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Experimental Thermal and Fluid Science 30 (2006) 725–731

Using viscous heating to determine the friction factor inmicrochannels – An experimental validation

G.P. Celata a,*, G.L. Morini b, V. Marconi a, S.J. McPhail a, G. Zummo a

a ENEA, Institute for Thermal Fluid Dynamics, Via Anguillarese 301, 00060 S.M. Galeria, Rome, Italyb DIENCA, Universita degli Studi di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

Abstract

Many experimental works on forced convection through microchannels have evidenced deviations from conventional theory in fluid-dynamic and diabatic behaviour. Whereas often these anomalies were ascribed to ‘‘new’’ micro-effects, it can usually be shown that theexplanations for different behaviour at microscale are hidden in the conventional theory. There is just the issue of scaling effects thatcause certain phenomena that are negligible at macroscale to become influential when channel geometry is reduced below a certain limit.Such an effect also pertains to viscous dissipation, which becomes important especially in liquid microchannel flows. This paper focuseson the role of viscous heating in such flows, explaining its occurrence in the Navier–Stokes equations and verifying its presence in exper-imental validation. A criterion will be presented to draw the limit of significance for viscous dissipation effects in microchannel flows. Theconnection between viscous heating and the friction factor will be demonstrated, which will then provide the basis for comparison in anexperimental context. The friction factor obtained through pressure measurements and through evaluation of the viscous heating areconfronted. The results show an exceedingly good description of fluid-dynamic behaviour by the viscous heating method for diametersbelow 100 lm, proving the validity of the principle of scaling effects. This implies that at extremely small diameters temperaturemeasurements suffice to describe head loss, and that in diabatic experiments the viscous heating should be accounted for.� 2006 Elsevier Inc. All rights reserved.

1. Introduction

The miniaturization of many appliances in biomedic,chemical and computer technology has brought with itincreased demands for space-efficient high-performanceheat dissipation and catalytic devices. Though muchresearch on microscale level has already been done inrecent years in the relevant fields, especially as regardshydrodynamic and heat transfer characterization, there isstill much diversion of results to be discerned in the variousexperimental and numerical reports.

The main difficulty in this new dimension of appliedphysics is the reliability of measurements, as conventionalmeasuring apparatus is simply too big or too coarse toimplement in the tested system. Conditions are fairly

0894-1777/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.expthermflusci.2006.03.002

* Corresponding author. Tel.: +39 6 30483905; fax: +39 6 30483926.E-mail address: celata@casaccia.enea.it (G.P. Celata).

extreme at microscale, especially when using liquid work-ing fluids. The large viscous deformation of the flow innarrow channels is a phenomenon that does not appear sig-nificantly in conventionally sized ducts, and is the cause ofthe large pressure drops involved in driving the liquid. Inaddition to this, the reduced geometry of the system is espe-cially difficult to penetrate with pressure measuring deviceswithout altering flow conditions, and especially so in thecase of circular microtubes. As progress in instrumentationis being made, however, the tendency is for publishedresults to be more orderly, and to converge towards theclassical theories more than was originally expected.

In fact, from careful study of the conventional theory itis possible to identify so-called scaling effects, that deter-mine the difference in importance of certain effects formicro-systems compared to macro-systems, though derivedfrom a common theoretical basis. This idea was coined byHerwig and Hausner [1] in studying the forced convectionof liquids in laminar microscale flow. Guo and Li [2]

Nomenclature

A area, m2

cp fluid specific heat, J/kg KD tube diameter, mDh hydraulic duct diameter (= 4C/X), mey absolute error of parameter y

Ec Eckert number, W2/2cpDhref

f Darcy friction factor, –k fluid thermal conductivity, W/m KL channel length, mL* dimensionless channel length, L/Dh

_m mass flow, kg/sp pressure of fluid in duct, Pap* dimensionless pressure defined in Eq. (3)Pe Peclet number, RePr

Pr Prandtl number, m/aq linear heat flux, W/mT dimensionless fluid temperatureu axial fluid velocity, m/sV dimensionless fluid velocityW fluid average velocity, m/sx, y, z dimensionless Cartesian co-ordinates

Greek symbols

a thermal diffusivity (= k/qcp), m2/sC wetted perimeter, m

C* dimensionless perimeter, C/Dh

e body emissivityH body temperature, Kh fluid temperature, Kl dynamic viscosity, Pa sm kinematic viscosity, m2/sq fluid density, kg/m3

r Stefan–Boltzmann constant, 5.669 · 10�8 W/m2 K4

U* dimensionless viscous-energy-dissipation func-tion, Eq. (6)

X cross-sectional area, m2

X* dimensionless cross-sectional area, W =D2h

n, g, f Cartesian co-ordinates, m

Subscripts

app apparentb bulke externalexp experimentalf pertains to the friction factori internalw wall

726 G.P. Celata et al. / Experimental Thermal and Fluid Science 30 (2006) 725–731

remarked that, since different forces have different lengthdependencies, the surface forces (like surface tension, vis-cous force or electrostatic force) become more importantand even dominant as the scale is reduced.

The effect of viscous force has been investigated byseveral authors. Tso and Mahulikar [3,4] carried out a the-oretical and experimental analysis of circular microchan-nels considering the effect of viscous dissipation by meansof the Brinkman number. The effect of the Brinkman num-ber is related to the reduction of the dynamic viscositybetween inlet and outlet of a microchannel due to theincrease in flow bulk temperature. The authors concludedthat axial variations of the Brinkman number affects theconvective heat transfer in microchannels, but found thatexperimental data on the friction factor for laminar flowcan be adequately correlated to this number.

Judy et al. [5] confirmed experimentally that viscous dis-sipation can be invoked to explain deviations from Stokesflow behaviour repeated in many other works on micro-channels; they evidenced that viscous heating has a signif-icant effect of increasing the temperature of the flowingfluid along the microchannel axis.

Tunc and Bayazitoglu [6] studied the effects of viscousdissipation for rarefied gas flow in slip-flow regime in circu-lar and rectangular microchannels. They concluded thatthe Brinkman number plays an important role in the heattransfer through microdevices also for gas flows.

Xu et al. [7] investigated viscous dissipation effects forliquid flows, and stated that deviations could be expectedfrom predictions using conventional approximation thatneglects viscous dissipation. The high velocity gradientsexisting in channels with small hydraulic diameters makethis a non-negligible phenomenon, and they proposed a cri-terion to draw the limit of significance of viscous dissipa-tion effects in microchannel flows.

More recently, Koo and Kleinstreuer [8] did a numericalinvestigation of viscous dissipation effects on the tempera-ture field and friction factor in circular and rectangularmicrochannels. They demonstrated that viscous dissipationis strongly dependent on hydraulic diameter and channelaspect ratio and concluded that ignoring viscous dissipa-tion could affect accurate flow simulation in microchannels.

In this paper, the conventional theory (Navier–Stokesequations) will be used to analyse those effects that arenot important in macro-scale, but that become so whenthe dimensions of the system decrease. In other words, aconsideration will be effected of the so-called scaling effectsregarding the friction factor, and in particular focusing onthe role of viscous heating in fluids flowing through micro-ducts. A criterion will be suggested to assess the signifi-cance of the effects of viscous dissipation in microchannelflows and validated through experimental verification,comparing the result with the criterion proposed by Xuet al. in [7].

G.P. Celata et al. / Experimental Thermal and Fluid Science 30 (2006) 725–731 727

2. Viscous heating in microchannels

The detailed derivation of the following theory is givenin [9].

Let us consider a microchannel with axially uniformcross-section, with area X, perimeter C and channel lengthequal to L. To model the fluid flow and heat transferprocesses, conventional Navier–Stokes and energy equa-tions will be used, based on the following simplifyingassumptions:

1. the fluid is Newtonian, incompressible and with a lami-nar fully developed profile of velocity u(n,g) and a uni-form inlet temperature hn;

2. the transport processes are considered to be steady-stateand bi-dimensional (fully developed velocity and tem-perature profiles);

3. thermal radiation is neglected;4. all channel walls are rigid and non-porous;5. there is no fluid slip (zero velocity at the wall);6. axial thermal conduction (Pe� 1), natural convection

(Gr/Re2� 1), and interior heat sources are neglected;7. fluid thermophysical properties are assumed as constant.

Under the mentioned hypotheses the conservation equa-tions of momentum and energy can be written as follows:

r2u ¼ � 1

ldpdf;

r2hþ lk½ru � ru� ¼ uðn; gÞ

aohof:

8>><>>: ð1Þ

Here, the right-hand-side of the energy equation representsconvective heat transport and the left-hand-side is com-posed of a conduction term and the heat generation due toviscous dissipation. Now, if we assume a fixed linear heatflux qw to be present at the wall (H1 boundary condition),and utilise the Stokes Theorem, it can be shown that whenintegrated over the cross-sectional area, the conduction termis described by the fixed linear heat flux qw only. Remember-ing furthermore that we are only considering stable, fullydeveloped flow, the second of the equations in (1) becomes:

ohof¼ ohb

of¼

qw þ lR

X½ru � ru�dX

qcpW X. ð2Þ

To rewrite the momentum and energy balance equationsin dimensionless form, we make use of the followingdimensionless parameters:

x ¼ nDh

; y ¼ gDh

; z ¼ fL; C� ¼ C

Dh

; X� ¼ X

D2h

;

r� ¼ Dhr; V ¼ uW; T ¼ ðh� hinÞ

Dhref

; L� ¼ LDh

;

p� ¼ � D2h

lWdpdf; Ec ¼ W 2

2cpDhref

;

Pe ¼ WDh

a¼ RePr; q�w ¼

qw

kDhref

.

ð3Þ

With the bulk temperature introduced, defined as

hb ¼1

X

ZX

uðn; gÞhðn; gÞdX; ð4Þ

problem (1) is now described in dimensionless form by

r�2V ¼ �p�;oT b

oz¼ q�wL�

PeX�þ 2

EcL�

ReX�U�;

8<: ð5Þ

where we have defined a dimensionless viscous-energy-dis-sipation function U*:

U� ¼Z

X�½r�V � r�V �dX. ð6Þ

By considering the thermophysical properties as con-stant, the momentum and energy balances are uncoupled.The momentum conservation equation can then be solvedby using the no-slip boundary condition at the wall (whichis a justified assumption for liquid flow in channels largerthan 1 lm hydraulic diameter). From the velocity distribu-tion V(x,y) for laminar flow, it is possible to derive thevalue assumed by the main flow parameters as a functionof the microchannel aspect ratio. In particular, it is possibleto determine the value of the Poiseuille number, i.e., theproduct of the Darcy friction factor for fully developedlaminar flow, f, and the Reynolds number, Re, using thefollowing relation:

fRe ¼ � 2

X�

ZX�r� � ðr�V ÞdX� ¼ 2p�. ð7Þ

It can be demonstrated that the Poiseuille number andthe viscous-energy-dissipation function U* are related.Using assumptions 4 and 5, the momentum balance inEq. (5) and the properties of the Laplacian operator, itcan be shown that:

U� ¼Z

X�½r�V � r�V �dX ¼ �p�X� ¼ 2fReX�. ð8Þ

Eq. (8) states that the heat generation due to viscous dis-sipation in the 1D average model, can be traced back tofriction at the channel walls. Then, combining Eqs. (8)and (5), the axial variation of the bulk temperature isrelated to the Poiseuille number:

oT b

oz¼ q�wL�

PeX�þ Ec

Re½fReL��. ð9Þ

In heated or cooled macro-size channels, the first term isalways predominant over the second term, especially inlaminar regime. As an example, for a tube with inner diam-eter (ID) equal to 1 cm, and 1 m long, through which waterflows in laminar regime (Re = 1000) the temperature risedue to viscous heating is of the order of 10�5 K; for anID equal to 100 lm the temperature rise in same condi-tions becomes of the order of 10 K. This is evidence thatviscous heating can be considered a scaling effect formicrochannels.

Fig. 1. Schematic of the experimental test-loop.

728 G.P. Celata et al. / Experimental Thermal and Fluid Science 30 (2006) 725–731

To analyze the contribution of viscous dissipation aloneon the temperature rise, an adiabatic microchannel will beconsidered; so that by integration of Eq. (9) over the lengthof the channel, where entrance effects are negligible, thevalue of the dimensionless temperature at the outlet canbe calculated as

T bðz ¼ 1Þ ¼ Dhb

Dhref

¼ EcRe½fReL��. ð10Þ

Hence, the outlet temperature increases if the mean veloc-ity of the fluid increases and if the microchannel hydraulicdiameter decreases; the role of the microchannel geometryis taken into account by means of the term in bracketsof Eq. (10). The temperature gradient along the micro-channel due to viscous heating can now be expressed asfollows:

ohb

of¼ Ec

Re½fRe�

� �Dhref

Dh

. ð11Þ

In Eq. (11) Dhref is a reference temperature rise; thisvalue can be fixed by considering the temperature sensitiv-ity of fluid viscosity; for instance, it could be defined asthe temperature rise for which the dynamic viscositydecreases, say, 2–3%. Thus, in order to neglect the effectsof viscous dissipation in adiabatic flow through micro-channels, the temperature rise so induced, between inletand outlet, shall be less than the reference temperaturerise. This enables us to formulate a criterion to establishwhen viscous dissipation effects cannot be ignored. FromEq. (10) it follows that this is the case if the following con-dition is satisfied:

EcRe½fReL��P 1. ð12Þ

It is evident that for different fluids the viscous dissipa-tion effect will be different. Considering Eq. (11), the tem-perature gradients of two different fluids, c and d, arelinked to their thermophysical properties (for a fixed valueof Reynolds number, hydraulic diameter and cross-sectiongeometry) as follows:

ohb

of

� �c

ohb

of

� �d

¼ mc

md

� �2 cpd

cpc

� �. ð13Þ

Finally, it must be noted that Eq. (10) above can beemployed to determine the value assumed by the apparentfriction factor:

fapp ¼Dhb

Dhref

1

EcL�

� �. ð14Þ

Eq. (14) enables us to determine the friction factor with-out measuring the pressure drop along the microchannel,but by means of temperature and flow rate measurementalone. This can be of considerable advantage at microscale,where pressure tappings and transducers are usually coarsecompared to channel dimensions and can be disrupting tothe flow. Temperature measurement on the other hand, can

be effectuated with comparatively non-invasive means. Forlarge-scale ducts, pressure measurement is far more conve-nient, as also the temperature effect of viscous dissipationwould be negligible in macrochannels.

3. Experimental procedure

The reasoning set out in the above section has been putto the test in a train of experiments, where the friction fac-tor calculated through ‘‘classic’’ pressure measurementswas confronted with the viscous heating analogy definedin Eq. (14). The test rig is schematized in Fig. 1.

Experiments are carried out using demineralised water,which is degassed by passing through a very small, but con-tinuous quantity of Helium. Being insoluble in water, anatmosphere of only Helium is created above the liquidlevel, so that all dissolved gases are driven out by theirrespective partial pressures in the water. A piston pumpwith damper (Gilson 305 with 25.SC Head, 806 Manomet-ric Module) is used to drive the flow, which passes througha 10 lm filter and is controlled with a microvalve regulator,before entering the test section. The capillaries tested areFused Silica tubes, mounted with high-pressure, Near-ZeroDead Volume gas chromatography fittings (UpchurchScientific).

The microtube under investigation is mounted inside astainless steel capsule (see Fig. 2) which is sucked vacuumby a turbo-molecular vacuum pump (Alcatel ATS-100) tocreate an environment free of natural convection. To thiseffect, the level of vacuum must be less than 10�3 mbar.The level obtained in our set-up is 2 · 10�4 mbar (EdwardsPenning Gauge Model 6), so that we can consider the heatloss to the surroundings through convection inexistent.

The heat loss due to radiation is evaluated by consider-ing the formula for two concentric cylindrical surfaces [10],simplified for the limiting case where the surface of theinternal body (OD of the test section is 0.4 mm) is muchsmaller than the external, concave surface (ID of the vac-uum chamber is 100 mm):

Fig. 2. Close-up of vacuum chamber and flange with mounted test section(circled), thermocouples and pressure ducts.

G.P. Celata et al. / Experimental Thermal and Fluid Science 30 (2006) 725–731 729

qradL ¼ rAieiðH4i �H4

eÞ. ð15Þ

The entity of this loss is of the order of 1% of the viscousheating in the case of the highest measured temperature dif-ference between the two bodies (equating to 1 mW for the50 lm capillary at Re = 700), and thus considerednegligible.

A 250 lm K-type thermocouple measures the fluid tem-perature on entrance into the test section, and pressuretransducers on either side of the microtube investigatedallow the pressure drop over the channel to be established(Druck PTX100/IS, 0–35 bar; Transamerica 0–160 bar).

At the outlet of the channel, a 200 lm K-type thermo-couple is made to be inserted inside the (Near-Zero) deadvolume of the fitting so that the fluid exit temperature ismeasured as closely as possible (see Fig. 3). The mass flowrate is measured with a high precision scale.

The validation of the friction factor calculated withEq. (14) is effected by comparison with the pressure descrip-tion of f:

Fig. 3. Thermocouple positioning inside the channel fitting at the outletextremity.

f ¼ 2DpDL

Dqu2

; ð16Þ

which is the classical relation of Darcy–Weisbach for lam-inar fully developed incompressible flow. The main diffi-culty in determining the friction factor in this way, is thecorrect measurement of the pressure drop, Dp. First ofall, it is almost impossible to create pressure tappings insidea (circular) channel of microscopic size without signifi-cantly disrupting the flow. And in measuring the pressurebefore and after the duct, the concentrated inlet and outleteffects cannot be accounted for. The methodology for by-passing the latter problem was described by us elsewhere[11,12], but is based fundamentally on the superpositionof two lengths of the same capillary which at equal massflow rates must have equal concentrated pressure losses,so that the difference between the two pressure drops rep-resents frictional head loss only. Thanks to this method,an accurate evaluation of the friction factor according toEq. (16) can take place.

4. Uncertainty analysis

In order to evaluate the influence of experimentalparameters on the measurement of the friction factor, itis convenient to perform an uncertainty analysis on the var-iable of interest, f.

With the assumption of zero heat loss from the channel,we can rewrite Eq. (14) in terms of measurable quantitiesonly, yielding:

f ¼ p2

8

q2cpD5

_m2LDhb. ð17Þ

This means that the uncertainty on the friction factor mea-sured through viscous heating is, according to the classicalmethod described by Holman [13]:

U f ¼ef

f

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eq

q

� �2

þecp

cp

� �2

þ 5eD

D

� �2

þ eDhb

Dhb

� �2

þ 2e _m

_m

� �2

þ eL

L

� �2

s.

ð18Þ

With the certified errors on the various parameters in Eq.(17) and an error on the measurement of the channel diam-eter of 1 lm, this uncertainty becomes dependent mainlyon the diameter of the tube under consideration, D (herein lm), and the viscous heating, Dhb:

Uf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:027þ 5

D

� �2

þ eDhb

Dhb

� �s. ð19Þ

Therefore, as expected, with a constant error of ±0.1 Kon the sum of thermocouple read-outs, the larger the vis-cous heating obtained, the more precise the measurementwill be. The fundamental cause for erroneous conclusionsat microscale, however, is always the small diameter, withrelatively large uncertainty.

730 G.P. Celata et al. / Experimental Thermal and Fluid Science 30 (2006) 725–731

5. Results and discussion

Three diameters of capillaries have been investigated:100, 70 and 50 lm. The results are presented in the formof a diagram confronting the measured and predicted vis-cous heating (Fig. 4), and of a friction factor versus Rey-nolds number plot, where the different elaborations forthe apparent friction factor (Eqs. (14) and (16)) are com-pared (Fig. 5).

From Fig. 4 the validity of the principle of viscous heat-ing is evident. Trends of the measured values of the temper-ature rise per unit of length are in good agreement with thebehaviour predicted by Eq. (11). (The correlation suggestedby Xu et al. [7], is added for comparison (dashed lines), butit can be seen that there is a gross underestimation of theviscous heating on their part.) An important conclusionto draw, is that viscous dissipation – as was predicted –starts to become important only below, say, 100 lm ID,

1

10

100

10 100 1000 104

50 μm_exp

VH_MoriniVH_Xu

70 μm_exp100 μm_exp

Δ b /L

Re

θ

Fig. 4. Gradient of the viscous heating (Dhb/L) versus Reynolds number;comparison of measurements (exp) and two correlations [9,7].

0.1

1

100 1000

64/Ref_ΔP 50 μmf_VH 50 μmf_ΔP 70 μmf_VH 70 μmf_ΔP 100 μmf_VH 100 μmf

Re

- error

+ error

Fig. 5. Friction factor versus Reynolds number for three diameters;pressure approximation (Eq. (16)) and viscous-heating approximation(Eq. (14)).

which is reflected by the accuracy with which experimentaldata are able to reproduce the scaling effect: as the tubediameter decreases, the theoretical curve correlates themeasurements more closely. At larger diameter the flowexperiences less viscous deformation, which, combinedwith the high value of the specific heat for water, which‘‘absorbs’’ heat production better, the temperature risebecomes imperceptible. What does remain very clear, alsoat 100 lm diameter, is the point of transition from laminarto turbulent flow. At Re � 2000 we can see clearly how themeasured temperature gradient deviates sharply from theslope predicted for viscous heat production in steady statelaminar flow.

The elaboration of the data for determining the appar-ent friction factor according to the two methodologiesdescribed in this article (using pressure drop, Eq. (16), orviscous heating, Eq. (14)), is shown in Fig. 5. The errorbands represent Eq. (18), and therefore include the relativeuncertainty on the measurement of the temperature riseand are also dependent on the microtube diameter. At afixed diameter, the former uncertainty increases at lowerReynolds numbers (where the temperature rise is of thesame order as the error on the thermocouple calibration)and thus is explained the large margin for error wheresmaller viscous deformation causes less conversion of fric-tional energy into heat.

It can be noted that the approximation of the frictionfactor based on the principle of viscous heating providesvery good agreement with the tried-and-tested pressuredrop approximation and with the reference line ofHagen–Poiseuille (the analytical solution to the momentumequation, f = 64/Re). In general, the data obtained fromthe elaboration of the viscous dissipation equation are onthe lower limit of the experimental error band, indicatingthat the measured temperature rise is slightly lower thanwhat is required for perfect agreement with the pressuredata. This is especially evident for the 100 lm diametermicrotube, where we can see how the viscous heating ofthe flow rapidly becomes negligible at lower Reynoldsnumbers, thereby magnifying the influence of the giveninaccuracy of the temperature measurement. At high flowrates fluid-dynamic behaviour is more precisely described,but still remaining slightly below the friction factorobtained from the Darcy equation (16). This means thatthere is a very slight, underestimating, systematic erroron the temperature measurement, which could be explainedby minimal contributions to heat loss by conduction, radi-ation and flow expansion or separation effects, as well as acertain level of thermal inertia of the test equipment.Nevertheless, the agreement is qualitatively excellent, andwith a more accurately chosen fluid (a refrigerant for exam-ple), the effect would be much easier to isolate and mapthan when using water, which in our range of experimenta-tion did never exceed 2.5 K of viscous heating. For thesmallest, 50 lm diameter channel the level of viscous dissi-pation is such as to allow a highly reliable quantification ofthe pressure drop, and thereby goes to prove that this effect

G.P. Celata et al. / Experimental Thermal and Fluid Science 30 (2006) 725–731 731

can be judged a valid scaling effect for microtubes below100 lm diameter.

6. Conclusions

The effect of viscous dissipation has been demonstratedto be a typical scaling effect for microchannel flows; thiseffect can become very important for liquid flows whenthe (hydraulic) diameter is less than 100 lm. Based on con-ventional theory a model to predict the viscous dissipationeffects in a microchannel with an axially unchanging cross-section was developed, that expresses the temperature risein an adiabatic microchannel as a function of the Eckert,Reynolds and Poiseuille numbers. A criterion has been pre-sented to discern the limit for which the effect of viscousdissipation can no longer be neglected.

In experimentally validating the model, use has beenmade of the model to describe the apparent friction factorfor laminar flow in microtubes, which was then comparedwith the friction factor evinced from pressure drop mea-surements and classical theory (Hagen–Poiseuille). Threediameters were tested: 50, 70 and 100 lm, all presumedto be within the range of applicability of the viscous dissi-pation analysis.

The measured heating of the fluid was very congruentwith what is predicted by the model, and the scaling effectis readily visible, as the smaller diameters show most accu-racy. Another model suggested by Xu et al. was added inthe comparison, but was found to significantly underesti-mate the level of viscous heating.

The values of the friction factor predicted by the modelfall within the margin for experimental error for all Rey-nolds numbers, and the advent of turbulence is clearly vis-ible. Though the use of a more suitable fluid (with low cp)would manifest the effect even more accurately, it can beconcluded that viscous dissipation is a valid method offluid-dynamic characterization of microtube flow. This isextremely convenient especially in microscale applicationswhere excessively intrusive pressure measurements are notpossible. Furthermore, viscous heating should thereforebe accounted for in heat transfer applications where thisscaling effect is present.

Acknowledgements

Gratitude is due to technicians Michele Sica and Mari-ano Morlacca for their continuous inventive contributions,and to Dr. Maurizio Diamanti for his invaluable aid in thequest for the turbomolecular pump. This work was madepossible in part by MIUR-URST and the EU, respectively,through Italian National Project COFIN 03 and FP6HMTMIC, RTN Contract HPRN-CT-2002-00204.

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