Experimental Analysis of Pressure Drop and Laminar to Turbulent Transition for Gas Flows in Smooth...

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Experimental Analysis of Pressure Drop and Laminar toTurbulent Transition for Gas Flows in SmoothMicrotubes

To cite this Article: Morini, Gian Luca, Lorenzini, Marco, Colin, Stéphane andGeoffroy, Sandrine , 'Experimental Analysis of Pressure Drop and Laminar toTurbulent Transition for Gas Flows in Smooth Microtubes', Heat TransferEngineering, 28:8, 670 - 679To link to this article: DOI: 10.1080/01457630701326308URL: http://dx.doi.org/10.1080/01457630701326308

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Heat Transfer Engineering, 28(8–9):670–679, 2007Copyright C©© Taylor and Francis Group, LLCISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457630701326308

Experimental Analysis of PressureDrop and Laminar to TurbulentTransition for Gas Flows inSmooth Microtubes

GIAN LUCA MORINI and MARCO LORENZINIDIENCA, Universita di Bologna, Bologna, Italy

STEPHANE COLIN and SANDRINE GEOFFROYLGMT, Institut National des Sciences Appliquees, Toulouse, France

The results of numerical and experimental works dealing with the behavior of gas flow through microchannels are by nomeans univocal, sometimes agreeing with the classical correlations and other times contradicting them. It is now agreed uponthat the effects due to both rarefaction and compressibility must be accounted for. In addition, the experimental works havedemonstrated that sometimes compressibility and rarefaction effects can be coupled in microchannels: because these twoactions contrast each other, the scatter of the friction factor data for gaseous flows is remarkably large. This paper is aimedat determining the friction factor for commercial short and long Peek microtubes with nominal internal diameters between300 and 100 µm and values of the length-to-diameter ratio, L/D, ranging between 167 and 5000. Nitrogen flows inside themicrotubes, with a maximum value of the supply pressure equal to 10 bar. Very low Knudsen numbers (Kn < 0.001) areconsidered in order to uncouple the rarefaction effects from the compressibility effects. The role of the minor losses related tothe inlet and outlet of the test section and of the gas compressibility on the friction factor are analyzed and discussed in orderto draw their limit of significance in microchannels. In addition, the effects of the gas compressibility and of the L/D ratioon the critical Reynolds number for which the laminar to turbulent transition takes place will be analyzed and discussed bycomparing the experimental results with the other data published in the literature.

INTRODUCTION

Gas flows through microchannels have been studied by dif-ferent authors in the last years, and a complete and critical reviewof the main results of these investigations can be found in [1,2].These reviews underlined that the reduction of the hydraulic di-ameter of the channels influences the frictional characteristics ofgas flows because rarefaction and compressibility effects gainin importance. Pfhaler et al. [3] conducted extensive researchon circular microchannels with a very low hydraulic diameter(1.6–3.4 µm) and on rectangular specimens 100 µm wide and0.5 to 50 µm deep, with liquids, gases, and Reynolds numbervarying between 50 and 300. The experimental friction factor

Address correspondence to Prof. Gian Luca Morini, DIENCA, Universitadi Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy. E-mail: [email protected]

was found to be substantially lower than what the classical the-ory would predict, and a dependence of the Poiseuille number(defined as the product of the Fanning friction factor times theReynolds number) on the Reynolds number was evidenced inthe channels with the smallest diameter. Pong et al. [4] measuredthe pressure along the 4.5 mm length of rectangular microchan-nels 5 to 40 µm wide and 1.2 µm deep for flows of heliumand nitrogen. This may have been the first experimental work inwhich local measurements of the pressure distribution along amicrochannel were made. They found that the pressure distribu-tion was non-linear due to compressibility effects but exhibitedvalues lower than those predicted by the conventional theorybecause the rarefaction effects were not negligible. Afterwards,Liu et al. [5], Lee et al. [6], Zohar et al. [7], and Jang and Were-ley [8] have experimentally confirmed these findings. Shih et al.[9] conducted experiments with helium and nitrogen for thesame geometry; they demonstrated that the rarefaction effects

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G. L. MORINI ET AL. 671

on the pressure drop can be well captured using a first-order slip-flow model that could predict the data up to an inlet pressure of0.25 MPa but underestimated them for higher values. Similarly,Arklic et al. [10] found that their experimental results for themass flow rate of helium and nitrogen in deep rectangular siliconmicrochannels (Dh = 2.6 µm) could be predicted by the con-ventional theory, provided slip boundary conditions at the wallswere used. Harley et al. [11] analyzed the flow of argon, helium,and nitrogen through 10 mm-long trapezoidal channels 100 µmin width and 0.5–20 µm in depth. The data were plotted as afunction of the reduced Poiseuille number C* = (Po)exp/(Po)th,which increased with increasing Reynolds numbers and fell be-low 1 for the smallest microchannel. This trend was attributedto slip at the walls, as confirmed by calculations with the slipboundary condition. Araki et al. [12] studied the flow of nitrogenand helium through three different trapezoidal microchannelswith hydraulic diameters between 3 and 10 µm. They attributedthe lower values of the friction factor to the rarefaction effect.The findings of Li et al. [13] evidenced again a non-linear distri-bution of the pressure obtained by cutting five microtubes withdiameters from 80 to 166.6 µm when the average Mach num-ber between the inlet and the outlet exceeded 0.3. In this case,the friction factor was found to be higher than that predictedby the conventional theory and compressibility was invoked toexplain these results. The paper by Yang et al. [14] aimed atinvestigating the flow of air in circular microtubes of diametersbetween 4001 and 173 µm and of length ranging from 15 to1000 mm. The agreement with the conventional theory was re-markable for the laminar regime, whereas in the turbulent case,the smaller the tube diameter was, the increasingly lower the val-ues of the friction factor were than those predicted by the Blasiusor the Filonenko correlations. The measurements by Lalondeet al. [15] of air flow in a 52.8 µm microtube showed a goodagreement with the predictions of the conventional theory, asdid those of Turner et al. [16], who investigated the laminar flowof nitrogen, helium and air for smooth and rough rectangularmicrochannels with hydraulic diameters ranging from 4 to 100µm. Hsieh et al. [17] investigated the behavior of nitrogen flowin a 24-mm long, 200-µm wide, and 50-µm deep microchannelfor Reynolds numbers between 2.6 and 89.4 and a value of theKnudsen number ranging from 0.001 to 0.02. They concludedthat the effects of compressibility were more important thanthose due to the rarefaction and found a friction factor lowerthan that predicted by the conventional theory. Celata et al. [18]studied the gaseous flow of helium through microtubes of diame-ter between 30 and 254 µm and high L/D ratio (between 300 and2400) in order to ascertain the effect of compressibility on thefriction factor for Reynolds numbers up to 500. They concludedthat the quantitative behavior is satisfactorily predicted—withinthe experimental uncertainty, which amounted to a maximumof ±20%—by the incompressible theory, and that the frictionallosses at the inlet and outlet only give a minor contribution tothe total pressure drop. Kohl et al. [19] studied the flow of waterand air in rectangular silicon microchannels with a hydraulicdiameter varying between 24.9 and 99.8 µm using internal pres-

sure measurements and could predict their data quite well in thecase of laminar flow employing the classical correlations. Tangand He [20] measured friction factors in fused silica micro-tubes and square microchannels with diameters ranging from50 to 201 µm. They concluded that the theoretical predictionfor conventional tubes were in good agreement with the mea-sured friction factor, but for smaller microtubes, the effects re-lated to compressibility, roughness, and rarefaction cannot beneglected.

These experimental results underline the importance of therarefaction and compressibility effects on the frictional char-acteristics of gas flows in microchannels. Some experimentalworks demonstrated that sometimes these effects can be coupledin microchannels, and these two actions contrast each other andincrease the scatter of the friction factor data for gaseous flows.

It is opinion of the authors that another fluid-dynamic aspect,the laminar to turbulent transition in microchannels, requiresfurther experimental analyses because the experienced devia-tions from the conventional theory seem to remain large. Recentreviews of the experimental results related to the laminar to tur-bulent transition are from Morini [1,21] and Hetsroni et al. [22].These studies indicate that the transition from the laminar tothe turbulent flow in micro-scale passages could take place atcritical Reynolds numbers ranging from 70 [23] up to 10,000for liquid flows [24]. This large spread of the critical Reynoldsvalues has stimulated the present investigation of the laminar toturbulent transition for gaseous flows in microtubes.

This paper aimed to determine the friction factor for smoothmicrotubes of Peek, with nominal internal diameters equal to300, 200, 175, 150, and 100 µm and different values of thelength-to-diameter ratio (167 < L/D < 5000), through whichnitrogen flows with a maximum value of the supply pressureequal to 10 bar. The range of the hydraulic diameters investi-gated and the values of the inlet pressure permit one to considerthe rarefaction effects on the friction factor (Kn < 0.001) asnegligible. In addition, as confirmed by means of a ScanningElectron Microscope (JEOL JSM 5200), the microtubes in Peekare characterized by a regular cross-section with very low val-ues of the absolute surface roughness (less than 0.05 µm); thisfact allows one to consider the tested microtubes as smooth andneglect the roughness effects on the friction factor.

The experimental results will be used to comment on the roleof compressibility effects on the frictional characteristics of gasflows in microchannels. The importance of a correct evaluationof the minor losses related to the inlet and outlet of the mi-crochannels on the determination of the friction factor and therole of the acceleration term on the total pressure drop will beshown.

The laminar to turbulent transition will be critically analyzedin order to highlight the effects of gas compressibility, quantifiedby the Mach number at the outlet, and of the L/D ratio on thecritical Reynolds number for which the transition takes place.

The experimental results will be compared with recent exper-imental data published in the literature and with the correlationsproposed to link the friction factor to the Mach number.

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672 G. L. MORINI ET AL.

Figure 1 Layout of the test rig.

EXPERIMENTAL APPARATUS

The sketch of the test rig is shown in Figure 1: nitrogen isstored in a high pressure flask (200 bar) (1) and is brought toapproximately 11 bars and ambient temperature before enteringa 7 µm particle filter (2, Hamlet) used to prevent possible im-purities from clogging the microchannels or the flow meters. Athree-way valve (3) directs the flow to the proper flow sensor,as the facility is equipped with two Bronkhorst EL-Flow E7000operating in the 0–5000 nml/min (4a) and in the 0–500 nml/min(4b) ranges, respectively. The flow sensors can also impose themass flow rate by means of a computer-steered valve: this allowsan indirect regulation of the pressure at the inlet of the micro-tube. The gas enters the test section (5), before which a pressuresensor is connected. The test section allows tubes of varyinglength (1 to 100 cm) and variable inner diameters to be tested,providing they have the same external diameter, namely 1/16′′.After exiting the microtube, the gas vents to the atmosphere. Thetotal pressure drop between the inlet and the outlet of the micro-tube is measured by means of a differential pressure transducer(6, Validyne DP15) with an interchangeable sensing element thatallows accurate measurements over the whole range of pressuresencountered. To measure the temperature at the entrance and exitof the microtube, two K-type, calibrated thermocouples are used(7): through these values and that of the pressure, the densities ofthe gas at the channel inlet and outlet can be calculated. First, themicrotube to be tested is mounted on the facility, and the circuitis checked for leakages; then, measurements are performed byimposing a certain inlet Reynolds number (which can be relatedto the mass flow rate, the parameter of which can be controlled)and recording for every measurement pressure drop, time, in-let and outlet temperatures, and flow rate by means of a dataacquisition system and a PC.

Twenty different Peek microtubes (Upchurch Inc.) weretested in the experiment with the following internal diameters:100, 150, 175, 200, and 300 µm. For each diameter, four dif-ferent lengths were considered: 50, 20, 10, and 5 cm. The tubeshaving a different length with a fixed diameter were obtained bycutting the same tube. The inner diameter of the frontal cross-section of each microtube has been verified by means of a SEMapparatus (JEOL JSM 5200).

Figure 2 Location of the pressure taps in the test rig.

Data Reduction and Uncertainty Analysis

Figure 2 shows how the pressure taps are located at the in-let and outlet of the microchannel. The measured pressure drop(�pm) indicates the pressure difference existing between thepressure before the inlet manifold (p1) and the atmospheric pres-sure (p2). The measured pressure drop (�pm) includes the net(�pn), inlet (�pin), and exit (�pex) pressure drop contributions:

�pm = p1 − p2 = �pin + �pn + �pex (1)

The minor losses at the inlet and at the outlet of the microtubecan be estimated using the following relations:

�pin = p1 − pin = Cin2

(mA

)2(

RTpin

)�pex = pex − p2 = Cex

2

(mA

)2(

RTpex

) (2)

where A is the cross-sectional area of the microtube, and Cin

and Cex are the minor losses coefficients equal to 1.5 and 0.88,respectively [25], for a curvature radius r (see Figure 2) equal to0. Equation (2) allows the determination of the pressure at theinlet (pin) and at the exit (pex) of the microtube.

The net pressure drop (�pn) can be calculated by combiningEqs. (1) and (2).

For channels that are extremely long compared to the hy-draulic diameter, there is sufficient area for heat transfer to makethe flow approximately isothermal [26]; because the microtubestested in this work are characterized by large values of the L/Dratio (>166), the flow can be modeled as isothermal. Shapiro[26] deduced the following equation to calculate the frictionfactor for an isothermal compressible flow in a constant-areaduct:

f = D

L

1 −

(1 − �pn

pi

)2

(m

√RT

Api

)2

− 2 ln

(1

1 − �pn

pi

) (3)

For compressible flows, the Reynolds number could changebetween the inlet and outlet of the tube. In this work, theReynolds number is defined in terms of measured quantities


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Table 1 Characteristics and uncertainties of the measurement instruments

Label Instrument Range Uncertainty

(4a) Flowmeter (BronkhorstEL-Flow E7000)

0–5000 [nml/min] ±0.6%FS

(4b) Flowmeter (BronkhorstEL-Flow E7000)

0–500 [nml/min] ±0.5%FS

(6) Differential pressuretransducer (Validyne DP15)

0–35 [kPa]0–86 [kPa]0–220 [kPa]0–860 [kPa]0–1440 [kPa]

±0.5%FS

(8) Thermocouple (K type) 0–200 [◦C] ±0.25%FS

as follows:

Re = mD

Aµ(4)

It is evident that for isothermal flows, the Reynolds numberdefined by Eq. (4) is constant along the microtube [26,27].

To assess the accuracy of measurements presented in thiswork, the uncertainty associated to each measurement device isreported in Table 1. The uncertainty in measuring f and Re canbe estimated by applying the theory on the propagation of errorsto the uncertainty associated with the measurement of the singlequantities p, D, L, and so on. It has been demonstrated that themeasurement of the diameter is critical for the overall measure-ment uncertainty at moderate to large Reynolds numbers. In thepresent case, the uncertainty of the inner diameter evaluated byusing a SEM is of the order of 2%; the uncertainty on the lengthis of the order of 0.3%. The uncertainty in measuring f and Reis estimated to be equal to ±10% and 3%, respectively.

RESULTS AND DISCUSSION

In Figures 3 and 4, the experimental values of the frictionfactor measured for the commercial Peek microtubes are shownas a function of the Reynolds number in a Moody chart. Thedata in Figures 3 and 4 put in evidence that:

• in the laminar regime, the agreement with the conventionaltheory seems to be very good, especially for the longer micro-tubes tested;

• for the shorter microchannels (L = 0.05 m), when Re isgreater than 1000, the friction factor tended to deviate fromthe Poiseuille law; the friction factors are higher than thosepredicted by the conventional theory;

• in the transition region, the experimental data are scattered;this fact can be explained by observing that, in general,the laminar to turbulent transition has to be considered ametastable phenomenon, though no earlier transition wasobserved;

• in the turbulent regime the experimental values of the frictionfactor follow the Blasius equation; the agreement worsensfor the smaller microtubes for which the friction factor seems

Figure 3 Friction factor as a function of the Reynolds number for microtubeswith D = (a) 100, (b) 150, and (c) 175 µm.


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Figure 4 Friction factor as a function of the Reynolds number for microtubeswith D = (a) 200 and (b) 300 µm.

to be less than the values predicted by the Blasius correlationfor smooth tubes.

Now, the experimental data obtained in this work are used toflesh out the main effects that tend to influence the value of thefriction factors for compressible flows in smooth microtubes.

The Role of the Minor Losses

The fittings at the inlet and outlet of the microtube can bemodeled as an abrupt inlet and an abrupt exit. The proper fittingloss coefficients are taken in agreement with the values indicatedby Idelchik [25]. In Figure 5, the minor losses at the inlet (�pin)and outlet (�pex) are compared with the value of the net pressuredrop (�pn) for two microtubes (5 cm and 50 cm long) with aninternal diameter equal to (a) 100 µm and (b) 300 µm. As a ruleof thumb, as seen in Figure 5, it is evident that the minor lossesdue to the exit and the inlet of the microtubes are of one orderof magnitude less than the net pressure drop.

Figure 5 Minor losses and net pressure drop for microtubes 5 cm long (emptysymbols) and 50 cm long (full symbols) with an internal diameter equal to (a)100 and (b) 300 µm.

The significance of the minor losses increases for large val-ues of the Reynolds number. It can be observed that the minorlosses at the inlet become more important for short microtubes.On the contrary, the value of the minor losses at the outlet isindependent on the length of the microtube because in the tests,the gas at the exit of the microtube vented directly into the at-mosphere and hence the pressure at the exit was imposed equalto the atmospheric value. Figure 5 underlines that the role of theminor losses on the evaluation of the friction factor in laminarregime can reasonably be considered of secondary importance,especially for long microtubes having a low value of the internaldiameter. In fact, by comparing Figure 5a with Figure 5b, it isevident that for the same net pressure drop, the minor losses arelower for the smaller microtubes. On the other hand, the dataof Figure 5 highlight that the minor losses can be significantin the determination of the friction factor in short microtubes,especially in turbulent regime.


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Figure 6 The effect of the acceleration losses on the friction factor as a func-tion of the Reynolds number.

The Role of the Acceleration Losses

Conventional theory [26,27] underlines that compressibilityeffects become important for a gaseous flow through a channelwhen at least one of these inequalities is satisfied:{

Maavg > 0.3�pn

pin> 0.05

(5)

where Maavg is the average value of the Mach number along themicrotube and pin is the value of the pressure at the inlet of themicrotube. When the first inequality is satisfied, the gas flowcannot be considered locally incompressible, and the momen-tum and energy equations have to be solved as coupled. Whenthe second inequality is satisfied, even if the gas flow can belocally modeled as incompressible, the density variation alongthe tube cannot be ignored. In this case, the effects due to thegas acceleration along the tube become important, even if theMach number is low. The gas acceleration losses are taken intoaccount in the evaluation of the friction factor by means of thesecond term of the right-hand side of Eq. (3). It is evident thatthe importance of the acceleration losses increases for large val-ues of the ratios �pn/pin and D/L. In Figure 6, the importanceof the acceleration losses is emphasized by calculating the fric-tion factor with (empty symbols) and without (full symbols) thesecond term of the right hand side of Eq. (3) for a microtubewith D = 100 µm, 5 cm long. It is evident that by neglectingthe acceleration term in the expression of the friction factor, fseems to deviate from the Poiseuille law in laminar regime.

As a rule of thumb, for microchannels, it is possible to ig-nore the effect due to the acceleration losses when the followinginequality is satisfied:

�pn

pin

D

L< 0.001 (6)

which corresponds to a variation less than 4–5% on the frictionfactor.

The Role of the Mach Number

Theoretical and numerical analyses on the behavior of com-pressible flows in microtubes have highlighted that the localvalue of the friction factor is dependent on the Mach numberas well as the Reynolds number. The dependence of the frictionfactor on the Mach number is a consequence of the momentumand energy equations coupling. However, Eq. (5) underlines thatthe dependence of the friction factor on the average value of theMach number can be neglected for low values of the Mach num-ber. This rule has been confirmed numerically by Guo and Wu[28]. Recently, Kohl et al. [19], Asako et al. [29] and Li et al.[30] have obtained numerical and experimental results on thefrictional characteristics of compressible flows through straightmicrochannels having a rectangular and circular cross-section,respectively. Li et al. [30] and Asako et al. [29] performed com-putational investigations on the effect of the compressibility fora wide range of Reynolds numbers and Mach numbers. Thenumerical results obtained by Li et al. [30] by solving the cou-pled momentum and energy equations for microtubes with aperturbation method have been condensed in the following cor-relation between the local Poiseuille number and the local Machnumber:

fRe = 64

[1 + Ma2

1.5 − 0.66Ma − 1.44Ma2

](7)

The correlation proposed by Li et al. [30] has been comparedwith the experimental results obtained in this work by replacingin Eq. (7) the local value of the Mach number with its averagevalue, defined as follows:

Maavg = m

A

(2

pin + pex

) √RT

γ(8)

in order to test the dependence on the Mach number of thePoiseuille number in laminar regime. The results of this com-parison are shown in Figure 7 for microtubes having a diameterequal to 100, 175, and 300 µm. Eq. (7) is plotted with a contin-uous line in Figure 7.

It is possible to note that in laminar regime, the increase ofthe average Poiseuille number with increasing the average valueof the Mach number is confirmed by the experimental data.

Because for a fixed microtube, the average Mach number in-creases by reducing the length of the channel, different ranges ofthe average Mach number were tested by changing the L/D ratiofor each diameter. The average value of the Poiseuille numberas a function of the average Mach number seems to be quite in-dependent of the value of the L/D ratio; in fact, the scattering ofthe data for different values of the L/D ratio is in general withinthe experimental uncertainty.

The correlation proposed by Li et al. [30] seems to be usefuleven if applied to correlate each of the average values of f Reand Ma. In fact, the experimental values made in this work forlong and short microtubes agree with the numerical correlationof Li et al. [30] within 10%.


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Figure 7 Poiseuille number as a function of the Mach number for microtubeswith D = (a) 100, (b) 175, and (c) 300 µm.

The dependence of the Poiseuille number on the Mach num-ber in the laminar regime can explain the deviation from thePoiseuille law of some experimental results especially for shorttubes, where the average values of Ma reached in the experi-mental test were larger (shorter microtubes), as evidenced byFigure 7. Another conclusion that can be drawn from Figure 7 isthat in order to evidence the effect of the average Mach number,careful experiments must be designed; in fact, a value of the un-

Table 2 Critical Reynolds numbers for isothermal gas flows in microchannels

CriticalAuthors Cross-section L/Dh Reynolds number

Wu and Little [31] Trapezoid 100–720 1000–3000Acosta et al. [32] Rectangle — 2770Choi et al. [33] Circle 640–8100 500–2000Yu et al. [34] Circle — 1700–6000Stanley [24] Rectangle — 1500–2000Li et al. [13] Circle 145–715 2300Li et al. [35] Circle 145–715 1700–2000Yang et al. [14] Circle 690–3984 1200–3800Li et al. [36] Circle 690–3984 1700–1900Tang and He [20] Rectangle/circle 498–995 1900–2500Faghri and Turner [37] Rectangle/trapeziod 263–5717 2000–2500Kohl et al. [19] Rectangle 220–533 2300–6000

certainty on f Re equal to 10% is sufficient to mask its influenceup to Ma equal to 0.2.

Laminar to Turbulent Transition

The main experimental results in terms of critical Reynoldsnumbers obtained for isothermal gas microflows quoted in theopen literature are summarized in Table 2. From the analysis ofthe works quoted in Table 2, in which the laminar to turbulenttransition for gas flows in microchannels has been investigated,it is possible to note that the laminar to turbulent transition seemsto be influenced by the following factors:

• the geometry of the cross-section;• the roughness of the microchannels walls;• the gas compressibility quantified by the Mach number at the

outlet;• the length-to-hydraulic diameter (L/Dh) ratio of the mi-

crochannel.

Figure 8 is a plot of the critical Reynolds number, Rec, atwhich transition occurs against the L/D ratio. Two different kindsof laminar to turbulent transition have been experienced whenthe gas flow rate is gradually varied:

1. Smooth laminar to turbulent transition (ST), in which thetransition occurs gradually.

2. Abrupt laminar to turbulent transition (AT), in which thetransition occurs suddenly.

The squared symbols in Figure 8 represent the smooth tran-sition (ST), whereas the circles indicate an abrupt jump fromthe laminar to the turbulent flow (AT). The value of Rec for thesmooth transition was chosen as that corresponding to a mini-mum of the friction factor in the laminar regime, and so was thatfor the abrupt transition. The data by Kohl et al. [19] are alsoplotted in the same chart.

Observation of the data reveals how the values of the criticalReynolds number fall in the range indicated by the classical the-ory for macro-tubes (2100–6000), and no marked dependence on


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Figure 8 Critical Reynolds number as a function of the L/D ratio for smooth(squares) and abrupt (circles) laminar to turbulent transition.

the value of the length-to-diameter ratio can be unmistakably dis-cerned. This is in agreement with the conventional theory as un-derlined by Prandtl and Tietjens [38] by critically analyzing thepioneeristic works of Reynolds, Hagen, and Schiller in this field.

It is interesting to note how the agreement between the presentresults and the data of Kohl et al. [19] worsens for low values ofthe L/D ratio.

Because the number of experimental data reported by [19]is quite limited, it seems fair to say that any statement basedonly on the data of Kohl et al. [19] about the dependence of thecritical Reynolds number on L/D is not completely motivated.In addition, it is important to underline that the present data onthe laminar-to-turbulent transition for an L/D ratio close to 200seem to confirm the results obtained by Li et al. [13], for whichthe critical Reynolds number was 2300.

If one considers the mode of transition instead, it can benoticed that, in general, the smooth transition is experiencedfor lower critical Reynolds number (from 2100 to 2900) thanthat of the abrupt transition (which goes from 2500 to 3900).Moreover, the channels tested in [19] had a rectangular cross-section, and this trend might be due to geometrical effects notimmediately captured by the equivalent hydraulic diameter, asevidenced theoretically by Morini [21] and Obot [39].

As the L/D ratio increases, the abrupt transition seems tobecome the prevailing mode; yet, there seems to be no clearreason why this should be the case; further experimentation onthis issue is surely mandatory.

Finally, in order to investigate the possible influence of theMach number on the critical Reynolds number, the present studyuses the value of the Mach number at the outlet, defined as:

Mao = m

pexA

√RT

γ(9)

which was chosen because it can be calculated directly frommeasured quantities and is related to the point (the exit) wherecompressibility effects are most severe.

Figure 9 Critical Reynolds number as a function of the outlet Mach numberfor ST (full circles) and AT (empty circles) laminar to turbulent transitions.

The critical Reynolds number was plotted in Figure 9 as afunction of the outlet Mach number, Mao. Again, no particulardependence of Rec on the outlet Mach number was remarked.The two types of transition are fairly spread over the whole rangeof Mach numbers, from 0.33 to 0.71; the results of Figure 9confirms that the abrupt transition seems to be coupled to thelarger values of the critical Reynolds number. The scatter of thecritical Reynolds number evidenced in Figures 8 and 9 confirmsthe metastable nature of the laminar to turbulent transition. Inparticular, this scatter can be due to the influence of the initialdisturbances related to the fittings and to the geometry of mi-crotube ends, sometimes altered after the cutting. By observingFigures 8 and 9, it is evident that no earlier transition is evi-denced in microtubes; in any condition, the critical Reynoldsnumber is always greater than 2100.

CONCLUSIONS

In this work, experimental tests on the frictional characteris-tics of compressible flows in microtubes with diameters rangingbetween 100 and 300 µm are described. The experimental resultsin terms of friction factors evidenced a good agreement with theprediction of the conventional theory validated for large tubes.This analysis has permitted one to systematically discuss all themain potential sources of unusual and often conflicting resultspreviously reported in literature. The main conclusions of thisstudy can be summarized as follows:

• in the laminar regime, the agreement with the conventionaltheory is very good, especially for the longer microtubestested; for the shortest microchannels (L = 0.05 m) when Reis greater than 1000, the friction factor tends to deviate fromthe Poiseuille law, but this fact can be explained by taking intoaccount the dependence of f on the Mach number;

• the effect of the minor losses can be very important especiallyfor a large Reynolds number. The minor losses at the exit are


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more important than those at the inlet for long microchannels;• the effect of the acceleration of the compressible flow along

a microtube is negligible in the determination of the frictionfactor only if �pn/pin is less than 0.1% of the L/D ratio;

• there is no evidence of an earlier transition than those pre-dicted by the classical theory for macro-tubes; the values ofthe critical Reynolds number fall between 2100 and 3900 forall the tested specimens;

• both smooth and abrupt laminar to turbulent transition modeshave been experienced, but, as the L/D ratio increases, theabrupt transition seems to become the prevailing mode;

• the dependence of the critical Reynolds number on the length-to-diameter ratio is questionable at best;

• the critical Reynolds number seems to be uncorrelated withthe outlet Mach number over the values of the Mach numberat the outlet investigated in this work.

ACKNOWLEDGMENTS

This work has been funded through the Italian NationalProject PRIN05 by MIUR-URST and the Italian-FrenchGALILEO program. Their support is gratefully acknowledged.

NOMENCLATURE

A cross-sectional area, m2

C minor losses coefficientcp specific heat capacity at constant pressure, J/(kgK)cv specific heat capacity at constant volume, J/(kgK)D inner diameter, mf Darcy Weisback friction factorsKn Knudsen numberL microtube length, mm mass flow rate, kg/sMa Mach numberp pressure, PaPo Poiseuille number (= fRe)R specific gas constant, J/kgKRe Reynolds numberT gas temperature, KU experimental uncertaintyW average velocity, m/s

Greek Symbols

ε average roughness, mγ specific heat ratio, cp/cv

µ dynamic viscosity, kg/m sρ gas density, kg/m3

Subscripts

avg averageex exit

h hydraulicin inletm measuredn netth theoretical

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Gian Luca Morini received his Ph.D. in nuclear en-gineering in 1996 from the University of Bologna,Italy. He started as an assistant professor at the Fac-ulty of Engineering of the University of Ferrara in1996. Since 2002, he has been an associate professorin Applied Thermal Engineering at the University ofBologna. He is Italian Member of the EUROTHERMCommittee. He is member of UIT, Italian Society ofThermo-Fluid-dynamics, and AICARR, the ItalianSociety of Air Conditioning. His main research in-

terests lie in microfluidics and heat transfer in micro-devices. He has publishedtechnical papers in the areas of single-phase heat transfer and enhancementtechnologies.

Marco Lorenzini graduated in mechanical engi-neering at Bologna University in 1995. In 2001, heachieved a Ph.D. in Energy Engineering at the Uni-versity of Pisa. Since 2002, he has been a researchfellow at the DIENCA in Bologna. He is a member ofUIT, the Italian Society of Thermo-Fluid Dynamics.He is co-author of some forty scientific papers, mainlyexperimental in nature, dealing with heat transfer andfluid dynamics issues. His interests include heat andfluid flow in microchannels and industrial applica-tions of thermal management.

Stephane Colin is a professor of mechanical engi-neering at the National Institute of Applied Sciences(INSA) of Toulouse, France. He obtained his engi-neer degree from ENSEEIHT in 1987 and receivedhis Ph.D. in fluid mechanics from the PolytechnicNational Institute of Toulouse in 1992. In 1999, hecreated and currently leads the Microfluidics Groupof the Hydrotechnic Society of France (SHF).

Sandrine Geoffroy is an assistant professor inthe Department of Mechanical Engineering at theNational Institute of Applied Sciences (INSA) ofToulouse, France. She obtained her engineer degreefrom ENSEM of Nancy in 1998 and received herPh.D. from Paris XI University at the FAST Lab-oratory, Orsay, France, in 2001. Her main researchinterests are heat and mass transfer in gaseous andliquid micoflows. She is a member of the Hydrotech-nic Society of France (SHF).


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