
⎥
⎦

(21)

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).
beff,exp =

−12µLQ
a (P2 − P1 )

Figure 3 Experimental test setup: apparatus schematic.

1/3

(22)

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

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


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 Kakac¸ 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

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

vol. 31 no. 8 2010


642

T. P. BRACKBILL AND S. G. KANDLIKAR

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.

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

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

REFERENCES
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[5] Brackbill, T. P., and Kandlikar, S. G., Effects of Low Uniform
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ICNMM2007–30031, 2007.
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Water Flow in Smooth and Rough Silicon Microchannels, Journal
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on Nanochannels, Microchannels, and Minichannels. Limerick,
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Proc. International Conference on Microchannels and Minichannels, Rochester, ICNMM2003–1037, 2003.
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228, 2004.
Kandlikar, S. G., Joshi, S., and Tian, S., Effect of Channel Roughness on Heat Transfer and Fluid Flow Characteristics at Low
Reynolds Numbers in Small Diameter Tubes, Proc. National Heat
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Convective Heat Transfer, John Wiley and Sons, New York, pp.
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Tim 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 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


Heat Transfer Engineering, 31(8):646–657, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903466613

Experimental Investigation of
Friction Factor in the Transition
Region for Water Flow in Minitubes
and Microtubes
AFSHIN J. GHAJAR, CLEMENT C. TANG, and WENDELL L. COOK
School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, Oklahoma, USA

A systematic and careful experimental study of the friction factor in the transition region for single-phase water flow in
mini- and microtubes has been performed for 12 stainless-steel tubes with diameters ranging from 2083 µm to 337 µm. The
pressure drop measurements were carefully performed by paying particular attention to the sensitivity of the pressure-sensing
diaphragms used in the pressure transducer. Experimental results indicated that the start and end of the transition region
were influenced by the tube diameter. The friction factor profile was not significantly affected for the tube diameters between
2083 µm and 1372 µm. However, the influence of the tube diameter on the friction factor profile became noticeable as

the diameter decreased from 1372 µm to 337 µm. The Reynolds number range for transition flow became narrower with
decreasing tube diameter.

INTRODUCTION
Due to rapid advancement in fabrication techniques, the
miniaturization of devices and components is ever increasing
in many applications. Whether it is in the application of miniature heat exchangers, fuel cells, pumps, compressors, turbines,
sensors, or artificial blood vessels, a sound understanding of
fluid flow in micro-scale channels and tubes is required. Indeed,
within this last decade, countless researchers have been investigating the phenomenon of fluid flow in mini-, micro-, and even
nanochannels. One major area of research in the phenomenon
of fluid flow in mini- and microchannels is the friction factor.
This is an extended version of paper ICNMM-62281: An Experimental
Study of Friction Factor in the Transition Region for Single Phase Flow in
Mini- and Micro-Tubes, presented at the ASME Sixth International Conference
on Nanochannels, Microchannels, and Minichannels, Darmstadt, Germany, June
23–25, 2008.
This work was partially funded by the Sandia National Laboratories, Albuquerque, New Mexico. Sincere thanks are offered to Micro Motion for generously donating one of the Coriolis flow meters and providing a substantial
discount on the other one. Thanks are also due to Martin Mabry for his assistance in procuring these meters. The assistance of Rahul Rao in the experimental
part of this study is greatly appreciated.
Address correspondence to Professor Afshin J. Ghajar, School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK
74078, USA. E-mail:

However, amid all the investigations in mini- and microchannel
flow, there seems to be a lack in the study of the flow in the transition region. One obvious question is the location of the transition region with respect to the hydraulic diameter of the channel
and the roughness of the channel. To successfully understand
friction factor and the location of the transition region, a systematic experimental investigation on various hydraulic diameters
of mini- and microchannels is necessary. However, the science
behind these advanced technologies seems to be controversial,
especially fueled by the experimental results of the fluid flow

and heat transfer at these small scales.
On one hand, researchers have found that the friction factors
to be below the classical laminar region theory [1, 2]. Meanwhile, some have reported that friction factor correlations for
conventional sized tubes to be applicable for mini- and microtubes [3–5]. However, many recent experiments on small-sized
tubes and channels have observed higher friction factors than
the correlations for conventional-sized tubes and channels [6–
11], and the cause of this discrepancy was attributed to surface
roughness. Literature also highlights the importance of diameter measurement and difficulties associated with quantifying the
effect of roughness. These difficulties are primarily due to the
large number of parameters used in describing various roughness geometries.
In this study, mini- and microtubes are chosen over other
noncircular channels to negate the effect of aspect ratio, which

646


A. J. GHAJAR ET AL.

may serve to alter flow characteristics at these small scales.
The major objectives of this study are to accurately measure
the pressure drop in mini- and microtubes over a wide range of
Reynolds numbers from laminar to the turbulent region and to
explore the start and end of the transition region in these small
sized tubes.

LITERATURE REVIEW
A brief investigation of literature ranging from early papers
to those that are more current presents us with highly contradictory results. In fact, these contradictions may better be labeled
as widespread disparity. The early researchers observed lower
friction factors while the later ones observed higher friction factors than predicted by theory. Despite this, it should be noted

that the majority of the more recent researchers tend to observe
results that agree with theory within calculated uncertainties.
In spite of all the contradicting results available, the role that
roughness, instrumentation, measurements, and dependence of
diameter bring about in altering the flow characteristics at these
micro-scales has been more or less acknowledged. Despite this
acknowledgment, it is still not clear exactly what role these
parameters play in influencing the flow characteristics.
Choi et al. [1] performed pressure drop measurements on
fused-silica microtubes with dry nitrogen gas as the test fluid.
The diameters ranged from 3 to 81 µm and the roughness
measurements confirmed that the microtubes were essentially
smooth. They found the f·Re value to be around 53, which
was considerably less than the theoretical value of 64. Similar
results were obtained for the turbulent flow data. The authors
also observed that the measurements were not influenced by the
roughness of the microtubes.
Similar results were obtained by Yu et al. [2] in their experiment using water and nitrogen gas. The microtubes used were
from the same manufacturer (Polymicro Technologies) as for
Choi et al. [1]. They found the f·Re product to be 50.13, which
is considerably lower than the classical value of 64. Both Choi
et al. [1] and Yu et al. [2] used compressible flow analysis for
the nitrogen test fluid. Friction factor was calculated using the
Fanno-line expression in both cases.
Hwang and Kim [3] investigated the pressure drop characteristics of R-134a in stainless-steel tubes with diameters of 244,
430, and 792 µm. They found that within experimental uncertainty, conventional theories are able to predict the experimental
friction factors. The authors found no evidence of early transition and they reported the onset of transition Reynolds number
occurred slightly below 2,000.
Yang and Lin [4] investigated water flow through stainlesssteel tubes with diameters ranging from 123 to 962 µm. They
found that the friction factor results correlate well with correlations for conventional tubes. There was no significant effect of

size on their results within the diameter range of their reported
work. Transition range was observed from Reynolds number of
2,300 to 3,000.
heat transfer engineering

647

Rands et al. [5] measured the frictional pressure drop and
temperature induced by viscous heating for water flowing
through fused-silica microtubes with diameters from 32.2 to
16.6 µm. The results from their work were confirmed with classical laminar flow behavior at low Reynolds number. The onset
of transition region was observed at the Reynolds numbers of
2,100 to 2,500.
Mala and Li [6] analyzed water flowing through fused-silica
and stainless-steel tubes ranging from 50 to 254 µm. Contrary to
the previous researchers, they found friction factor values larger
than what the theory predicted. Moreover, they also observed
a dependence of the f·Re product on Reynolds number. Early
transition in Reynolds number range of 300 to 900 was reported,
and surface roughness was proposed as a significant cause of
that early flow transition.
Celata et al. [7] performed pressure drop tests using R-114
in a 130 µm microtube. The Reynolds numbers investigated
ranged from 100 to 8,000. Transition was observed to be in the
Reynolds number range of 1,880 to 2,480. In the laminar region, the experimental values matched well with the theoretical
predictions until the Reynolds number of 585. For Reynolds
numbers greater than 585, higher friction factor values were
observed. The authors attributed this deviation from theory to
roughness of the stainless-steel microtube.
Kandlikar et al. [8] investigated the effect of roughness on

pressure drop in microtubes. The roughness was changed by
etching the tubes with different acids. They observed that for
larger tubes (1067 µm), the effect of roughness is negligible.
For smaller tubes (620 µm), increases in roughness resulted in
higher pressure drop accompanied by early transition.
Li et al. [9] investigated flow through glass microtubes (79
to 449 µm in diameter), silicon microtubes (100 to 205 µm
in diameter), and stainless-steel microtubes (129 to 180 µm in
diameter). They found that the f·Re in laminar region for smooth
tubes was nearly 64, while the results for rough tubes with peak–
valley roughness of 3 to 4% showed 15 to 37% higher than the
classical f·Re value of 64. Based on flow characteristics, Li et al.
[9] concluded that the onset of transition region occurred at the
Reynolds numbers of 1,700 to 2,000.
Zhao and Liu [10] conducted pressure drop studies on smooth
quartz-glass tubes and rough stainless-steel tubes of varying diameters. They observed that in the laminar regime, experimental
results agreed well with theoretical values. However, early transition at Reynolds numbers ranging from 1,100 to 1,500 (for
smooth microtubes) was recorded. For rough microtubes (with
ε/D = 0.08), laminar theory agrees only until the Reynolds
number of 800, where similar early transition was observed.
Tang et al. [11] investigated the flow characteristics of nitrogen and helium in stainless-steel and fused-silica tubes of
various diameters. They observed that the friction factors in
stainless-steel tubes are much higher than the theoretical correlation for the laminar region, deviating by as much as 70% for
a tube diameter of 172 µm. Friction factors for the smoother
walled fused-silica tubes were found to be in relative agreement with the theory for conventional-sized tubes. The positive
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A. J. GHAJAR ET AL.

deviation was attributed to the roughness and was found to increase with decreasing diameter, bringing up questions of both
diameter and roughness effects. They also acknowledged the
fact that accurate measurement of the inner diameter is essential, citing it as a possible factor in leading to higher friction
factors.
In a review by Kandlikar [12], he suggested that the uncertainties in the experiments by Nikuradse [13] in the laminar region were very high, and the conclusion regarding the absence of
roughness effects in the laminar region is questionable. Noticing that mini- and micro-fluidic systems routinely violate the
5% relative roughness threshold set by Moody, Colebrook, and
Nikuradse, Kandlikar et al. [14] and Taylor et al. [15] proposed
modifying the Moody diagram to reflect new experimental data.
Kandlikar et al. [14] proposed a new effective flow diameter
based on the effect of flow constriction due to roughness elements,
Dcf = D − 2ε

(1)

where Dcf is the constricted flow diameter, D is the tube inner
diameter, and ε is the roughness height. One may consider that
the constricted flow diameter (Dcf ) corresponds to the free flow
area. This concept proposed by Kandlikar et al. [14] is very
much like the effect of vena contracta seen in orifice meters,
where a contraction coefficient is used to relate the orifice area
to the vena contracta area. The relation of the friction factor (f)
with the friction factor based on the constricted flow diameter
(fcf ) is
fcf = f

Dcf
D

5

(2)

Based on the constricted flow diameter, the Reynolds number is
then expressed as
Recf =

˙
4m
πDcf µ

Figure 1 Friction factor versus Reynolds number plotted by Brackbill and
Kandlikar [16] for channels with various relative roughness: (a) without using
constricted flow parameters, (b) with using constricted flow parameters.

(3)

Brackbill and Kandlikar [16] experimentally investigated
the effect of relative roughness on friction factor and critical
Reynolds number for mini- and microchannels. In their experiments, the Reynolds numbers were varied from 30 to 7,000 for
hydraulic diameters ranging from 1,084 to 198 µm with relative
roughness ranging from 0 to 5.18%. To obtain uniform roughness on the channel surface, a systematic approach was taken by
sanding the surface 45 degrees in both directions from the axis
along the channel length [16]. An in-depth discussion in the
parameterization of relative roughness for different machined
surfaces using this surface roughening method is documented
by Young et al. [17]. Contrary to the findings of Nikuradse [13],
Brackbill and Kandlikar [16] observed that there was indeed the

effect of roughness in the laminar region. Figure 1a illustrates
the friction factor versus Reynolds number plot by Brackbill
and Kandlikar [16] for channels with varying relative roughness. Clearly, as shown in Figure 1a, roughness effects played a
role in the laminar region, and the effects increased with higher
heat transfer engineering

relative roughness. By including the constricted flow hydraulic
diameter (Dh,cf ) to the experimental data, Brackbill and Kandlikar [16] observed that the agreement between the experimental
data and laminar flow theory for friction factor was significantly
improved (Figure 1b). In addition, they also observed a trend
relating the critical Reynolds number and the relative roughness.
To predict the onset of transition region, Brackbill and Kandlikar [16] recommended a correlation for the critical Reynolds
number and the relative roughness based on the constricted flow
hydraulic diameter,
Rec,cf = Rec,o −

Rec,o − 800
0.08

ε
Dh,cf

for

0 ≤ ε Dh,cf ≤ 0.08
Rec,cf = 800 − 3270(ε Dh,cf − 0.08) for
0.08 ≤ ε Dh,cf ≤ 0.15
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(4)

A. J. GHAJAR ET AL.

where the critical Reynolds number for smooth channels
(Rec,o = 2,500) was used [16]. Equation (4) was developed
for channels and has only been verified with data for miniand microchannels from [16] and [18]. The correlation has an
average absolute error of 13% [16].
Recently, Celata et al. [19] conducted experimental studies
for compressible flow of nitrogen gas inside microtubes ranging from 30 to 500 µm with relative roughness of 1% or less.
The results they found indicated that the agreement of friction
factor in laminar flow with theory for conventional sized tubes
is excellent. For microtubes with diameter of 100 µm or less,
Celata et al. [19] reported that when Re > 1,300 the friction
factor tends to deviate from the Poiseuille law and attributed the
deviation to acceleration associated with compressibility effect.
Furthermore, their studies observed no evidence of early transition, with respect to conventional-size pipes, with the critical
Reynolds number for transition ranging from 2,160 to 4,430,
and critical Reynolds number showed no dependence on tube
length to diameter ratio.
In most mini- and micro-fluidic systems, the flow regions are
likely to be mainly laminar and transitional. The other question
that needs to be addressed is the location (start and end) of the
transition region and its shape for different diameters. Literature
has reported the onset of transition to be either early [6, 8, 10,
16] or in agreement with conventional-sized tubes and channels [4, 5]. The discrepancies in whether size and roughness
effects contribute to the increase of friction factors, and lower
critical Reynolds numbers (early transition) may be attributed to
inadequacies in instrumentation. While accurate measurement
of inner diameter is certainly acknowledged to be of great importance, it is shown in this article that the sensitivity of the

instrument providing pressure drop measurements should be of
equal if not greater concern. This is discussed in detail in the
Results and Discussion section of this article.

EXPERIMENTAL SETUP
Experimental Apparatus
The experimentation for this study was performed using a
relatively simple but highly effective apparatus. The apparatus used was designed with the intention of conducting highly
accurate pressure drop measurements. In addition to accurate
measurements, the apparatus was also designed to be versatile,
accommodating the use of multiple diameters and lengths of
test sections. The apparatus consists of four major components.
These are the fluid delivery system, the flow meter array, the
test section assembly, and the data acquisition system. Each
of these different components is discussed independently. An
overall schematic for the experimental test apparatus is shown
in Figure 2.
The fluid delivery system is a pneumatic and hydraulic combination, consisting of a high-pressure cylinder filled with ultraheat transfer engineering

649

Figure 2 Schematic of the experimental setup.

high-purity nitrogen in combination with a stainless-steel pressure vessel. The system is an open loop. Thus, after the working
fluid passes through the apparatus, it is passed into a sealed collection container and recycled manually. Nitrogen in the highpressure cylinder is pressurized to 17.2 MPa by the distributor.
This pressurized nitrogen is then fed to the stainless-steel pressure vessel via a two-phase regulator and line. The working
fluid, distilled water for the purposes of this research, is stored
in the stainless-steel pressure vessel. As the pressurized nitrogen is fed into the stainless-steel pressure vessel, the working
fluid is forced up a stem extending to the bottom of the vessel,
out of the pressure vessel, and through the flow-meter array and

test section. An Airgas regulator is used for the purposes of controlling the pressure of the nitrogen inlet to the stainless-steel
pressure vessel. This dual-stage regulator is capable of providing pressures ranging from 0 to 1.72 MPa. The stainless-steel
pressure vessel used is an Alloy Products model 72–05, providing a maximum working pressure of 1.37 MPa and a capacity
of 19.0 L.
After exiting the pressure vessel, the distilled water travels
to the flow-meter array. The flow rate of the water entering
the array is further regulated using a Parker N-Series model
6A-NLL-NE-SS-V metering valve, which allows fine-tuning of
the flow rate. Fluid passes through the metering valve and into
one of the two Micro Motion Coriolis flow meters. Two flow
meters are necessary in order to accommodate the large range
of flow rates that are studied using the experimental apparatus.
The larger of the two meters used is a CMF025 coupled with a
model 1700 transmitter. This meter is designed to measure mass
flows ranging from 54 to 2,180 kg/h for liquids. Within this
range of mass flows, this meter is accurate to 0.05%. However,
much smaller flow rates can be measured with very little loss
in accuracy. The smaller of the two meters is a Micro Motion
model LMF3M, coupled with an LFT transmitter. This second
meter is designed to measure mass flows ranging from 0.001 to
1.5 kg/h.
After passing through the flow-meter array, fluid enters the
test section assembly via a second section of PFA tubing to the
test section assembly. The test section assembly contains the test
section as well as the equipment necessary for measurement of
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650

A. J. GHAJAR ET AL.

inlet and outlet fluid temperatures and pressure drop. This test
section was constructed for the incorporation of a very broad
range of test section diameters, encompassing both mini and
micro tube sizes. In experimentation to date, research has been
conducted on 12 different tube sizes. The inner diameters of
these tubes vary from 2,083 to 337 µm. All of the tubes used are
available from Small Parts, Inc. The tubes used are stainlesssteel type 304 hypodermic tubes with factory-cut lengths of
30.5 cm (for 2083 ≤ D ≤ 667 µm) and 15.2 cm (for 559 ≤ D
≤ 337 µm). Since the friction factor measurements are conducted for fully developed flow, the length of the tube bears
no effect on the results. As pointed out by Krishnamoorthy and
Ghajar [20], the effect of tube length on the friction factor is
negligible as long as the flow is fully developed.
The pressure transducer used for pressure drop measurements
is a Validyne model DP15. This pressure transducer utilizes a
series of interchangeable diaphragms to provide the ability to
measure a very large range of differential pressures. The research facilities used for experimentation have different pressure transducer diaphragms to encompass a range of differential
pressures from 1.38 to 1,380 kPa. The Validyne pressure transducer has an accuracy of ±0.25% full scale of the diaphragm
used. Careful attention is given to ensure that the range of the
diaphragm used is conducive to the pressure being measured.
The use of the numerous interchangeable diaphragms is an important factor in ensuring the accuracy of the pressure drop
measurements.
All data from the thermocouples and pressure transducer are
acquired using a National Instruments data acquisition system
and recorded with the laboratory PC (personal computer) and
LabView software.

Calibration of Instruments
Nine different pressure transducer diaphragms are used to

cover differential pressures ranging from 1.38 to 1,380 kPa.
Calibrations are performed at the beginning of each experiment
with the appropriate diaphragms. During calibration, the voltage output of the differential pressure transducer at numerous
applied pressures is compared against the reading of one of the
four research grade test gauges. Of these four gauges, the highest rated in terms of pressure is a Perma-Cal 2070 kPa test gauge
with an accuracy of ±0.25% full scale. For lower pressures, two
Wika test gauges, also with accuracy of ±0.25% full scale, are
used. The first of these has a pressure rating of 1,100 kPa and
the second is rated up to 103 kPa. For low-pressure diaphragm
calibration a Cole-Palmer digital manometer is used. This instrument has a pressure rating of 103 kPa, an accuracy of ±0.3%
full scale, and a resolution of 69 Pa.
The Micro Motion Coriolis flow meters are factory calibrated
as well. For the CMF-025, the manufacturer’s specified tolerance for calibration error is ±0.1%. For the LMF3M, the manufacturer’s specified calibration tolerance is ±1.0%. For both of
these meters, in-laboratory calibration consisted of checking the
heat transfer engineering

manufacturer’s calibrations over a range of flow rates via timed
collection of fluid passing through the meters. In addition, the
maximum and minimum milliamp outputs of the CMF-025 were
tuned to improve the resolution of the meter.

Experimental Uncertainty
Developing an understanding of the experimental uncertainty
in the calculated friction factor is absolutely necessary. From the
measured pressure drop data, the friction factor can be calculated
with
f=

2 pD
=

ρLV2

pD5 π2 ρ
˙2
8Lm

(5)

The uncertainty in the pressure drop measurement can be
readily obtained from the manufacturer’s specifications on the
Validyne pressure transducer. As has been previously stated,
this specification is given as ±0.25% of the full-scale reading
of each diaphragm used. Diaphragms were carefully selected
during experimentation in order to obtain the highest accuracies
possible. The worst-case scenario occurs with small tube size
and low Reynolds number. In this region, the uncertainty in
the pressure drop measurement can be estimated at ±1.0%.
It is more representative to look at intermediately sized tubes
and/or flow rates through the transition and turbulent regions.
Uncertainty of the pressure drop measurement in these areas
drops to ±0.4%.
The uncertainty in mass flow rate measurement given by
the Micro Motion flow meter specifications for the CMF-025
meter is ±0.05%. For the LMF3M meter, an uncertainty of
reading of ±0.50% is given. However, it has to be taken into
consideration that the larger meter is being utilized at flow rates
lower than its range in order to cover the entire range of flow
rates for all of the tubes under research. Based upon uncertainty
equations given in the Micro Motion specifications, the worstcase scenario accuracy between the two flow meters is ±1.8%.
Both the use of the LMF3M and the under-ranging of the CMF025 occur at smaller tube sizes and lower Reynolds numbers.

Thus, it is necessary to calculate uncertainty for either of the
meters running within their specified mass flow ranges and to
estimate uncertainty when the CMF-025 is pushed to its lowest
range of measurement.
Uncertainty in tube length is determined by the accuracy
of the cutting of the high-density polyethylene tube cradles.
The cradles serve as a reference point for the mounting of the
different tube sections in order to ensure consistency. Measured
uncertainty in the cradle lengths is ±0.26% of length.
Due to the fact that uncertainty in both the Validyne pressure
transducer and the Micro Motion meters is dependent upon tube
size and Reynolds number, three different uncertainty values
have been established. In order to quantify the overall uncertainty, analysis was conducted using the method described by
Kline and McClintock [21]. In the case of larger tube sizes and
higher Reynolds numbers, the CMF-025 meter is used and is
vol. 31 no. 8 2010