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

The topics covered are basic fluid flow in plain and rough channels, application of lubrication theory for periodic roughness struc-tures, laminar, transition, and turbulent region frict

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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

Confer-ence on Nanochannels, Microchannels, and Minichannels, held

in the newly built and environmentally friendly modern

confer-ence 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

par-ticipation 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

appli-cations, 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

struc-tures, 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

under-standing of some of the new areas in microscale transport that

are being pursued worldwide

Address correspondence to Professor Satish G Kandlikar, Mechanical

En-gineering Department, Rochester Institute of Technology, James E Gleason

Building, 76 Lomb Memorial Drive, Rochester, NY 14623-5603, USA E-mail:

sgkeme@rit.edu

The conference organizers are thankful to all authors forparticipating enthusiastically in this conference series Specialthanks are due to the authors of the papers in this special is-sue The authors have worked diligently in meeting the reviewschedule and responding to the reviewers’ comments The re-viewers have played a great role in improving the quality ofthe papers The help provided by Enrica Manos in the ME De-partment at RIT in organizing this special issue is gratefullyacknowledged

I would like to thank Professor Afshin Ghajar for his ication to this field and his willingness to publish this specialissue highlighting the current research going on worldwide Hehas been a major supporter of this conference series, and I amindebted to him for this collaborative effort

ded-Satish Kandlikar is the Gleason Professor of

Me-chanical Engineering at Rochester Institute of nology (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

Tech-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

Engi-neering 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

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ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457630903463404

Laminar Fully Developed Flow in

Periodically Converging–Diverging

Microtubes

MOHSEN AKBARI,1DAVID SINTON,2and MAJID BAHRAMI1

1Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey,

British Columbia, Canada

2Department 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,

theoret-ical works for such ducts often focus on the periodtheoret-ically 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

equip-ment 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

Engi-neering, School of Engineering Science, Simon Fraser University, Surrey, BC,

V3T 0A3, Canada E-mail: maa59@sfu.ca

Many researchers have conducted experimental or cal investigations on the flow and heat transfer in streamwise-periodic wavy channels Most of these works are based on nu-merical methods Sparrow and Prata [1] performed a numericaland experimental investigation for laminar flow and heat trans-fer in a periodically converging–diverging conical section forthe Reynolds number range from 100 to 1000 They showedthat the pressure drop for the periodic converging–divergingtube is considerably greater than for the straight tube, while

numeri-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 flowand heat transfer in periodic sinusoidal passages Their resultsrevealed that for steady laminar flow, pressure drop increasesmore significantly than heat transfer The same result is reported

in Niceno and Nobile [3] and Wang and Chen [4] numericalworks for the Reynolds number range from 50 to 500 Hydro-dynamic and thermal characteristics of a pipe with periodicallyconverging–diverging cross section were investigated by Mah-mud et al [5], using a finite-volume method A correlation wasproposed for calculating the friction factor, in sinusoidal wavytubes for Reynolds number ranging from 50 to 2,000 Stalio

628

M AKBARI ET AL 629

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

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

en-gineering 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

devel-oped, 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

PROBLEM STATEMENT

Consider an incompressible, constant property, Newtonian

fluid which flows in steady, fully developed, pressure-driven

a0)2],

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

Figure 1 Geometry of slowly varying cross-section microtube.

The governing equations for this two-dimensional (2-D) floware:

r

∂

+∂2v

∂z2

(3)with boundary conditions

ity profile u(r, z) remains parabolic To satisfy the requirements

of the continuity equation, the magnitude of the axial velocitymust change, i.e.,

using conservation of mass as

PRESSURE DROP AND FLOW RESISTANCE

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.

a converging–diverging module can be obtained by integrating

Eq (2) The final result after simplification is

where
P is the difference of average pressure at the module

Defining flow resistance with an electrical network analogy

in mind [22],

diverging module becoms

R f =16µL

πa4 0

of a fixed-cross-section tube of radius a0, i.e

to complex geometries by constructing resistance networks to

analyze the pressure drop

becomes small, and thus Eq (13) reduces to

R∗

The maximum difference between the dimensionless flow

Figure 2 Schematic of the periodic converging–diverging microtube.

locally Poiseuille approximation With this approximation, thefrictional resistance of an infinitesimal element in a graduallyvarying cross-section microtube is assumed to be equal to theflow 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 mod-ules considered in the numerical study Two geometrical pa-rameters, taper angle, φ, and minimum–maximum radius ratio,

The working fluid was considered to be Newtonian with stant fluid properties A Reynolds number range from 0.01 to

con-100 was considered Despite the model is developed based on

100) were also investigated to evaluate the limitations of themodel with respect to the flow condition A structured, mappedmesh was used to discretize the numerical domain Equations(1)–(3) were solved as the governing equations for the flow forsteady-state condition A uniform velocity boundary conditionwas applied to the flow inlet Since the flow reaches streamwisefully developed condition in a small distance from the inlet, thesame boundary conditions as Eq (4) can be found at each mod-ule inlet A fully developed boundary condition was assumed

to ensure accuracy of the numerical results Calculations were

for each module for various Reynolds numbers and geometrical

was monitored since the velocity profile in any cross sectionremained almost unchanged with the mesh refinement Figure 3

R∗

all calculations to optimize computation cost and the solutionaccuracy

The effect of the streamwise length on the flow has been

u

umax(z), is plotted at β = a0

M AKBARI ET AL 631

Figure 3 Mesh independency analysis.

a1

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

number, Re= 2ρu m,0a0

rep-resent the bounds of nondimensional flow resistance for the

Figure 4 Effect of the streamwise length.

Figure 5 Effect of module number on the dimensionless flow resistance.

value is unity R∗f,1stands 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 R f,∗1is higher than the value of R∗f,0.Both numerical and analytical results show the flow resistance

to be effectively independent of Reynolds number, in keepingwith low Reynolds number theory For low Reynolds numbers,

in the absence of instabilities, flow resistance is independent ofthe Reynolds number

Table 1 lists the comparison between the present model, Eq.(14), and the numerical results over the wide range of minimum–

Figure 6 Variation of R∗

fwith the Reynolds number, φ= 10, and ε = 0.95.

as can be seen in Table 1, the proposed model can be used for wall

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

PARAMETRIC STUDIES

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= a1

Table 2 Input parameters for two typical microtubes

in Figure 7, both numerical and analytical results indicate that

the minimum–maximum radius ratio, ε For a constant taper

as the average fluid velocity Hence, higher flow resistance can

be observed in Figure 7 for smaller values of ε For better ical interpretation, flow resistances of two straight microtubes

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

M AKBARI ET AL 633

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

hand, the flow resistance of the microtube with the minimum

smaller Keeping in mind that the flow resistance is inversely

Variation of the flow resistance with respect to the taper angle

a0, was kept

case, the only parameter that has an effect on the flow resistance

is the variation of the module length with respect to the taper

Figure 9 Effect of φ and ε on R∗

nondi-As can be seen, the taper angle φ effect is negligible while thecontrolling parameter is the minimum–maximum radius ratio, ε

SUMMARY AND CONCLUSIONS

Laminar fully developed flow and pressure drop in graduallyvarying cross-sectional converging–diverging microtubes havebeen investigated in this work A compact analytical modelhas been developed by assuming that the axial velocity pro-file 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 parametersconsidered in this work, numerical observations show that theparabolic assumption of the axial velocity is valid The follow-ing results are also found through analysis:

less than 6% error and the local Poiseuille approximation can

be used to predict the flow resistance

flow becomes fully developed after less than five modules oflength

data shows good accuracy of the model to predict the flow

1 for more details

be more significant than taper angle, φ on the frictional flowresistance

As an extension of this work, an experimental investigation tovalidate the present model and numerical analysis is in progress

NOMENCLATURE

[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

Jour-nal 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

Me-chanics, 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

[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 ConcentrationGradient Generation Using a Converging–Diverging Microchan-

nel, 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.

[20] Chung, C K., and Shih, T R., Effect of Geometry on Fluid Mixing

of the Rhombic Micromixers, Microfluids and Nanofluids, vol 4,

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 Univer- sity 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 sity, British Columbia, Canada Research interests include modeling and characterization of transport phenomena in microchannels and metalfoams, con- tacting surfaces and thermal interfaces, development

Univer-of compact analytical and empirical models at cro and nano scales, and microelectronics cooling.

mi-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

Uni-versity 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 mechani- cal engineering He is currently an associate profes- sor 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.

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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

in-ternal 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

struc-tures 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

ac-cepted into modern engineering textbooks on this topic, as is

Address correspondence to Satish G Kandlikar, Mechanical Engineering

Department, Rochester, NY 14623, USA E-mail: sgkeme@rit.edu

evidenced through the Moody diagram Previous work [4, 5]has shown that the instrumentation used in Nikuradse’s experi-ments had unacceptably high uncertainties in the low Reynoldsnumbers range Additionally, all experimental laminar frictionfactors 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 laminarfriction factor; however, the results were mixed and contradic-tory 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 roughnessstructures of varying separation, or pitch They showed that thelaminar friction factors are affected with varying pitch Thiseffect has not been studied in the literature, and is an open area.Several models have attempted to characterize the effect ofroughness on laminar microscale flow Chen and Cheng [7]

635

636 T P BRACKBILL AND S G KANDLIKAR

Table 1 Previous experimental studies

higher Re numbers

1,881–2,479 is transition region

Li et al [20] 2000 0.1% RR to 4% RR 79.9–449 Smooth tubes follow macroscale,

rough have 15–37% higher f

1,700–1,900 for rough tubes Kandlikar et al [29] 2001 1.0–3.0 620 and 1,067 No effect on Dh 1067, highest f

and Nu from roughest 620

Lowered w/ roughness Bucci et al [12] 2003 0.3% to 0.8% RR 172–520 Re < 800–1000 follows classical 1,800–3,000, abrupt transition for

high RR Celata et al [27] 2004 0.05 µm smooth, 0.2–0.8 µm

rough

31–326 Tentatively propose higher than

normal friction

Pfund et al [8] 2000 Smooth 0.16 and 0.09, rough 1.9 252.8–1,900 Higher, highest for rough Approach 2,800 w/ larger

Tu et al [13] 2003 Ra < 20 nm 69.5–304.7 RR < 0.3%, conventional, RR=

0.35%, f is 9% higher

2,150–2,290 w/ RR < 0.3%, 1,570

for 0.35%

Hao et al [21] 2006 Artificial 50 × 50 µm RR 19% 153–191 Follows theory until Re = 900,

then higher, indicating transition

Transition ∼900

with Re, nothing at low Re

N/A Wibel et al [24] 2006 1.3 µm ( ∼0.97% RR) ∼133 Near classical values 1,800–2,300;varies with aspect ratio

Wu et al [19] 1983 0.05–0.30 height 45.5–83.1 Greater than predicted

created a model for pressure drop in roughened channels based

on a fractal characterization and an additional empirical

modi-fication The additional experimental data was drawn from the

results by Pfund [8] Bahrami et al [9] used a Gaussian

distri-bution 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

diffi-culty 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

Roughness Characterization

Recently, Kandlikar et al [16] proposed new roughness

pa-rameters of interest to roughness effects in microfluidics These

parameters are illustrated graphically in Figure 1 The eters are listed next, as well as how they are calculated Thesevalues are established to correct for the assumption that different

may have different effects on flows with variations in other file characteristics For example, a roughness surface with twice

of all the points from the raw profile, which physically relates

to the height of each point on the surface Note that Z is theheight of the scan at each point, i It is calculated from thefollowing equation:

n

n

i=1

T P BRACKBILL AND S G KANDLIKAR 637

translates to the highest point in the profile sample minus the

mean line

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

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= Ziiff Zi<Mean Line

nz

n

i=1

the mean line:

the following equation:

Using these parameters, Kandlikar et al [16] replotted the

Dtis as follows:

fMoody,cf= fMoody

(Dt− 2εFP)

Dt

5

(6)The constricted-diameter-based friction factor and Reynolds

number yielded a single line in the laminar region on the

Moody-type plot In the turbulent region, all values of relative roughness

high Reynolds numbers

Objectives of the Present Work

The objectives of the present work are summarized here:

1 Investigate the applicability of lubrication theory and

exam-ine it as a basis of constricted diameter

2 Laminar flow—Examine effects of both sawtooth and

uni-form 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

5 Turbulent flow—In the turbulent regime, sawtooth samplesare tested to high Reynolds numbers

Application of Lubrication Theory

The application of the constricted parameter set is based ontheory, in addition to being a practical method for predictingchannel performance A simple derivation from the Navier–Stokes (NS) equation with lubrication approximations yields avery similar concept Originally intended for looking at hydro-dynamic effects in fluid bearings, lubrication theory allows one

to account for slight wall geometry variances while keeping thesolution analytical The structure of the problem is as follows

A rectangular duct is formed in two dimensions using unknownfunctions f(x) for the bottom face and h(x) for the top face Thesimple 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 thanthe length, and the slope of the roughness is also assumed to besmall The gravity effects are negligible compared to pressuredrop in the x direction The flow is assumed to be incompressibleand steady, with entry and exit regions ignored, since the analysis

is applied to the fully developed flow It is also assumed thatthere is no velocity in the y direction Referring to Figure 2, thefollowing assumptions are made:

1 (h – f) << L for all x.

9 Flow is unidirectional and fully developed

Using the assumption of incompressibility and no flow in ydirection, the continuity equation, Eq (7), simplifies to Eq (8):

Figure 2 Illustration of lubrication problem.

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:

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:

With the NS equations, continuity equation, and boundary

con-ditions (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

Now to account for the velocity in the z direction, we integrate

the continuity equation over the gap spacing

h

f

∂ux

h

f

∂uz

h

f

∂ux

∂xdz+ uz|h

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):

∂ux

dxux|h+df

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

(13) in a form that is easy to integrate:

ddx

h

f

This equation is integrated once to get the form shown in

Eq (16) It can be intuitively seen that integrating x velocityacross the gap will give volumetric flow rate (Q) per width ofthe channel (a) As such, the constant of integration is expressed

as Q/a:

h

f

The expression derived in Eq (16) is substituted in for uxfrom

Eq (12) and then integrated The result of this integration is:

1

For analysis purposes, we can now define a channel height

when two samples of known roughness profiles are placed intothe test apparatus If we look back to Eq (18) and use beffdefined

T P BRACKBILL AND S G KANDLIKAR 639

we are left with the expression in Eq (21):

0

1(h −f) 3dx

needed is a rearrangement of Eq (20) into the form of Eq

it is easy to find beffin Eq (22)

roughness elements of low slope Once we surpass the

assump-tions 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

the-ory 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

sili-cone 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)

dif-ferential 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

am-plifiers built into the NI SCXI chassis to increase the accuracy

The separation of the samples is controlled by two Mitutoyo

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

Figure 3 Experimental test setup: apparatus schematic.

a known period of time Thermocouples are mounted on theinlet and outlet of the test section Fluid properties are calculated

at the average temperature All of the data is acquired and thesystem is controlled by a LabVIEW equipped computer with

an SCXI-1000 chassis Testing equipment allows for fully tomated acquisition of data at set intervals of Reynolds number

au-Samples

For this testing, multiple roughness structures machined intodifferent sets of samples are used The two types of roughnessexamined were a patterned roughness with repeating structuresand a less structured cross-hatch design For samples with saw-tooth roughness elements, a ball end mill cutter is used in aCNC (computer numerical control) mill to make patterned cutsacross the sample at very shallow depths The remaining pro-trusions 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-hatchpattern 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 andflat and then lapping the testing surface to reduce roughness Theroughness parameters for the surfaces studied in this work aresummarized in Table 2 Figure 4 shows high-resolution images

of some of these surfaces using an interferometer and a confocalmicroscope along with the traces normal to the flow direction

Table 2 Summary of roughness on samples

640 T P BRACKBILL AND S G KANDLIKAR

Uncertainties

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

anal-ysis, 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

RESULTS Smooth Channel Validation

The smooth channel friction factors are plotted againstReynolds number over a range of hydraulic diameters tested

in Figure 5 Note that not all data points for each hydraulic ameter are shown for simplicity of the plot To acquire this range

di-of hydraulic diameters, the lapped samples are held at varyinggap spacing Laminar theoretical friction factor is plotted as asolid black line (Eq (23)) and is given as by Kakac¸ et al [30]

in Eq (23) The agreement is quite good as expected, within theexperimental uncertainties of 7% Transition to turbulence isdeduced as a departure from the laminar theory line The range

of turbulent transition Reynolds numbers is between 2,000 and2,500, as is also expected For accurately calculating turbulenttransition, 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.

T P BRACKBILL AND S G KANDLIKAR 641

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

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

saw-tooth and uniform samples are plotted against Reynolds number

in Figure 6 On the left side the data are plotted using the

un-constricted 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

dispar-ity is seen with respect to the theory As the relative roughness

increases, the disparity between theory and experiment also

increases, regardless of whether the roughness structures are peating or uniform roughness At the highest relative roughness

re-of 27.6%, the data is far above the theory These data also tradict Nikuradse’s finding that roughness less than 5% relativeroughness (RR) has no effect on laminar flow

con-When the experimental data are replotted with the constricted

theoretical curve quite well This confirms the validity of usingthe constricted flow diameter in predicting the laminar frictionfactors as recommended in [16]

The other interesting feature of Figure 6 is that the transition

to turbulence decreases dramatically and systematically as therelative 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 perturbationsnear the channel walls will increase chaos in the flow even beforesmooth channel turbulence

Laminar Regime—Varying Pitches

The preceding roughened results hold for roughness that hasstructures that are close to each other, similar to the resultantsurface profiles of machined parts It is apparent that as thepitch of roughness elements becomes larger and larger, even-tually the channel will more resemble a smooth channel withwidely spaced protrusions into the flow At large enough separa-tions between roughness structures, or large pitches, the use ofconstricted 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 aretested in the laminar regime The roughness element height inall cases is close to 50 µm and has the same sawtooth shape Thesamples are tested at two constricted separations, 400 µm and

500 µm The plots of friction factor versus Reynolds numbersfor both separations are shown in Figure 7 and are plotted withconstricted parameters

Figure 6 Data plotted with (a) root parameters and (b) constricted parameters.

642 T P BRACKBILL AND S G KANDLIKAR

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

con-stricted theory that worked well with more closely spaced

ele-ments Not only does the friction factor depart more from the

constricted theory, but the transition Reynolds number also

in-creases 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

pa-rameters 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

Figure 8 Comparison of largest pitch results of constricted versus root

pa-rameters.

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:

Re calculated using the unconstricted parameters for each datapoint versus β The data shows a downward trend with increas-ing β This shows that the effect of pitch lies in between the twoextreme limits, one with closely spaced elements represented bythe constricted flow diameter, and the other with infinite spacingrepresented by the smooth channel

re-sulting correlation to determine the transition Reynolds number

T P BRACKBILL AND S G KANDLIKAR 643

Figure 10 Transition Reynolds numbers for all the tests.

chan-nel 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

transi-tion to turbulence decreases from its smooth channel transitransi-tion

value of around 2700 The lowest relative roughness in Figure

10 is 1.4%, which yielded an experimental critical Reynolds

These data serve as one of the first systematic study of channels

with the exact same roughness structures and varying hydraulic

diameters

Turbulent Regime

Additional experiments are run to look at the roughness

ef-fects 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

Figure 11 f–Re characteristics for sawtooth samples.

turbulent regime For the lower pitch of 405 µm, the data verge to a single friction factor value in the upper series of points.The second set of data points from the 1,008 µm pitch samplesare also shown in Figure 11 in the lower series of data points.Again, the turbulent regime appears to be converging to a singlevalue for friction factor, although to a lower value from the 405

con-µm samples The effect of pitch is thus clearly seen It is lated that as β tends to infinity, the constricted-diameter-basedfriction factor approaches the smooth channel values depicted

postu-in the origpostu-inal Moody diagram postu-in the fully developed turbulentregion As β approaches zero, the constricted-diameter-basedfriction factor approaches the constant value of 0.042 as de-picted in the modified Moody diagram in [16] For intermediatevalues of β, the converged friction factors based on the con-stricted parameters lie in between these two extreme values.This work is the first study that reports experimental datawith systematic variation of roughness height and pitch in theturbulent region In order to gain a complete understanding ofthe effect of these parameters, further experimental study with awide range of β values is recommended This work is currently

in progress in the second author’s laboratory at the RochesterInstitute of Technology

Results of Lubrication Theory

The results from lubrication theory are applied to see whichparameter is best able to represent the laminar flow friction data

Figure 12 Parameters for separation normalized with experimental results.

644 T P BRACKBILL AND S G KANDLIKAR

ob-tained 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

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

separa-tion, 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

CONCLUSIONS

1 By comparing an idealized version of Nikuradse’s 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%

rela-tive roughness (RR) were shown to have appreciable effects

on laminar flow

3 Uniform roughness less than 5% RR also led to earlier

tran-sition 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

ele-ments 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

fac-tor and transition data approach those of a channel without

roughness elements The ratio of roughness pitch to

rough-ness 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

[2] Nikuradse, J., Forschung auf dem Gebiete des Ingenierwesens,

Verein Deutsche Ingenieure, vol 4, p 361, 1933.

[3] Moody, L F., Friction Factors for Pipe Flow, ASME Trans Journal

of Applied Mechanics, vol 66, pp 671–683, 1944.

[4] Kandlikar, S G., Roughness Effects at Microscale—Reassessing

Nikuardse’s Experiments on Liquid Flow in Rough Tubes, letin of the Polish Academy of Sciences, vol 53, no 4, pp 343–

Bul-349, 2005

[5] Brackbill, T P., and Kandlikar, S G., Effects of Low UniformRelative Roughness on Single-Phase Friction Factors in Mi-

crochannels and Minichannels, Proc International Conference

on Nanochannels, Microchannels, and Minichannels, Puebla,

T P BRACKBILL AND S G KANDLIKAR 645

Roughness, Microfluidics and Nanofluidics, Springer-Verlag, DOI

10.1007/S10404–005-0064–5, 2005

[7] Chen, Y., and Cheng, P., Fractal Characterization of Wall

Rough-ness on Pressure Drop in Microchannels, International

Commu-nications in Heat and Mass Transfer, vol 30, no 1, pp 1–11,

2003

[8] Pfund, D., Pressure Drop Measurments in a Microchannel, AlChe

Journal, vol 46, no 8, pp 1496–1507, 2000.

[9] Bahrami, M., Yovanovich, M M., and Cullham, J R., Pressure

Deop of Fully-Developed, Laminar Flow in Rough Microtubes,

Proc International Conference on Minichannels, and

Microchan-nels., Toronto, ICMM2005–75108, 2005.

[10] Zou, J., and Peng, X., Effects of Roughness on Liquid Flow

Be-havior in Ducts, ASME European Fluids Engineering Summer

Meeting., FEDSM2006–98143, pp 49–56, 2006.

[11] Mala, G M., and Li, D., Flow Characteristics of Water in

Micro-tubes, International Journal of Heat and Fluid Flow, vol 20, pp.

142–148, 1999

[12] Bucci, A., Celata, G P., Cumo, M., Serra, E., and Zummo, G

Wa-ter Single-Phase Fluid Flow and Heat Transfer in Capillary Tubes,

Proc International Conference on Microchannels and

Minichan-nels, Rochester, ICNMM2003–1037, 2003.

[13] Tu, X., and Hrnjak, P., Experimental Investigation of Single-Phase

Flow Pressure Drop Through Rectangular Microchannels, Proc.

International Conference on Microchannels and Minichannels,

Rochester, ICNMM2003–1028, 2003

[14] Baviere, R., Ayela, F., Le Person, S., and Favre-Marinet M.,

An Experimental Study of Water Flow in Smooth and Rough

Rectangular Micro-Channels, Proc International Conference on

Microchannels and Minichannels, Rochester, pp 221–228,

IC-NMM2004 2004

[15] Celata, G P., Cumo, M., Gugielmi, M., and Zummo, G.,

Experi-mental Investigation of Hydraulic and Single-Phase Heat Transfer

in 0.13-mm Capillary Tube, Microscale Thermophysical

Engi-neering, vol 6, pp 85–97, 2002.

[16] Kandlikar, S G., Schmitt, D., Carrano A L., and Taylor, J B.,

Characterization of surface Roughness effects on Pressure Drop

in Single-Phase Flow in Minichannels, Physics of Fluids vol 17,

no 10, 2005

[17] Wu, H Y., and Cheng, P., An Experimental Study of Convective

Heat Transfer in silicon Microchannels with different surface

con-ditions, International Journal of Heat and Mass Transfer, vol 46,

pp 2547–2556, 2003

[18] Wu, P., and Little, W A., Measurement of Heat Transfer

Char-acteristics in the Fine Channel Heat Exchangers Used for

Micro-miniature Refrigerators, Cryogenics, vol 24, pp 415–420, 1984.

[19] Wu, P., and Little, W A., Measurement of Friction Factors for the

Flow of Gases in Very Fine Channels used for Microminiature

Joule–Thomson Refrigerators, Cryogenics, vol 23, pp 273–277,

1983

[20] Li, Z., Du, D., and Guo, Z., Experimental Study on Flow

Charac-teristics of Liquid in Circular Microtubes, Proc Intl Conference

on Heat Transfer and Transport Phenomena in Microscale., pp.

162–167, 2000

[21] Hao, P., Yao, Z., He, F., and Zhu, K., Experimental Investigation of

Water Flow in Smooth and Rough Silicon Microchannels, Journal

of Micromechanics and Microengineering, vol 16, pp 1397–

1402, 2006

[22] Shen, S., Xu, J L., Zhou, J J., and Chen, Y., Flow and

Heat Transfer in Microchannels With Rough Wall Surface,

En-ergy Conversion and Management, vol 47, pp 1311–1325,

2006

[23] Peng, X F., Peterson, G P., and Wang, B X., Frictional Flow acteristics of Water Flowing Through Rectangular Microchannels,

Char-Experimental Heat Transfer, vol 7, pp 249–264, 1994.

[24] Wibel, W and Ehrhard, P., Experiments on Liquid Drop in Rectangular Microchannels, Subject to Non-Unity As-

Pressure-pect Ratio and Finite Roughness, Proc International Conference

on Nanochannels, Microchannels, and Minichannels Limerick,

Minichan-[26] Weilin, Q., Mala, G M., and Dongquing, L., Pressure-Driven

Water Flows in Trapezoidal Silicon Microchannels, International Journal of Heat and Mass Transfer, vol 43, pp 353–364, 2000.

[27] Celata, G P., Cumo, M., McPhail, S., and Zummo, G., drodynamic Behaviour and Influence of Channel Wall Rough-

Hy-ness and Hydrophobicity in Microchannels, Proc International Conference on Microchannels and Minichannels, Rochester,

Tim Brackbill is currently a mechanical

engineer-ing 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

Me-chanical 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 fo- cuses on the heat transfer and fluid flow phenomena

in microchannels and minichannels He is involved in advanced single-phase and two-phase heat exchang- ers incorporating smooth, rough, and enhanced mi- crochannels He has published more than 180 journal and conference papers He is a Fellow of the ASME, associate editor of a num-

ber 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 standing Scholarship Award in 2006 Currently he is working on a Department

Out-of Energy-sponsored project on fuel cell water management under freezing conditions.

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

minia-ture 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

investi-gating 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,

Al-buquerque, New Mexico Sincere thanks are offered to Micro Motion for

gen-erously 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

assis-tance in procuring these meters The assisassis-tance of Rahul Rao in the experimental

part of this study is greatly appreciated.

Address correspondence to Professor Afshin J Ghajar, School of

Mechan-ical and Aerospace Engineering, Oklahoma State University, Stillwater, OK

74078, USA E-mail: afshin.ghajar@okstate.edu

However, amid all the investigations in mini- and microchannelflow, there seems to be a lack in the study of the flow in the tran-sition region One obvious question is the location of the transi-tion region with respect to the hydraulic diameter of the channeland the roughness of the channel To successfully understandfriction factor and the location of the transition region, a system-atic experimental investigation on various hydraulic diameters

of mini- and microchannels is necessary However, the sciencebehind these advanced technologies seems to be controversial,especially fueled by the experimental results of the fluid flowand 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] while, some have reported that friction factor correlations forconventional sized tubes to be applicable for mini- and micro-tubes [3–5] However, many recent experiments on small-sizedtubes and channels have observed higher friction factors thanthe correlations for conventional-sized tubes and channels [6–11], and the cause of this discrepancy was attributed to surfaceroughness Literature also highlights the importance of diame-ter measurement and difficulties associated with quantifying theeffect of roughness These difficulties are primarily due to thelarge number of parameters used in describing various rough-ness geometries

Mean-In this study, mini- and microtubes are chosen over othernoncircular channels to negate the effect of aspect ratio, which

646

A J GHAJAR ET AL 647

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

contradic-tory 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

fac-tors 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

exper-iment 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

character-istics of R-134a in stainless-steel tubes with diameters of 244,

430, and 792 µm They found that within experimental

uncer-tainty, conventional theories are able to predict the experimental

friction factors The authors found no evidence of early

transi-tion and they reported the onset of transitransi-tion Reynolds number

occurred slightly below 2,000

Yang and Lin [4] investigated water flow through

stainless-steel tubes with diameters ranging from 123 to 962 µm They

found that the friction factor results correlate well with

correla-tions 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

Rands et al [5] measured the frictional pressure drop andtemperature induced by viscous heating for water flowingthrough fused-silica microtubes with diameters from 32.2 to16.6 µm The results from their work were confirmed with clas-sical laminar flow behavior at low Reynolds number The onset

of transition region was observed at the Reynolds numbers of2,100 to 2,500

Mala and Li [6] analyzed water flowing through fused-silicaand stainless-steel tubes ranging from 50 to 254 µm Contrary tothe previous researchers, they found friction factor values largerthan what the theory predicted Moreover, they also observed

transition in Reynolds number range of 300 to 900 was reported,and surface roughness was proposed as a significant cause ofthat early flow transition

Celata et al [7] performed pressure drop tests using R-114

in a 130 µm microtube The Reynolds numbers investigatedranged from 100 to 8,000 Transition was observed to be in theReynolds number range of 1,880 to 2,480 In the laminar re-gion, the experimental values matched well with the theoreticalpredictions until the Reynolds number of 585 For Reynoldsnumbers greater than 585, higher friction factor values wereobserved The authors attributed this deviation from theory toroughness of the stainless-steel microtube

Kandlikar et al [8] investigated the effect of roughness onpressure drop in microtubes The roughness was changed byetching the tubes with different acids They observed that forlarger tubes (1067 µm), the effect of roughness is negligible.For smaller tubes (620 µm), increases in roughness resulted inhigher 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 indiameter) They found that the f·Re in laminar region for smoothtubes was nearly 64, while the results for rough tubes with peak–valley roughness of 3 to 4% showed 15 to 37% higher than theclassical f·Re value of 64 Based on flow characteristics, Li et al.[9] concluded that the onset of transition region occurred at theReynolds numbers of 1,700 to 2,000

Zhao and Liu [10] conducted pressure drop studies on smoothquartz-glass tubes and rough stainless-steel tubes of varying di-ameters They observed that in the laminar regime, experimentalresults agreed well with theoretical values However, early tran-sition at Reynolds numbers ranging from 1,100 to 1,500 (forsmooth microtubes) was recorded For rough microtubes (with

number of 800, where similar early transition was observed.Tang et al [11] investigated the flow characteristics of ni-trogen and helium in stainless-steel and fused-silica tubes ofvarious diameters They observed that the friction factors instainless-steel tubes are much higher than the theoretical corre-lation for the laminar region, deviating by as much as 70% for

a tube diameter of 172 µm Friction factors for the smootherwalled fused-silica tubes were found to be in relative agree-ment with the theory for conventional-sized tubes The positive

648 A J GHAJAR ET AL.

deviation was attributed to the roughness and was found to

in-crease 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

essen-tial, citing it as a possible factor in leading to higher friction

factors

In a review by Kandlikar [12], he suggested that the

uncer-tainties in the experiments by Nikuradse [13] in the laminar

re-gion were very high, and the conclusion regarding the absence of

roughness effects in the laminar region is questionable

Notic-ing 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

ele-ments,

diameter, and ε is the roughness height One may consider that

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

DcfD

5

(2)Based on the constricted flow diameter, the Reynolds number is

then expressed as

Brackbill and Kandlikar [16] experimentally investigated

the effect of relative roughness on friction factor and critical

Reynolds number for mini- and microchannels In their

experi-ments, 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

rough-ness 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

rough-ness Clearly, as shown in Figure 1a, roughness effects played a

role in the laminar region, and the effects increased with higher

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.

relative roughness By including the constricted flow hydraulic

Kand-likar [16] observed that the agreement between the experimentaldata and laminar flow theory for friction factor was significantlyimproved (Figure 1b) In addition, they also observed a trendrelating the critical Reynolds number and the relative roughness

To predict the onset of transition region, Brackbill and likar [16] recommended a correlation for the critical Reynoldsnumber and the relative roughness based on the constricted flowhydraulic diameter,

0.08

ε

Dh,cf

for

A J GHAJAR ET AL 649

where the critical Reynolds number for smooth channels

for channels and has only been verified with data for

mini-and microchannels from [16] mini-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

rang-ing 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

tran-sition, 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

chan-nels [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

im-portance, 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

appara-tus 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

com-bination, consisting of a high-pressure cylinder filled with

ultra-Figure 2 Schematic of the experimental setup.

high-purity nitrogen in combination with a stainless-steel sure vessel The system is an open loop Thus, after the workingfluid passes through the apparatus, it is passed into a sealed col-lection container and recycled manually Nitrogen in the high-pressure cylinder is pressurized to 17.2 MPa by the distributor.This pressurized nitrogen is then fed to the stainless-steel pres-sure vessel via a two-phase regulator and line The workingfluid, distilled water for the purposes of this research, is stored

pres-in the stapres-inless-steel pressure vessel As the pressurized gen is fed into the stainless-steel pressure vessel, the workingfluid is forced up a stem extending to the bottom of the vessel,out of the pressure vessel, and through the flow-meter array andtest section An Airgas regulator is used for the purposes of con-trolling the pressure of the nitrogen inlet to the stainless-steelpressure vessel This dual-stage regulator is capable of provid-ing pressures ranging from 0 to 1.72 MPa The stainless-steelpressure vessel used is an Alloy Products model 72–05, provid-ing a maximum working pressure of 1.37 MPa and a capacity

nitro-of 19.0 L

After exiting the pressure vessel, the distilled water travels

to the flow-meter array The flow rate of the water enteringthe array is further regulated using a Parker N-Series model6A-NLL-NE-SS-V metering valve, which allows fine-tuning ofthe flow rate Fluid passes through the metering valve and intoone of the two Micro Motion Coriolis flow meters Two flowmeters 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 amodel 1700 transmitter This meter is designed to measure massflows ranging from 54 to 2,180 kg/h for liquids Within thisrange 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 Motionmodel LMF3M, coupled with an LFT transmitter This secondmeter is designed to measure mass flows ranging from 0.001 to1.5 kg/h

After passing through the flow-meter array, fluid enters thetest section assembly via a second section of PFA tubing to thetest section assembly The test section assembly contains the testsection as well as the equipment necessary for measurement of

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

stainless-steel type 304 hypodermic tubes with factory-cut lengths of

≤ 337 µm) Since the friction factor measurements are

con-ducted 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

re-search facilities used for experimentation have different

pres-sure transducer diaphragms to encompass a range of differential

pressures from 1.38 to 1,380 kPa The Validyne pressure

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

im-portant 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

volt-age 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

high-est rated in terms of pressure is a Perma-Cal 2070 kPa thigh-est gauge

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

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

these meters, in-laboratory calibration consisted of checking the

manufacturer’s calibrations over a range of flow rates via timedcollection of fluid passing through the meters In addition, themaximum and minimum milliamp outputs of the CMF-025 weretuned 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 themeasured pressure drop data, the friction factor can be calculatedwith

of each diaphragm used Diaphragms were carefully selectedduring experimentation in order to obtain the highest accuraciespossible The worst-case scenario occurs with small tube sizeand low Reynolds number In this region, the uncertainty in

It is more representative to look at intermediately sized tubesand/or flow rates through the transition and turbulent regions.Uncertainty of the pressure drop measurement in these areas

The uncertainty in mass flow rate measurement given bythe Micro Motion flow meter specifications for the CMF-025

consideration that the larger meter is being utilized at flow rateslower than its range in order to cover the entire range of flowrates for all of the tubes under research Based upon uncertaintyequations given in the Micro Motion specifications, the worst-

Both the use of the LMF3M and the under-ranging of the

CMF-025 occur at smaller tube sizes and lower Reynolds numbers.Thus, it is necessary to calculate uncertainty for either of themeters running within their specified mass flow ranges and toestimate uncertainty when the CMF-025 is pushed to its lowestrange 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 thedifferent tube sections in order to ensure consistency Measured

Due to the fact that uncertainty in both the Validyne pressuretransducer and the Micro Motion meters is dependent upon tubesize and Reynolds number, three different uncertainty valueshave been established In order to quantify the overall uncer-tainty, analysis was conducted using the method described byKline and McClintock [21] In the case of larger tube sizes andhigher Reynolds numbers, the CMF-025 meter is used and is

A J GHAJAR ET AL 651

functioning at the manufacturer’s specified uncertainty level

The pressure transducer is operating at the better of its two

cal-culated uncertainty levels Taking this into consideration, the

and Reynolds number decrease, the CMF-025 meter begins to

operate under range In this area, the pressure transducer is

con-sidered to be operating at the lesser of its two uncertainty levels

In this range, the overall uncertainty associated with the

lowest ranges of tube size and Reynolds number, the LMF3M

meter is used In this area, the pressure transducer is still

oper-ating at the lesser of its two uncertainty levels For this lowest

Reynolds number and smallest tube size situation, the overall

this section are for the experimental results when the

pressure-sensing diaphragms were used appropriately for the measured

pressure drop ranges

Diameter and Surface Roughness Verification

Since the mini- and microtubes under research were

pur-chased from an outside source, data obtained from these tubes

are only as accurate as the manufacturer’s specifications The

diameters of the tubes as well as the roughness of the inner

walls of the tubes are of particular concern due to the type

of research being undertaken In order to ensure that the data

recorded were of the highest quality possible, it was deemed

nec-essary to determine the degree of accuracy of the manufacturer’s

specifications In order to do this, both the scanning electron

mi-croscope (SEM) and the scanning probe mimi-croscope (SPM) at

the Oklahoma State University Microscopy Laboratory were

utilized Diameter measurements were taken using the SEM,

while roughness measurements were taken using the SPM

Two different tube sizes were examined using the SEM in

order to check the accuracy of the manufacturer’s tolerances

The first of these two tubes had an inner diameter and tolerance

size was covered between the two tubes examined Imaging was

done using the JEOL JXM 6400 scanning electron microscope

system in combination with a digital camera system The

reso-lution of the microscope ranged from 30 to 50 nm Once images

had been captured, it was possible to determine image pixel size

in terms of length scale With a known pixel-to-length scale, the

inner diameter of the tubes could be estimated from the SEM

images

For the first tube with the manufacturer-specified inner

to be 5,280 µm from the SEM image For the second tube with

average inner diameter was estimated to be 574 µm from the

SEM image (see Figure 3) The SEM imaging of these two tubes

verified that the manufacturer’s specifications of the tube

diam-eters and tolerances are verifiable and reasonably dependable

Figure 3 SEM image of a 584 ± 38 µm (manufacturer’s specification) eter stainless-steel tube; based on this SEM image, the tube diameter was found

diam-to be 574 µm.

Roughness measurements were conducted using a SPM tion in combination with Digital Instruments Multimode V elec-tronics and an optical microscope for tip positioning The systemused is capable of three-dimensional (3-D) spatial mapping and

sta-an ultimate resolution of 0.1 nm laterally sta-and 0.01 nm cally Scans were taken of multiple sections of two stainless-steel tubes with different inner diameters: 5,330 µm and 2,390

verti-µm Roughness data were taken from three different sections

of each of these tubes In order to negate the effect of the vature of the tubes upon the roughness measurement generated

cur-by the SPM, a flattening feature was utilized From the SPM,

section of the 5,330 µm diameter tube is shown in Figure 4.From the SPM measurements, the inner surface of the 5,330

re-spectively In similar manner, the inner surface of the 2,390 µm

1,710 nm, and 194 nm, respectively Some variability was foundbetween the two tubes, though this was to be expected The man-ufacturer specified an inner wall root mean square roughness of

410 nm Thus, the root mean square roughness measured by theSPM for each of the two tubes was within the manufacturer’sspecifications

When compared with the roughness results documented byYoung et al [17], the maximum roughness profile peak height

roughness profile peak height of milled stainless-steel surface

999 nm) It should be noted that measurements by Young et al.[17] were from surface roughness that was created systemati-cally to be uniform and aligned On the other hand, the tubes

652 A J GHAJAR ET AL.

Figure 4 SPM topographic image for a section of a 5,330 µm inner diameter

stainless-steel tube (Ra = 240 nm, Rmax = 2,628 nm, Rq = 292 nm).

used in this study were obtained commercially, and thus the

uni-formity and alignment of the surface roughness in these tubes

are uncertain

Based on the inner diameter tolerances specified by the

manu-facturer and the results from both SPM and SEM measurements,

[14] can be determined Having known the constricted flow

esti-mated using the concept of constricted flow diameter—see Eq

(1)—proposed by Kandlikar et al [14] As shown in Figure 3,

the irregularities in the tube and the tolerances of the inner

di-ameter can result in much larger roughness height (ε) than the

RESULTS AND DISCUSSION

The review documented by Krishnamoorthy and Ghajar [20]

pointed out the need for further experiments to confirm the start

and end of transition region in mini- and microtubes In large

part, this need for further experimentation is exemplified by the

highly contradictory observations that have been reported by

various investigators The disparity found in the literature may

be attributed to factors such as tube diameter, surface roughness,

experimental facilities, and instrumentation Without any doubt,

the sensitivity of the instruments used in the measurement of

pressure drop plays one of the most crucial roles in collecting

accurate data In addition, in properly addressing the effect of

tube diameter on pressure drop for flow in mini- and microtubes,the importance of systematically investigating various tube sizescannot be overlooked

In order to be able to clearly pick up the transition regionalong with the laminar and turbulent regions, the sensitivity

of the pressure-sensing diaphragms used in the Validyne DP15pressure transducer had to be given meticulous consideration.Even the numerous studies that covered the laminar and turbu-lent regions fail to explicitly capture the transition region Inmany cases, this failure may be attributed to the questionablesensitivity of the instrumentation used The dilemma that arisesfrom this is that if one is not confident with the results for thetransition region, then the confidence in the results for laminarand turbulent regions is also questionable

To properly recognize sensitivity for each pressure-sensingdiaphragm, it is necessary to collect pressure drop data for theentire pressure-sensing range of each diaphragm before chang-ing to the next diaphragm Collecting pressure drop data throughthe entire pressure-sensing range of each diaphragm further en-hances collection of accurate data The overall uncertainty asso-ciated to each pressure-sensing diaphragm was estimated to be

the pressure transducer, the pressure gages used for ing the pressure diaphragms, and the standard deviation of the

diaphragm’s pressure-sensing range implies that a 345-kPa

Figure 5 illustrates the comparison of the friction factor datapoints measured using various pressure-sensing diaphragms for1,600 and 1,067 µm diameter stainless-steel tubes Figure 5abrings out the obvious scenario that using both 55.2 and 138

kPa pressure diaphragms for 700 < Re < 3,500 would easily

bring one to conclude that the higher friction factor values aredue to surface roughness The appropriate error bars, based on

pressure-sensing range, are attached on two selected data points obtained

by 55.2 and 138 kPa pressure diaphragms to illustrate theiruncertainties The data point measured by the 55.2 kPa pressure

the data point measured by the 138 kPa pressure diaphragm at

data point measured by the 138 kPa pressure diaphragm at Re

= 1,400 shows the error bar with a 14% extension below the

a wrong conclusion that the value of f·Re is 55 rather than theconventional value of 64 For comparison purposes, in Figure 5aerror bars are also attached on two selected data points obtained

by the 3.45 and 13.8 kPa pressure diaphragms, which showsignificantly lower uncertainties than the error bars on the datapoints obtained by the 55.2 and 138 kPa pressure diaphragms

As shown in Figure 5a, pressure diaphragms with ratings of

13.8 kPa or lower would be appropriate for Re < 3,500, and

pressure diaphragms with ratings of 55.2 kPa or higher would

be appropriate for Re > 3,500 For 500 < Re < 1,700, data points

Figure 5 Comparison of results measured by various pressure-sensing

di-aphragms for two different tubes: (a) 1,600 µm diameter tube, (b) 1,067 µm

diameter tube.

measured by the 13.8 kPa pressure diaphragm were also verified

by the 3.45 kPa pressure diaphragm to be within experimental

uncertainties Similarly, data points measured by the 55.2 kPa

pressure diaphragm were also verified by the 138 kPa pressure

diaphragm to be within experimental uncertainties for 3500 <

Re < 5,500.

Figure 5(b) illustrates that the proper selection of

pressure-sensing diaphragms is essential to accurately measure friction

factor in the transition region The appropriate error bars, based

pressure-sensing range, are attached on a data point measured by

(Figure 5b) The data point measured by the 138 kPa pressure

the data point measured by the 345 kPa pressure diaphragm at

64/Re line As seen previously in Figure 5a, this scenario shown

in Figure 5b also implies the possibility of having a wrong

con-clusion that the value of f·Re is 55 rather than the conventional

value of 64 For comparison purposes, in Figure 5b error bars

are also attached on two selected data points obtained by the

34.5 and 55.2 kPa pressure diaphragms which show cantly lower uncertainties than the error bars on the data pointsobtained by the 138 and 345 kPa pressure diaphragms The dis-crepancies between the data points measured by the 138 kPaand 345 kPa pressure diaphragms show that these diaphragmscould not accurately capture the transition region The actualfriction factor values were accurately measured by the 55.2 kPadiaphragm and verified by the 34.5 kPa diaphragm to be withinexperimental uncertainties At the trough of the transition region

15% higher than the data measured by the 55.2 kPa diaphragm

8% higher than the data measured by the 55.2 kPa diaphragm.Based on the illustrations of Figure 5, improper use ofpressure-sensing diaphragm could easily lead to erroneous con-clusions about the flow phenomena in the microtubes tested

It should be noted that the ability to capture transition decaysquite rapidly when diaphragms inappropriate to the range ofpressure drop under investigation are utilized Thus, extremecare in diaphragm selection is imperative in order to capturethe transition region with the greatest possible accuracy Evensmall failures in terms of accuracy can lead to flatter transitionregions, leaving the actual physics of the flow unobserved Withthe effect of surface roughness and diameter in microtubes stilllargely unexplored, these types of failures are unacceptable Al-though it seems trivial to discuss the sensitivity of the pressurediaphragm, Figures 5a and b clearly illustrate the consequences

of ignoring it It should be noted that all the friction factor data

The experimental results of 12 different stainless-steel tubeswith diameters ranging from 2,083 to 337 µm were investigated

in detail with regard to the laminar, transition, and turbulentregions over Reynolds numbers ranging from 500 to 10,000 Theexperimentally determined friction factor in the laminar regionwas compared with the conventional friction factor equation for

the experimental friction factor was compared with the Blasius

representative of the experimental results for the friction factor

of the 1,372 µm diameter stainless-steel tube is shown in Figure

6 As illustrated in Figure 6, the onset of transition for the 1372

µm diameter tube is at Reynolds number of 1,900, and the end

of transition is at Reynolds number of 4,000

Based on the experimental friction factors, the transition

re-gion can be estimated by locating the Reynolds numbers where

the friction factor departs from the laminar line and merges

with the turbulent line The transition Reynolds number ranges

summarized in Table 1 were determined by using a 5%

devia-tion criterion from the laminar and turbulent lines According

to our experimental uncertainty analysis, the maximum error

in friction factor measurements was estimated to be no more

than 3%, and the 5% deviation criterion was used to encompass

the experimental uncertainty In the estimation of the transition

Reynolds number range for each tube diameter, the transition

region begins with the first data point that is 5% higher than the

laminar line, and ends with the data point that is 5% lower than

the perceived turbulent line The perceived turbulent line is a

straight line connecting the data points in the turbulent region

(Re > 4,000) on the base-10 logarithms friction factor versus

Reynolds number plot

The experimental friction factor results provided an

inter-esting observation The decrease in tube diameter from 2,083

to 667 µm actually delayed the onset of transition region For

µm is consistently located at a Reynolds number of 4,000 ever, for tube sizes of 732 and 667 µm, the end of the transitionregion shifted forward to a Reynolds number of 3,000 Furtherdecrease in the tube diameter from 667 to 337 µm caused theonset of the transition region to shift from a Reynolds num-ber of 2,200 to 1,300, while the end of the transition regionshifted from a Reynolds number of 3,000 to 1,700 The ex-perimental results indicated that the Reynolds number rangefor transition flow becomes narrower with the decrease in tubediameter

How-By focusing on the transition region, the effect of tube ameter on the friction factor profile can be clearly seen Thefriction factor profiles of the 12 stainless-steel tubes in the tran-sition region are shown in Figures 7 to 9 Figure 7 shows thatthe decrease in tube diameter from 2,083 to 1,372 µm did notsignificantly affect the profile of the friction factor, with theexception of the onset of transition region The decrease in thetube diameter from 2,083 to 1,600 to 1,372 µm showed the onset

di-of transition region shifted from Reynolds number di-of 1,500 to1,700 to 1,900, respectively

As the tube diameter is further decreased, the friction factorprofiles also shifted (see Figure 8) When the tube diameter isdecreased from 1,372 to 1,067 µm, another group of similar

A J GHAJAR ET AL 655

friction factor profiles in the transition region is seen Figure

8 shows that the decrease in tube diameter from 1,067 to 838

µm did not significantly affect the profile of the friction factor

Since the 991 µm diameter tube is only 7% smaller than the

1,067 µm diameter tube, it is expected that they have the same

friction factor profile and transition range (2,000 < Re < 4,000).

However, as the diameter is decreased from 991 to 838 µm, there

is a slight noticeable change in the friction factor profile, with

the onset of the transition region shifted from Reynolds number

of 2,000 to 2,200 and an increase in the depth of the trough in

the transition region

Figure 9 shows that further decrease in the tube diameter

from 838 to 337 µm caused the transition region to become

significantly narrower The onset of transition region for

732-and 667-µm tubes is the same as that of the 838-µm tube,

at Reynolds number of 2,200 However, the end of transition

region is shifted forward to Reynolds number of 3,000, making

the transition region of the 732 and 667 µm tubes narrower than

that of the 838 µm tube Further decrease in the tube diameter

from 667 to 337 µm caused the onset and end of the transition

region to shift to lower Reynolds numbers, while the transition

range became narrower The decrease in the tube diameter from

732 to 337 µm also caused the friction factor profile to shift

higher This suggests that the effect of surface roughness may be

beginning to influence the friction factor Also, the irregularities

in the stainless-steel tube at such small scale, as illustrated in

Figure 3, could have contributed to the shift in the friction factor

profile

et al [14], can be determined based on the inner diameter

tol-erances specified by the manufacturer and the results from both

SPM and SEM measurements Using the concept of constricted

flow diameter, as in Eq (1), the roughness height (ε) of the tubes

can be estimated using the determined constricted flow

work of others, friction factor data points plotted by Brackbill

and Kandlikar [16] for channels with varying relative roughness

(Figure 1a) were extracted and plotted with current experimental

data for tube with diameter of 413 µm (Figure 10) The

com-parison in Figure 10 verifies the notion of roughness affecting

the laminar friction factors It should be noted that the friction

factors measured by Brackbill and Kandlikar [16] were from

channels with surface roughness that was created systematically

to be uniform and aligned On the other hand, the tubes used in

this study were obtained commercially, and the uniformity and

alignment of the surface roughness in these tubes are uncertain

Thus, some discrepancies in the results of current study with the

results from Brackbill and Kandlikar [16] are to be expected In

addition, results from Li et al [9] have reported that results from

rough tubes with peak–valley roughness of 3 to 4% showed 15

to 37% higher than the f·Re value of 64 in laminar region, which

is in agreement with the findings of this work

con-stricted flow friction factor (fcf) and the Reynolds number (Recf)

can be determined using Eqs (2) and (3), respectively Figure

Re

500 600 800 1000 1500 2000 2500 3000 4000 5000

0.02 0.03 0.04 0.06 0.08

0.15 0.20

11 shows the experimental data plotted using the constricted

the experimental data plotted with the constricted flow ters showed better agreement than the data plotted in Figure 9.This is another confirmation of the observation by Brackbill andKandlikar [16] that roughness has effects on the friction factor inthe laminar region (Figure 1b) To improve the agreement withlaminar friction factor theory, the experimental data needs to beplotted with the constricted flow parameters, which in essencediscounts the roughness element height and only considers thefree flow area corresponding to the constricted flow diameter(Dcf)

parame-To verify that roughness affects the onset of transition fromlaminar flow, the correlation (Eq (4)) proposed by Brackbilland Kandlikar [16] was applied to the critical Reynolds num-bers summarized in Table 1 The comparison of Eq (4) withthe critical Reynolds numbers observed in current experimentalwork is shown in Figure 12 The correlation shows favorablecomparison with the observations of this experimental study

average error band reported in [16], while most of the data points

Recf

400 500 600 800 1000 1500 2000 2500 3000 4000

f cf

0.02 0.03 0.04 0.06 0.08

0.15 0.20

Figure 12 Comparison of critical Reynolds numbers observed in current

work with Eq (4) proposed by Brackbill and Kandlikar [16].

data from the current work agrees with the finding of Brackbill

and Kandlikar [16] that increase in the relative roughness

low-ers the Reynolds numblow-ers for the onset of the transition region

It should be noted that the correlation (Eq (4)) proposed by

Brackbill and Kandlikar [16] was recommended for channels

The use of the correlation here with data from tubes is to merely

demonstrate that increase in the relative roughness causes early

transition, which was seen in both small sized tubes and

chan-nels Further validation with data from both tubes and channels

is necessary to confirm the performance and feasibility of Eq

(4) for both tubes and channels

CONCLUSIONS

This study systematically investigated the experimental

re-sults for the single-phase flow characteristics of distilled water in

stainless-steel mini- and microtubes of diameters ranging from

2,083 to 337 µm The sensitivity of the instruments and careful,

systematic experimental methodology are the key to obtaining

the accurate measurements necessary for this type of research

Improper use of pressure-sensing diaphragms could easily lead

to erroneous conclusions about the flow phenomena in mini- and

microtubes in addition to improper representation of the

transi-tion region With so much left to explore in terms of the effects

of surface roughness and diameter on mini- and microtube flow,

neither of these outcomes can be deemed tolerable

Decrease in tube diameters and increase in relative roughness

have been found to influence friction factor, even in the

lami-nar region These findings were confirmed with results from [9,

14–16] In addition to friction factor, decrease in tube diameters

and increase in relative roughness have shown that the onset of

transition from laminar flow occurred at lower Reynolds

num-bers Also, the experimental results indicated that the Reynolds

number range for transition flow becomes narrower with the

decrease in the tube diameter

When measuring the friction factor and determining the onset

of transition flow for mini- and microtubes, both the diameter

and the roughness height have to be accounted for As shown

in this work, relative roughness affects the friction factor and

the critical Reynolds number, and both the diameter and theroughness height affect the relative roughness When the tubediameter is accounted for while the surface roughness is ignored,the discrepancy of the friction factor with classical theory cannot

be properly explained

NOMENCLATURE

(= fcf/4)

˙

[1] Choi, S B., Barron, R F., and Warrington, R O., Fluid Flow

and Heat Transfer in Microtubes, Proceedings of Winter Annual Meeting of the ASME Dynamic Systems and Control Division,

Atlanta, GA, vol 32, pp 123–134, 1991

[2] Yu, D., Warrington, R., Barron, R., and Ameel, T., Experimentaland Theoretical Investigation of Fluid Flow and Heat Transfer

in Microtubes, Proceedings of the 1995 ASME/JSME Thermal Engineering Joint Conference, Maui, HI, vol 1, pp 523–530,

1995

[3] Hwang, Y W., and Kim, M S., The Pressure Drop in

Micro-tubes and the Correlation Development, International Journal

of Heat and Mass Transfer, vol 49, no 11–12, pp 1804–1812,

2006

A J GHAJAR ET AL 657[4] Yang, C Y., and Lin, T Y., Heat Transfer Characteristics of Water

Flow in Micro Tubes, Experimental Thermal and Fluid Science,

vol 32, no 2, pp 432–439, 2007

[5] Rands, C., Webb, B W., and Maynes, D., Characterization of

Transition to Turbulence in Microchannels, International Journal

of Heat and Mass Transfer, vol 49, no 17–18, pp 2924–2930,

2006

[6] Mala, Gh M., and Li, D., Flow Characteristics of Water in

Mi-crotubes, International Journal of Heat and Fluid Flow, vol 20,

no 2, pp 142–148, 1999

[7] Celata, G P., Cumo, M., Guglielmi, M., and Zummo, G.,

Experi-mental Investigation of Hydraulic and Single-Phase Heat Transfer

in 0.130-mm Capillary Tube, Nanoscale and Microscale

Thermo-physical Engineering, vol 6, no 2, pp 85–97, 2002.

[8] Kandlikar, S G., Joshi, S., and Tian, S., Effect of Surface

Rough-ness on Heat Transfer and Fluid Flow Characteristics at Low

Reynolds Numbers in Small Diameter Tubes, Heat Transfer

En-gineering, vol 24, no 3, pp 4–16, 2003.

[9] Li, Z X., Du, D X., and Guo, Z Y., Experimental Study on Flow

Characteristics of Liquid in Circular Microtubes, Nanoscale and

Microscale Thermophysical Engineering, vol 7, no 3, pp 253–

265, 2003

[10] Zhao, Y., and Liu, Z., Experimental Studies on Flow Visualization

and Heat Transfer Characteristics in Microtubes, Proceedings of

the 13th International Heat Transfer Conference, Sydney,

Aus-tralia, MIC-12, 2006

[11] Tang, G H., Li, Z., He, Y L., and Tao, W Q., Experimental

Study of Compressibility, Roughness and Rarefaction Influences

on Microchannel Flow, International Journal of Heat and Mass

Transfer, vol 50, no 11–12, pp 2282–2295, 2007.

[12] Kandlikar, S G., Roughness Effects at Microscale—Reassessing

Nikuradse’s Experiments on Liquid Flow in Rough Tubes, Bulletin

of the Polish Academy of Sciences, vol 53, no 4, pp 343–349,

2005

[13] Nikuradse, J., Laws of Flow in Rough Pipes (English Translation),

NACA Technical Memorandum 1292, 1950.

[14] Kandlikar, S G., Schmitt, D., Carrano, A L., and Taylor, J B.,

Characterization of Surface Roughness Effects on Pressure Drop

in Single-Phase Flow in Minichannels, Physics of Fluids, vol 17,

100606, 2005

[15] Taylor, J B., Carrano, A L., and Kandlikar, S G.,

Characteri-zation of the Effect of Surface Roughness and Texture on Fluid

Flow—Past, Present, and Future, International Journal of

Ther-mal Sciences, vol 45, pp 962–968, 2006.

[16] Brackbill, T P., and Kandlikar, S G., Effect of Low Uniform,

Rel-ative Roughness on Single-Phase Friction Factors in

Microchan-nels and MinichanMicrochan-nels, Proceedings of the 5th International

Conference on Nanochannels, Microchannels and Minichannels,

Puebla, Mexico, ICNMM2007–30031, 2007

[17] Young, P L., Brackbill, T P., and Kandlikar, S G., Estimating

Roughness Parameters Resulting from Various Machining

Tech-niques for Fluid Flow Applications, Heat Transfer Engineering,

vol 30, no 1–2, pp 78–90, 2009

[18] Brackbill, T P., and Kandlikar, S G., Effect of Sawtooth

Rough-ness on Pressure Drop and Turbulent Transition in Microchannels,

Heat Transfer Engineering, vol 28, no 8–9, pp 662–669, 2007.

[19] Celata, G P., Lorenzini, M., Morini, G L., and Zummo, G.,Friction Factor in Micropipe Gas Flow under Laminar, Transition

and Turbulent Flow Regime, International Journal of Heat and Fluid Flow, vol 30, no 5, pp 814–822, 2009.

[20] Krishnamoorthy, C., and Ghajar, A J., Single-Phase FrictionFactor in Micro-Tubes: A Critical Review of Measurement,Instrumentation and Data Reduction Techniques from 1991–

2006, Proceedings of the 5th International Conference on Nanochannels, Microchannels and Minichannels, Puebla, Mex-

ico, ICNMM2007-30022, 2007

[21] Kline, S J., and McClintock, F A., Describing Uncertainties in

Single-Sample Experiments, Mechanical Engineering, vol 75,

no 1, pp 3–8, 1953

Afshin J Ghajar is a Regents Professor and Director

of Graduate Studies in the School of Mechanical and Aerospace Engineering at Oklahoma State University Stillwater, Oklahoma, and a Honorary Professor of Xi’an Jiaotong University, Xi’an, China He received his B.S., M.S., and Ph.D., all in mechanical engineer- ing, from Oklahoma State University His expertise

is in experimental and computational heat transfer and fluid mechanics Dr Ghajar has been a summer research fellow at Wright Patterson AFB (Dayton, Ohio) and Dow Chemical Company (Freeport, Texas) He and his coworkers have published over 150 reviewed research papers He has received several outstanding teaching/service awards, such as the Regents Distinguished Teach- ing Award; Halliburton Excellent Teaching Award; Mechanical Engineering Outstanding Faculty Award for Excellence in Teaching and Research; Golden Torch Faculty Award for Outstanding Scholarship, Leadership, and Service by the Oklahoma State University/National Mortar Board Honor Society; and re- cently the College of Engineering Outstanding Advisor Award Dr Ghajar is a

Fellow of the American Society of Mechanical Engineers (ASME), Heat

Trans-fer Series editor for Taylor & Francis/CRC Press, and editor-in-chief of Heat Transfer Engineering He is also the co-author of the fourth edition of Cengel and

Ghajar, Heat and Mass Transfer – Fundamentals and Applications,

McGraw-Hill, 2010.

Clement C Tang is a Ph.D candidate in the

School of Mechanical and Aerospace Engineering at Oklahoma State University, Stillwater, Oklahoma He received his B.S and M.S degrees in mechanical en- gineering from Oklahoma State University His areas

of specialty are single-phase flow in mini- and tubes and two-phase flow heat transfer.

micro-Wendell L Cook is a graduate student in the

School of Mechanical and Aerospace Engineering at Oklahoma State University, Stillwater, Oklahoma He received his B.S and M.S degrees in mechanical engineering from Oklahoma State University He re- cently started his Ph.D studies in mechanical engi- neering focusing on single-phase and two-phase flow

in mini- and microchannels.

Copyright
C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457630903466605

Flow in Channels With Rough

Walls—Old and New Concepts

H HERWIG, D GLOSS, and T WENTERODT

Hamburg University of Technology, Hamburg, Germany

In our study we determine the influence of wall roughness on friction for pipe and channel flows by numerically calculating

the entropy production in the flow It turns out that there is an appreciable influence of wall roughness in laminar flows,

though this effect often is neglected completely In addition to the friction factor results, we gain an understanding of the

physics since we have access to the dissipation distribution in the flow field close to the roughness elements For a concise

description of flows over rough walls there should be a reasonable choice of the wall location as well as the roughness

parameter Various options are discussed and assessed.

INTRODUCTION

Real flows of any kind of fluid are always subject to losses of

mechanical energy From a thermodynamic point of view this

is due to a dissipation process that converts exergy (available

work) into anergy by producing entropy [1] Therefore, entropy

production in a flow is directly linked to the losses and,

pro-vided it can be determined, may help to quantify these losses

[2, 3] Following this idea, we are able to determine losses by

calculating the corresponding entropy production rates We thus

can replace measurements by numerical calculations and/or give

a detailed physical explanation of how and where the losses

oc-cur

introduced that serve to quantify the losses of certain

compo-nents in a pipe or channel system like bends, tee junctions, or

flow exits, but also are used for straight pipes or channels [4–6]

Their definition is

u2 m

with ϕ as specific dissipation [in J/kg= m2/s2] and umas a

char-acteristic (mean) velocity of the component under consideration

When Eq (1) is applied to straight pipes or channels a friction

factor f is introduced with

(2)

Address correspondence to H Herwig, Hamburg University of Technology,

Denickestr 17, 21073 Hamburg, Germany E-mail: h.herwig@tuhh.de

diameter,

Dh=4A

with A being the fluid filled cross section and C its wetted

expression for f valid for all different kinds of cross sections

well for turbulent flows but gives poor results in the laminarcase [6]

Since micro flows, which are the scope of our study, arepredominantly laminar, we restrict ourselves to this case For

a more comprehensive study also including turbulent flows see[7]

is introduced Therefore, the Poiseuille number

is often used instead

For macro flows the widely accepted proposition is that

for laminar flows This, for example, is claimed in the famousMoody chart [8], among others, based on Nikuradse’s measure-ments made almost 75 years ago [9] In this context there is noinfluence of roughness as long as the flow is laminar—i.e., in the

range Taking this as a starting point, we address the followingtwo questions:

658

H HERWIG ET AL 659

Figure 1 Channel of arbitrary cylindrical shape with cross section A and

length L A, C, and umonly apply for the equivalent smooth channel defined

by De [see Eq (6)].

1 How strong is the influence of wall roughness for laminar

flows—i.e., is it small enough to be neglected normally?

2 Is there a special situation when scales are changed from

macro to micro size—i.e., are there scaling effects with

re-spect to the influence of wall roughness for laminar micro

flows?

Before we answer these questions with the help of entropy

production considerations, we want to address two fundamental

issues concerning the definitions of wall location and roughness

parameter for pipes and channels with rough walls in the next

two sections

A MISLEADING QUESTION: WHERE IS THE WALL?

In view of the surface profile of a rough wall the question

arises: Where actually is the wall? The answer to this question,

however, is strikingly simple: The wall is where it is, and it is

a rough wall! This question is better phrased as: Where is the

equivalent smooth wall with respect to the real rough wall? This

criterion, from which in turn it can be decided which smooth

wall representation is geometrically equivalent to the real rough

wall

Three choices (with the first and the third of them often used)

are:

De

assump-tions about the physics of the flow around the roughness

elements (e.g., [10–12])

De

to the corresponding volume of the equivalent smooth

hIII(e.g.,[9])

measured without access to the surface, i.e., without opening

volume of the real channel, respectively Then the real volume

The real volume V can be determined by measuring how much

fluid it takes to fill the rough channel, as Nikuradse [9] didalready

smooth channel Its shape has to be determined under the

con-straint that its volume is that of the real channel, i.e., V There

are two options to do this Either one presets the cross-sectionalform (circular, triangular, ) or one defines an equivalencecriterion (least standard deviation, ) with some smoothnesscondition to be met Setting a circular cross section, for example,

C = πDe

the friction law for rough pipes can be cast into the well-known

dx

2Dh

u2 m

dx

u2 m

u2 m

as a roughness number, defined with a roughness parameter k,

discussed in the next section

For fully developed and horizontal flows in pipes or

chan-nels, the downstream increase of the specific dissipation dϕ/dx exactly corresponds to the pressure drop dp/dx (which then

often is called pressure loss) and via a force balance also to the

AN OPEN QUESTION: HOW DOES ONE DEFINE THE ROUGHNESS PARAMETER?

next question is about the appropriate definition of the roughness parameter k that goes into the roughness number K according

660 H HERWIG ET AL.

to Eq (9) It is a matter of concept how the overall effect of wall

roughness is accounted for Three such concepts, each with its

own definition of a roughness parameter k, are:

param-eter kI The influence of this kind of wall roughness is

how-ever, cannot be transferred to other kinds of roughness

a unique representation of different kinds of roughness,

rough-ness with a roughrough-ness parameter kIII Its influence is

roughness are individually referred to this case The

equiv-alent standard roughness is determined case by case and

stored in a table of correspondence.

concept [9] used for turbulent flows Its shortcoming is the need

for a table of correspondence, which provides a very rough

estimate only When the influence of rough walls should also be

is an option

since the problem presumably is a multi-parameter problem

Nevertheless, there are many attempts in this direction, such as

[13]

straightforward approach But it also is the least attractive one,

since it just puts on record what is measured without any

gen-erality in applying these results

determine the influence of wall roughness on the total head loss

This can be done experimentally (as in the past) or by analyzing

the dissipation process with an analytical/numerical approach,

which we want to present here

THE DISSIPATION MODEL APPROACH

Losses with internal flows are often named pressure losses.

However, they should more accurately be called losses of total

pressure (or total head) since they occur when the total pressure,

i.e., the mechanical energy in a flow, is reduced In such a

process mechanical energy is converted into internal energy

(conserving the total energy in accordance with the first law of

thermodynamics) From a thermodynamic point of view this is

a dissipation process (dissipation of mechanical energy), so that

a nonzero friction factor f is due to finite dissipation rates in

the flow

Turning this argument around, an alternative approach is

straightforward: One can determine the local dissipation rates

in the flow, integrate them over the flow domain, and thus find

the corresponding friction factor If this is done within the cise geometry, i.e., including the details of the rough wall,

representation of the actual wall roughness

Dissipation of mechanical energy from a thermodynamicpoint of view is directly linked to the production of entropy

in a flow field Therefore, a second-law (of thermodynamics)analysis can give valuable information about losses in flows(see [14–16]) The entropy production occurs locally in thepresence of velocity gradients Mathematically it is represented

by one term in the balance equation for entropy For Newtonian

fluids it reads in Cartesian coordinates with T as thermodynamic temperature (see [2, 1]) assuming constant density
:

 

∂u

∂x

2+



∂v

∂y

2+

and can be evaluated once the flow field (u, v, w) is known in

detail The method is part of a post-processing step, imposing noadditional costs to the calculation itself Integration with respect

to the channel volume V gives the overall entropy production

rate,

˙

SD=

V

˙

S

specific energy dissipation rate between two cross sections 1and 2 in terms of

L12

2Dh

u2 m

(13)

Only for the special case of a fully developed flow in Eq.(13) (i.e., no changes in streamwise velocity profiles) flowing

imme-diately linked to the pressure drop, i.e.,

The general definition in Eq (13) of a friction factor f or

thermody-namic point of view: losses of exergy or available work) is stillapplicable when the flow is transient, not fully developed, or un-

dergoes changes in potential energy For example, f according

H HERWIG ET AL 661

to Eq (13) can be introduced for a radial and thus accelerated

channel flow, but for this case, neither can be linked to dp/dx

pressure (and thus for losses of mechanical energy)

However, even for the case of fully developed horizontal

flows, our dissipation model allows a look into the “black box of

f12= −(p2− p1)2Dh/
L12u2m.” The detailed entropy

produc-tion field yields informaproduc-tion about where and how losses occur

This information is the background for a physical interpretation

as well as for systematic modifications of wall roughness, for

example in heat transfer problems

To illustrate the dissipation model approach, we consider

the friction factor according to Eq (13) for the fully developed

laminar flow in a horizontal plane channel with smooth walls

Between the two walls of distance 2H the velocity profile is [4]

when y starts from the centerline Substituting this velocity in

Eq (10) and evaluating ˙SD, ϕ12and f12,f according to Eq (11),

(12), and (13) respectively, results in

1 0

complicated geometries, integration has to be performed

nu-merically For rough walls, this includes all fluid filled cavities

between the roughness elements

APPLICATION OF THE DISSIPATION MODEL

In this study we apply the dissipation model in a

two-dimensional and an axisymmetric version, representing a

chan-nel and a pipe flow respectively As far as wall roughness is

concerned, roughness elements then are grooves in the wall,

perpendicular to the streamwise direction

For our calculations, we use three types of regular roughness

elements shown in Figure 2: triangular (T-type), quadratic

(Q-type), and sinusoidal (S-type) roughness with the characteristic

length scale h.

Figure 2 Three types of regular roughness elements esw, Equivalent smooth

wall for the definition of Dh = De

hIII Note: The position of esw indicated here

is that for plane channels.

Figure 3 Details of the numerical solution: (a) solution domain with periodic boundary conditions; (b) numerical grid (three-knot, triangular elements) for the three types of roughness and for the smooth wall.

equal steps as

KI = kI

Since the flows under consideration are quasi-fully

pe-riodically repeated downstream), we can set periodic boundaryconditions on a section of the whole flow field shown in Fig-ure 3(a) for the S-type rough wall The numerical grids shown

in Figure 3(b) consist of two-dimensional, three-knot triangularelements locally refined toward the wall In order to guaranteegrid-independent solutions, calculations were performed with

at least two different grid refinements and accepted only whendeviations in the solutions on two different grids were less than

0.1%.

specific dissipation rate in the section of the flow field that is

662 H HERWIG ET AL.

Figure 4 Poiseuille number for T-type wall roughness.

covered by the numerical grid From Eqs (10)–(12) we get

˙

mϕ12= T ˙SD

12 = µ

2

∂x

2+

which must be determined numerically in the solution domain

for the plane channel, and the corresponding form for the

ax-isymmetric pipe flow

Figure 4 shows results in terms of Po(ReDh, KI) gained by the

CFD code FLUENT 6.3 for the T-type roughness in channels

features are obvious:

15% for the pipe flow

the convective terms (inertia forces) in the Navier–Stokes

close to the rough wall)

Figure 5 Poiseuille number as a function of KI(wall roughness) for two Reynolds numbers ReDh.

The overall effect is shown in Figure 5 for the channel and the

curves are interpolations with respect to the calculated values.Obviously the Q-type roughness elements have the strongest

followed by the S- and T-type elements The influence of Re on

Po, however, is lowest for the Q-type roughness

The distribution of the entropy production is shown in ures 6 and 7 for all three types of wall roughness at two different

of all three geometries is that almost no entropy productionoccurs in the cavities between the elements, but it is rather con-centrated in a small band along the heads of the single roughness

pro-duction show a pattern of symmetry, this symmetry is lost forhigher Reynolds numbers due to the influence of the convection.The decreasing roughness effect (in the order Q-, S-, T-type)obviously corresponds to the decreasing percentage of a nearlyhorizontal wall in the small band of high entropy production

almost zero

The dissipation model, applied to regular roughness so far,can easily be used for arbitrary and irregular roughness distri-butions, once the geometry is known in detail As an example,Figure 8 shows an irregular roughness composed of varioussingle elements with rectangular, triangular, and round cross

H HERWIG ET AL 663

Figure 6 Distribution of the specific entropy production rate ˙S

D close to the rough wall for ReDh = 145 (dark: weak, light: strong): (a) T-type wall

roughness; (b) Q-type wall roughness; (c) S-type wall roughness.

Figure 7 Distribution of the specific entropy production rate ˙S

D close to the rough wall for ReDh = 2300 (dark: weak, light: strong): (a) T-type wall roughness; (b) Q-type wall roughness; (c) S-type wall roughness.

664 H HERWIG ET AL.

Figure 8 Distribution of the specific entropy production rate ˙S

D close to an irregular rough wall (dark: weak, light: strong): (a) flow field; (b) close-up.

sections of different size Again, as in Figures 6 and 7 entropy

production is concentrated close to the tips of the roughness

elements with almost no production in the cavities

As already shown the known entropy production rate

imme-diately can be “translated” into a friction factor f or a Poiseuille

number Po, respectively, which for the geometry of Figure 8 is

done in [17]

CONCLUSIONS

From our study we can draw the following conclusions, with

respect to the influence of surface roughness on friction in

lam-inar micro pipe or channel flows:

1 There is an appreciable increase in friction due to surface

roughness (in micro as well as in macro flows)—cf Figure

5

2 There is no special micro effect with respect to surface

rough-ness as long as the analysis is based on the Navier–Stokes

equations with standard no-slip boundary conditions Such

an effect could surface as a scaling effect with respect tothe Reynolds number This would be the case if there was

parameter range for micro flows (in contrast to macro flows

Our results show, however, that there is no special trend when

ele-4 For pipes and channels an equivalent smooth wall should

be defined based on an equivalence criterion The preferred

choice is that the real and the equivalent smooth channel havethe same fluid filled volume

5 In order to allow for general results, an equivalent roughnessparameter may be introduced, just like Nikuradse [9] did forturbulent flows with his sand roughness concept The table

of correspondence (between the real and the sand ness) that is part of Nikuradse’s concept, however, cannot beextended to laminar flows as shown in [7]

rough-6 With the dissipation model approach a table of dence can be exactly determined by numerical calculations

correspon-It can link a specific roughness to an equivalent roughness in

a general manner (for turbulent flows shown in [7])

7 Friction factors are better defined with dϕ/dx instead of

and totally due to dϕ/dx only in special cases (fully

devel-oped, horizontal flow)

8 In the dissipation model, head loss is determined by an tegration over the whole flow field This, however, is less

in-error-prone than the determination of p from two pressures

8τw/
u2m

NOMENCLATURE Arabic Symbols

˙

˙

[1] Herwig, H., and Kautz, C., Technische Thermodynamik, Pearson

Studium, M¨unchen, Germany, 2007

[2] Bejan, A., Entropy Generation Minimization, CRS Press, Boca

Raton, FL, 1996

[3] Herwig, H., and Kock, F., Direct and Indirect Methods of

Cal-culating Entropy Generation Rates in Turbulent Convective Heat

Transfer Problems, Heat and Mass Transfer, vol 43, pp 207–215,

2007

[4] Herwig, H., Str¨omungsmechanik, 2nd ed., Springer-Verlag,

Berlin, Heidelberg, 2006

[5] Munson, B., Young, D., and Okiishi, T., Fundamentals of Fluid

Mechanics, 5th ed., John Wiley & Sons, New York, 2005.

[6] White, F, Viscous Fluid Flow, 3rd ed., McGraw-Hill, New York,

2005

[7] Herwig, H., Gloss, D., and Wenterodt, T., A New Approach to

Un-derstand and Model the Influence of Wall Roughness on Friction

Factors for Pipe and Channel Flows, Journal of Fluid Mechanics,

vol 613, p 35–53, 2008

[8] Moody, L., Friction Factors for Pipe Flow, Trans ASME, vol 66,

pp 671–684, 1944

[9] Nikuradse, J., Str¨omungsgesetze in rauhen Rohren In

VDI-Forsch.-Heft, vol 361, pp 1–22 (VDI-Verlag, D¨usseldorf), 1933.

[10] Croce, G., and D’Agaro, P., Numerical Simulation of

Rough-ness Effect on Microchannel Heat Transfers and Pressure Drop

in Laminar Flow, Journal of Physics D: Applied Physics, vol 38,

pp 1518–1530, 2005

[11] Hu, Y., Werner, C., and Li, D., Influence of the Three-Dimensional

Heterogeneous Roughness on Electrokinetic Transport in

Mi-crochannels, Journal of Colloid and Interface Science, vol 280,

pp 527–536, 2004

[12] Kleinstreuer, C., and Koo, J., Computational Analysis of Wall

Roughness Effects for Liquid Flow in Micro-Conduits, Journal

of Fluids Engineering, vol 126, pp 1–9, 2004.

[13] Kandlikar, S., Schmitt, D., Carrano, A., and Taylor, J., acterization of Surface Roughness Effects on Pressure Drop in

Char-Single-Phase Flow in Minichannels, Physics of Fluids, vol 17,

pp 100606-100606–11 (2005)

[14] Hesselgreaves, J., Rationalisation of Second Law Analysis of Heat

Exchangers, International Journal of Heat and Mass Transfer,

vol 32, pp 4189–4204, 2000

[15] Rosen, M., Second-Law Analysis: Approaches and Implications,

International Journal of Energy Research, vol 23, pp 415–429,

[17] Gloss, D., K¨ocke, I., and Herwig, H., Micro Channel

Rough-ness Effects: A close-Up View, In Proceedings of the Fifth ternational Conference on Nanochannels, Microchannels and Minichannels, Darmstadt, Germany, 2007.

In-Heinz Herwig received the Ph.D degree from

Ruhr-University, Bochum, Germany, in 1981 From 1987 to

1992, he was a professor of theoretical fluid ics with the University of Bochum, Bochum There- after, he formed a private consulting bureau, Flow and Heat, and was its chief executive officer from

mechan-1992 to 1994 From 1994 to 1999, he was a sor of theoretical thermodynamics with the Chemnitz University of Technology, Chemnitz, Germany He is currently the head of the Institute of Thermo-Fluid Dynamics at the Hamburg University of Technology, Hamburg, Germany.

profes-Daniel Gloss received the Dipl.-Ing degree in

me-chanical engineering at the Hamburg University of Technology in Hamburg, Germany, in 2005 His main research interest is in microchannel flow, fluid me- chanics, and heat transfer He is currently working

as research assistant at the Institute for Thermo-Fluid Dynamics at the Hamburg University of Technology.

Tammo Wenterodt received the Dipl.-Ing degree in

naval architecture at the Hamburg University of nology in 2007 His research is focused on numeri- cal methods and the improvement of heat transfer Currently he is working as research assistant at the Institute for Thermo-Fluid Dynamics at the Hamburg University of Technology.