Morini 2011 Experimental Thermal and Fluid Science

Experimental Thermal and Fluid Science 35 (2011) 849–865

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Experimental Thermal and Fluid Science

journal homepage: www.elsevier .com/locate /et fs

A critical review of the measurement techniques for the analysis of gasmicroflows through microchannels

Gian Luca Morini ⇑, Yahui Yang, Habib Chalabi, Marco LorenziniDipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale, Alma Mater Studiorum, Università di Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 November 2010Received in revised form 21 February 2011Accepted 21 February 2011Available online 25 February 2011

Keywords:MicrofluidicsMicroconvectionFriction factors in microchannelsRarefied gasesCompressible gases

0894-1777/$ - see front matter � 2011 Elsevier Inc. Adoi:10.1016/j.expthermflusci.2011.02.005

⇑ Corresponding author. Tel.: +39 051 2093381; faxE-mail address: [email protected] (G.L. M

In Microfluidics, a large deviation in the published experimental data on the dynamic and thermal behav-ior of microflows has been observed with respect to the classical theory but, from a chronological analysisof these experimental results, it can be realized how the deviations in the behavior of fluid flows throughmicrochannels from that through large-sized channels are decreasing. Today, it seems to be clear thatsome of the inconsistencies in the data were originated from the experimental methods used for theinvestigation of convective microflows. This fact highlights the need for the development of specific mea-surement techniques for Microfluidics. In this work, we explore and categorize different approachesfound in literature for measuring microflow characteristics, especially for gas flows, and the geometryof the microchannels pointing out the advantages and disadvantages inherent to each experimental tech-nique. Starting from the operative definition of friction factor, the main parameters that must be checkedin an experimental work in order to characterize the flow are reviewed. A discussion based on uncer-tainty analysis will be presented in order to individuate the main operative parameters that one mustbe able to measure accurately to determine pressure drop in the microchannels with a low level of uncer-tainty. In the paper each measurement technique is critically analysed to evidence the important issueswhich may have been overlooked in previous researches. The main goal of this study is to give a summaryof experimental procedure and a useful guideline for experimental research in Microfluidics.

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1. Introduction

During the last 10 years a rapid development of new microflowdevices (MFD) in several scientific fields has taken place. Nowa-days, the manufacture of MFDs, like micropumps, microvalves,microcold plates, microheat exchangers, and other microcompo-nents and sensors used in chemical analysis, in biomedical diag-nostics or in flow measurements, is a consolidated reality. Thedesign of new MFDs requires a deep knowledge of the fluid-dynamic and heat transfer phenomena within microchannels inwhich a liquid or gas flows.

For this reason, many experimental studies have been con-ducted in order to analyze the behavior of convection throughmicrochannels, of which a review is given in [1–3]. In particular,the main goal of these studies was to determine the friction factorsand the convective heat transfer coefficients through microchan-nels in which a pressure-driven flow was imposed.

These experimental results have been used in order to verify ifthe laws governing transport phenomena within channels of mac-

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: +39 051 2093296.orini).

roscopic dimensions still hold at the microscale, and, if not, whichnew effects must be taken into account at the microscale.

A large scatter in published experimental data and inconsisten-cies have been observed with respect to the classical theory but,from a chronological analysis of these results, it is possible toextrapolate how the deviations between the behavior of fluidsthrough microchannels and through large-sized channels aredecreasing. The last experimental works in Microfluidics seem tohighlight that some of the observed discrepancies in the data wereoriginated from the experimental methods used for the investiga-tion of convective microflows.

In fact, in the last years a dramatic improvement of the tech-niques of microfabrication has enabled a more accurate controlover the geometry of microchannels and innovative and moreaccurate measurement techniques for microflows have been pro-posed with a general improvement of the reliability/accuracy ofthe experimental data reported in the literature: these latest dataseem to be in agreement with the classical theory.

This highlights the need for the development of specific mea-surement techniques for the Microfluidics field or a refinementand adaption of the ones used at larger scales.

In this work, we explore and categorize different approachesfound in literature for measuring microflow characteristics –especially for gas flows – and the geometry of the microchannels

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Nomenclature

d inner diameter of circular microtube (m)Dh hydraulic diameter of channel, 4 X/C (m)f friction factorh height of rectangular microchannel (m)t time (s)L channel length (m)K pressure loss coefficientKn Knudsen number, k/Dh_m mass flow rate (kg/s)

Ma Mach numberp pressure (Pa)R specific gas constant (J/(kgK))Re Reynolds numberT temperature (K)v average axial velocity (m/s)V volume (m3)w width of rectangular microchannel (m)W mean flow velocity (m/s)

Greek symbolsa coefficient defined in Table 3b coefficient defined in Table 3c specific heat ratioC wetted perimeter (m)k gas mean free path (m)l dynamic viscosity (Pa s)q gas density (kg/m3)r coefficient related to channel geometry in Eq. (12)s viscous stress (Pa)X area of cross-section (m2)

Subscriptscr critical valuein inlet valueout outlet valuew pertains to the channel wallz axial direction

850 G.L. Morini et al. / Experimental Thermal and Fluid Science 35 (2011) 849–865

by examining the advantages and disadvantages inherent to eachof them.

Starting from the operative definition of friction factor, the mainparameters that must be checked in an experimental campaign inorder to characterize the flow are highlighted.

A discussion based on uncertainty analysis is presented in orderto pinpoint the main operative parameters that must be measuredaccurately to determine the pressure drop in the microchannelswith a low level of uncertainty.

2. Pressure drop and friction factor

The behavior of isothermal pressure-driven single phase fluidflows in microchannels can be studied by determining the velocitydistribution in the fluid region with the help of the mass conserva-tion principle (continuity equation) and the equations of conserva-tion of fluid momentum (Navier–Stokes equation).

Consider a flow through a straight microchannel having an axi-ally unchanging and uniform cross-section with an area equal to Xand a wetted perimeter equal to C; it is possible to define thehydraulic diameter of the microchannel as:

Dh ¼4XC

ð1Þ

and the mean fluid velocity as the integrated average axial velocityover the flow area X:

W ¼ 1X

ZX

vzdX ¼_m

qX¼ RT

p

� �_mX

ð2Þ

In Eq. (2) the relationship between the average axial velocityand the measurable quantities, like mass flow rate _m, density qand cross-section area X or pressure p and temperature N for gasesis highlighted (R is the specific gas constant). It is important tohighlight that Eq. (2) has been written for an ideal gas: this meansthat we have restricted our analysis to pressures below the criticalone (pcr) and/or to temperature larger than two times the criticaltemperature (Tcr) for the gas considered. This hypothesis holds inMicrofluidics because the gas temperature is generally larger thanthe critical one. During experiments in microchannels it is difficultto directly derive the mean fluid velocity along the channel fromthe cross-sectional velocity profile, especially when the micro-channel is very small and without optical access which can enablethe use of non-intrusive specific optical techniques developed with

this aim as the microParticle Image Velocimetry (lPIV) [4]. Thelimitations of velocimetry techniques for gas flows in microchan-nels will be discussed in the next sections. Therefore, the meanvelocity of flow is often calculated measuring the microchannelcross-sectional area, the fluid density and the mass flow rate, asper the RHS of Eq. (2).

The pressure drop Dp along the channel and the viscous stressesat the channel walls can be related to the mean fluid velocity bymeans of the following expressions:

jswj ¼ fFqW2

2

!Dp ¼ fD

LDh

qW2

2¼ ð4f FÞ

LDh

qW2

2ð3Þ

where L is the microchannel length.These equations are used as definition for the Fanning friction

factor (fF) and the Darcy friction factor (fD = 4fF). Eq. (3) evidencesthat the pressure drop along a microchannel can be predicted ifone knows the value of the friction factor.

For an incompressible, isothermal flow the Darcy friction factorcan be operatively defined as:

fD ¼Dh

L2q_mX

� �2 ðpin � poutÞ ð4Þ

where pin and pout are the pressure at the channel inlet and outletrespectively.

For isothermal compressible gas flows, using the state equationof an ideal gas, the expression of friction factor can be re-written asfollows:

fD ¼Dh

Lp2

in � p2out

RT _mX

� �2 � 2 lnpin

pout

� � !ð5Þ

where T is the gas temperature, and R the gas constant respectively.Since in microchannels the pressure drop along the tube length

can be much larger than for conventional-size tubes, the effects ofgas acceleration cannot be neglected in general, even for very lowvalues of the Mach numbers as observed theoretically and experi-mentally by several authors [1,3,5]. The gas acceleration leads tothe velocity profile change not only in magnitude but also in shape;the magnitude increment produces an additional pressure dropwhile the continuous variation in shape of the velocity profilemeans no fully or locally fully developed flow occurs. In heatedmicrochannels, since the fluid radial temperature profile is

Fig. 1. Darcy friction factors for nitrogen flow through a microtube in adiabaticconditions as a function of the Reynolds number (from Yang et al. [6]).

G.L. Morini et al. / Experimental Thermal and Fluid Science 35 (2011) 849–865 851

strongly dependent on the radial velocity profile no fully developedtemperature profiles will occur as long as the flow is developing.

In tubes having an inner diameter less than 500 lm and with alarge length-to-diameter ratio (long tube) high pressure ratios be-tween inflow and outflow and high heat transfer ratios produce asignificant increase of the Mach number along the tube and highMach numbers yield higher friction factors and heat transfer rates.In adiabatic and in heated microchannels, the compressibility ef-fects can be monitored by using the diameter-to-lenght ratio (d/L) of the microtube and the outlet Mach number defined asfollows:

Maout ¼4 _m

pd2pout

ffiffiffiffiffiffiffiffiffiffiffiRTout

c

sð6Þ

where c is the specific heat ratio of the gas considered and Tout andpout are the gas bulk temperature and pressure at the outlet ofmicrochannel.

The conventional theory states that if the average value of theMach number (Maavg) along a channel is less than 0.3 the flowcan be modeled locally as incompressible. In addition, when thepressure drop along the tube length is lower than 5% of the inletpressure, the effect of the acceleration of the gas flow in the axialdirection can be neglected (in other words, the variation of thegas density along the channel can be neglected):

Maavg > 0:3pin�pout

pin> 0:05

(ð7Þ

When the first inequality is satisfied, the gas flow is not locallyincompressible and the momentum and energy equation have tobe solved as coupled. When the second inequality is satisfied, evenif the gas flow can be locally modeled as incompressible, the den-sity variation along the tube cannot be ignored. In this case, the ef-fects due to the gas acceleration along the tube become important,even if the Mach number is low. The gas acceleration losses are ta-ken into account in the evaluation of the friction factor through thesecond term of the right-hand side of Eq. (5). For microchannels, ithas been demonstrated by Yang et al. [6] that the friction factor isinfluenced by the value of the Mach number at the outlet (Eq. (6));when Maout is larger than 0.3 a deviation of the friction factor fromthe prediction of the conventional theory is evidenced. For this rea-son, the two constraints of Eq. (7) (with Maout instead of Maavg)have always to be verified when an experimental determinationof the friction factor in a microchannel for a gas flow is made in or-der to know whether compressibility effects can be considerednegligible.

In Fig. 1 the trend of the Darcy friction factor as a function of theReynolds number for a nitrogen flow through a microtube having acircular cross-section, with an inner diameter equal to 172 lm anda length of 10 cm is shown [6]. The error bars related to the exper-imental friction factors are not shown in order to improve readabil-ity; the maximum relative uncertainty on the friction factor isequal to ±10% as indicated in [6]. It is possible to see that the agree-ment between the experimental data and the Poiseuille law(fD = 64/Re) is very good in the laminar regime, at least up to Reequal to 1200. For Re > 1200 it is evident a departure of the frictionfactors from the Poiseuille law; this behavior can be explained byobserving the value taken by the outlet Mach number (Maout) inthis case, as it becomes larger than 0.3. This result is typical formicrotubes having an inner diameter smaller than 500 lm.

In the laminar regime, the deviation from the Poiseuille law canbe theoretically explained by solving the Navier Stokes equationfor compressible flows [5,7]. When the pressure ratio increases,the outlet Mach number increases, especially for short microtubes,and the friction factor increases with the Mach number. Asako

et al. [7] have calculated numerically how the friction factor de-pends on the Mach number in the laminar regime and proposedthe following correlation:

fD ¼64Reþ 2:703

MaReþ 93:89

Ma2

Reð8Þ

As can be noticed from Eq. (4), for incompressible flows the knowl-edge of the differential pressure between the microchannel inletand outlet is required only in order to determine the friction factor.On the contrary, Eq. (5) shows that for a compressible flow even theabsolute value of the gas pressure at the inlet or at the outlet isneeded. This aspect underlines that the experimental strategiesfor the determination of the friction factor must be differentwhether compressibility effects are negligible or not.

The local measurement of pressure is problematic for micro-channels: drilling pressure taps along the channel can be challeng-ing due to the small dimensions involved. For this reason, in themost accurate experimental works, special microchannels are de-signed with customized pressure taps along their length (Bavièreet al. [8]). On the other hand, when commercial microtubes orstandard microdevices are used in the tests, the measurement ofthe pressure along the microchannel becomes impossible. In thesecases, pressure measurements are usually performed by connect-ing with appropriate fittings both sides of the microchannel to alarger reservoir, where the pressure taps can be drilled withoutdifficulties.

However, in this way the pressure measured is not that at theinlet and outlet, which is associated with the change of section.These non-negligible minor losses (Dpin, Dpout) are due to theabrupt change in flow area between the manifolds and the channel.The total pressure drop, which is the measured pressure differencebetween the reservoirs, can be expressed as follows:

Dptotal ¼ Dpin þ ðpin � poutÞ þ Dpout ð9Þ

For perfect gas flows, the minor losses at the inlet and outlet can bedetermined as follows:

Dpin ¼Kin

2_mX

� �2 RTpin

� �; Dpout ¼

Kout

2_mX

� �2 RTpout

� �ð10Þ

For conventional-sized tubes the values of Kin and Kout have beenestablished for a number of simple entrance and exit geometriesand are quoted in many textbooks (e.g. Idelchik [9]). These coeffi-cients have been evaluated under the assumption of essentially uni-form velocity at the inlet but considering a fully established velocity


profile in the channels (that means L� Dh). These assumptions canbe considered verified for microchannels, where L > 100Dh as a rule.

In applying the data related to the values of Kin and Kout itshould be remembered that these coefficients already include thepressure change associated with the change in the velocity profileand depend on the geometry of the ends and of the channels. Inaddition, for the estimation of the minor losses it is mandatoryto know the absolute value taken by the pressure at the inlet orat the outlet, as evidenced by Eq. (10).

In Fig. 2 the minor losses at the inlet (Dpin) and at the outlet(Dpout) are compared with the value of the net pressure drop(Dpn = pin � pout) for two microtubes (5 cm and 50 cm long) withan internal diameter equal to 100 lm [10] when a nitrogen flowis imposed through the microtubes. From Fig. 2 it is evident thatthe minor losses at the outlet of the microtubes are one order ofmagnitude lower than the net pressure drop while those at the in-let are two order of magnitude less. The relative weight of the min-or losses increases for large values of the Reynolds number and thecontribution at the inlet becomes more important for short micro-tubes. On the contrary, the value of the minor losses at the outlet isindependent of the length of the microtube because in the experi-mental tests of Morini et al. [10] the gas at the exit vented directlyinto the atmosphere and hence the pressure at the exit was alwaysequal to the atmospheric value. Fig. 2 underlines that the role ofthe minor losses on the evaluation of the friction factor in laminarregime has to be considered of importance for short microtubes (d/L > 0.01) having a low value of the inner diameter.

The correct evaluation of these losses is a crucial point for theaccurate calculation of the friction factors for gas flows in micro-channels. For example, to the best of the Authors’ knowledge, theeffect of the rarefaction of gases on the Kin and Kout have not beensystematically investigated up to now and this fact can be animportant source of error for the experimental investigation ofthe pressure drop for rarefied flows through microchannels.

When the effects related to rarefaction become significant thefriction factor is influenced by the value of the Knudsen number,the ratio of the gas mean free path (k) and the characteristicdimension (usually Dh) of the channel:

Kn ¼ kDh¼ l

Dhp

ffiffiffiffiffiffiffiffiffipRT

2

rð11Þ

where l is the dynamic viscosity of the gas, p is the local value ofthe pressure and the gas mean free path is evaluated with the

Fig. 2. Minor losses and net pressure drop (Dpn = pin � pout) for nitrogen flowthrough microtubes 5 cm long (empty symbols) and 50 cm long (full symbols) withan internal diameter equal to 100 lm (from Morini et al. [10]).

formula kM ¼ffiffiffiffiffiffiffiffiffip=2

pl= q

ffiffiffiffiffiffiRTp� �

proposed originally by Maxwell in1879 [11].

The analysis of gas rarefaction effects in internal flows is a spe-cific feature of the analysis of convective flows through microchan-nels. When the Knudsen number is less than 0.001, rarefactioneffects can be considered negligible; for 0.001 < Kn < 0.1 the slipflow regime occurs. From a theoretical perspective, the friction fac-tor tends to be reduced when the Knudsen number increases be-cause of the slip existing between the gas and the channel walls.It has been demonstrated that in laminar regime the friction factordepends on the Knudsen number and on the cross-section geome-try of the microchannel as follows:

fDðKnÞ ¼ fDðKn ¼ 0Þ 11þ rKn

ð12Þ

where fD(Kn = 0) indicates the value of the Darcy friction factor inthe laminar regime for the channel geometry considered whenthe rarefaction is negligible (Kn = 0); the friction factor reductionis related to the channel geometry through the coefficient r andto the Knudsen number. The value of r for channels with differentgeometries, typical of microfluidic applications (i.e. circular, rectan-gular, trapezoidal and double-trapezoidal) and different aspect ra-tios was reported by Morini et al. [12].

Observing Eqs. (4)–(12) it is evident that, in order to character-ize and analyze a microchannel flow in terms of friction factor, thefollowing quantities should be determined:

(a) microchannel geometry, including the roughness of thechannel wall, the hydraulic diameter: this means to knowthe geometrical characteristics of the cross-section, thegeometry of the connections between the microchannelsand the reservoirs (if any) and the microchannel length;

(b) operating conditions including quantities such as fluid tem-perature, differential pressure between inlet and outlet,absolute pressure at one end of the channel and mass flowrate.

The above observations can be summarized by stating that thefriction factor for a microchannel can be considered in general afunction of the following quantities:

fD ¼ fDðDp;p; _m; T;C;X; L;Kin;KoutÞ ð13Þ

Among the other variables appearing in Eq. (13) the friction factorcan depend in turbulent regime also on the wall relative roughness(e/Dh) of the microchannel; it has been demonstrated that, even inlaminar regime, very large values of the relative roughness (e/Dh > 5%) can influence the value of the friction factor [13].

Using the physical variables recalled in Eq. (13) one can calcu-late also the Mach number by means of Eq. (6) and the Knudsennumber by means of the Eq. (11); through these values one cancheck the effect of compressibility and of rarefaction on the frictionfactor and compare the results with the theoretical predictions forcompressible and rarefied gases.

With these parameters in mind, special consideration must bepaid both to the design of the experimental setup and to measure-ments during a specific test round at such a small scale to ensure acertain accuracy for reasonable results. Focussing on the experi-mental determination of the friction factor in microchannels forgas flows, with the aim to discuss the role played by each param-eter on its determination, a detailed analysis of the most interest-ing experimental techniques for the measurement of the channeldimensions, gas pressure, temperature, velocity and mass flowrate, together with the analysis of the typical values of their uncer-tainty, is presented in the coming sections.


3. Measurement techniques for microchannels

In this section the most interesting measurement techniquesproposed for the determination of the physical quantities usefulfor the determination of the friction factor in microchannels aresummarized.

3.1. Measurement of microchannel dimension and of its roughness

In the experimental tests devoted to the pressure drop analysisin microchannels some methods have been used in order to esti-mate the inner diameter of microtubes. Poiseuille [14] in 1840used mercury to fill glass capillary tubes (down to 15 lm of diam-eter) of a fixed length in order to determine the average diameterby weighing the tube before and after filling. This technique is verysimple and enables an accurate determination of the average flow-ing area of a microchannel along its whole length. One limitation isthat, in order to obtain a good accuracy of the measurement forvery small tubes, one has to use very long microtubes. An advan-tage of this method is that only a precision balance is required toestimate the flowing area.

One interesting development of this technique is based on theaccumulation of a heavy liquid (water, mercury, etc.) flowingthrough a microtube under a constant pressure drop, which gener-ates a constant mass flow rate through the channel [7]. If oneknows the length of the microchannel, the fluid density and theviscosity, the cross-sectional area of the microchannel can bedetermined by a careful measurement of the liquid weight passingthrough the microtube over a certain time interval: this impliesthat the Poiseuille law is assumed to hold at the microscale too.This technique can be very useful to reduce the uncertainty onthe determination of the inner diameter of a microtube as demon-strated by Asako et al. [7], who reported an uncertainty of the orderof ±0.2 lm on average inner diameter measurement for a tube witha nominal diameter of 150 lm (±0.13%). For very small mass flowrates this measurement technique requires very long time for li-quid accumulation. As underlined by Asako et al. [7], when wateris used as test fluid, care must be used in order to estimate the lostweight of the accumulated liquid due to evaporation during theexperimental tests, which strongly depends on the environmentalcondition. This aspect can be very critical when lower mass flowrates are generated in the smaller microchannels.

Currently, one of the most common methods to measure micro-channel dimensions is the scanning electron microscopy (SEM),which images the profile of the channel cross-section with a highresolution. When the tested microchannels are closed (e.g. com-mercial microtubes), this technique is applicable to the inlet andoutlet sections only; no information can be derived on the axialvariation of the channel geometry and on the roughness distribu-tion along the channel. This can represent a strong limitation forthe correct evaluation of the average channel roughness becausethe inlet and the outlet sections may be non-representativeof the dominant cross-section along the length due to the cuttingwhich may deform the ends. In this case, only with destructivetests after the experimental investigation can one try to recon-struct the axial variation of the roughness and of the cross-sectionby grinding the channel open. Multiple cuts along the channel mayprove useless because the action can change the channel’s cross-section permanently. On the contrary, for open or optically acces-sible microchannels (like silicon rectangular microchannelsbonded with a silicon/glass cover) this technique can be appliedbefore the specimens are sealed. With magnification and imageprocessing, the dimensions of the channel cross-section and thetopology of the roughness can be determined with a good accuracy.Since this technique is convenient and exhibits satisfying accuracy,

it has been employed by many investigators (Tang et al. [15],Celata et al. [16], Zohar et al. [17], Morini et al. [10,18]). It is inter-esting to note that the Poiseuille [14] and Asako et al. [7] methodsallow to measure directly the average flowing area along the wholelength of the microchannel, which is a significant parameter for theevaluation of the friction factor as compared to the algebraic aver-age of the inlet and outlet area obtained by SEM imaging.

Still concerning the roughness, in non-circular microchannelsmade by chemical or ion etching on silicon wafers and sealed withdifferent materials (glass, Pyrex, silicon) the distribution of thewall roughness along the wetted perimeter of the channel can ex-hibit a strong non-uniformity. However, in this case the measure-ment of the wall roughness is generally possible by using scanningelectron microscope or profilometers along the channel beforesealing it.

Another characteristic dimension of the microchannels whichmust be evaluated for the determination of the pressure drop isthe channel length L. For long microchannels, the length L is in gen-eral larger than the inner diameter; typical values of the Dh/L ratiofall within the range 0.1–0.001. In this case, the measurement of Lis easier than the measurement of the inner diameter. On the con-trary, when the behavior of rarefied gases through microchannel isstudied in order to verify the predictions of the mathematical mod-els based on the Boltzmann equation for transitional and freemolecular regimes, very short microchannels must be tested whichare characterized by Dh/L ratios near unity (Marino [19]). In thiscase, the measurement of L can become problematic and, as gen-eral rule, characterization of the microchannel must take also thetube length into account, rather than just the hydraulic diameter,as is otherwise customary.

3.2. Measurement of pressure

In the experimental works devoted to pressure drop analysisthrough microchannels, pressure measurement has been carriedout: (i) only at the inlet and at the outlet of the channel (Kandlikaret al. [20], Asako et al. [7], Morini et al. [18], Celata et al. [16], Tanget al. [15], Demsis et al. [21]); (ii) and/or along the channel in orderto reconstruct the axial distribution (Zohar et al. [17], Hsieh et al.[22], Jang and Wereley [23], Turner et al. [24], Kohl et al. [25]). Inthe latter case, the knowledge of the axial pressure trend enablesto identify special effects, as those related to compressibility andrarefaction, which can play an important role in the determinationof the pressure drop; however, the microchannel must be specifi-cally provided with pressure taps along its length. If traditionalpressure transducers are used, these are generally placed in thereservoirs connected to the microchannel inlet and outlet. Theirpresence introduces pressure losses at the inlet and the outlet ofthe channel which can be determined with Eq. (10). In order toeliminate the effects of these minor losses the tube cutting methodhas been proposed by Du et al. [26] and other authors (Asako et al.[7], Celata et al. [16]). This technique consists of repeating the samemass flow rate conditions of a long tube in a short one with thesame inner diameter. Since the effect of the minor losses is approx-imately identical for both tubes, one can eliminate it by subtractingthe total pressure drop determined for the two. The short and thelong tube can be selected from a set of identical tubes or one canobtain the short one by cutting the long tube. In the latter case,care must be exerted in minimizing the effect of the cut on thegeometry of the microchannel. When the selection of the shorttube and the long one is made among commercial tubes with thesame nominal diameter, Asako et al. [7] demonstrated that the realinner diameter may actually vary markedly: in their work in orderto obtain a diameter difference in each pair chosen less than0.1 lm only three pairs of tubes were selected from precise innerdiameter measurement among more than 30 specimens, which


have a nominal diameter of 150 lm but actually vary from 148 lmto 154 lm. This kind of difference in diameter is stochastic anduncontrollable, which means that with the tube cutting method,the smaller the difference in tube pairs one desires, the larger thenumber of the sample one has to provide.

With this method, the value of the friction factor for a microtubecan be calculated as follows (as indicated in Morini et al. [18]):

f ¼ 2ql2

DptotalðL1Þ � DptotalðL2ÞL1 � L2

� �D3

h

Re2 ð14Þ

where L1 and L2 are the lengths of the longer and shorter tuberespectively.

It has been demonstrated that this method can only be appliedcorrectly when the pressure drop in the tube is linear, as for incom-pressible flows (liquids or gases under low pressure ratio pin/pout).When the pressure distribution along the channel is not linear, asin the case of compressible and/or rarefied flows, this method can-not be applied. In fact, in this case the effect of the minor lossescannot be eliminated completely, as demonstrated by Moriniet al. [18]. This means that for gas flows this method holds onlyfor low Reynolds numbers which corresponds to low Mach num-bers at the outlet.

When the goal is to determine the axial distribution of the pres-sure along a microchannel other techniques can be employed. Anon-intrusive one for the pressure determination along a micro-channel is based on Pressure-Sensitive Paints (PSP).

PSPs are optical ‘‘molecular-sensors’’ which enable the measureof the pressure over a surface. When excited by an outer lightsource of a certain wavelength, the luminescent molecules withwhich the surface of the channel cover is coated will emit lumines-cence of a longer wavelength. By appropriate filtering, the emittedluminescence can be detected. The luminescent intensity is sensi-tive to oxygen molecules near the cover surface and for this reasonthis technique has been proposed for the analysis of gas flows. Spe-cifically, an increase in the oxygen concentration causes a decreasein the intensity of the luminescence, which is known as oxygenquenching (Huang et al. [27] and Liu et al. [28]). After calibration,a relation between pressure and luminescent intensity can beestablished. This non-intrusive measurement technique can pro-vide pressure data with high spatial resolution both along thechannel and at the channel entrance and exit. On the other hand,as this technique is based on oxygen quenching, it cannot be ap-plied to test of other pure gases, such as N2 and He. Moreover,the pressure sensitive paint is coated on a transparent cover ofthe microchannel to ensure the direct contact with oxygen andexcitation of luminescent molecules from an external light source.It is evident that the main disadvantage of this technique is tied tothe shape of the microchannel tested: circular or elliptical micro-tubes make it impossible to coat the pressure sensitive paint ontothe inner wall, and passages which do not have a transparent sidecannot benefit from this optical technique.

The pressure sensitive paint (PSP) technique has been applied tothe measurement of the pressure distributions in high Knudsennumber regimes by Liu et al. [29] and Bell et al. [30]. Because thePSP works as a so-called ‘‘molecular sensor’’, it is considered suit-able for the analyses of high Knudsen number flows, which requirediagnostic tools at the molecular level. However, an application ofPSPs to microdevices is very difficult, because conventional PSPsare very thick compared to the dimension of microdevices owingto the use of polymer binders. Moreover, they do not have suffi-cient spatial resolution for the pressure measurement of micro-flows due to the aggregation of luminescent molecules inpolymer binders as indicated by Mori et al. [31].

More recently, for local measurement of pressure in high Knudsennumbers Matsuda et al. [32,33] developed a pressure-sensitive

molecular film method (PSMF) by using the Langmuir–Blodgett(LB) technique. Before applying PSMF to micro gas flows, theauthors have tested the dependence of temperature on pressuresensitivity, which is the main factor of the measurement errorfor this kind of pressure sensor. Matsuda et al. [33] have demon-strated that their paint composed of Pt(II) Mesoporphyrin IX andarachidic acid showed a small dependence of pressure sensitivityon temperature, while that of luminescent intensity is significant.In addition, the authors have demonstrated how, compared withthe classical PSP method, the PSMF method can offer more uniformpressure sensitivity and an enhanced intensity of luminescenceemission. By applying their technique to the determination of thepressure field for gaseous flow through a 2-D nozzle [32], they con-cluded that the spatial resolution of PSMF is high enough formicroscale flow measurements whose characteristic length is over50 lm.

In order to measure the local pressure along a microchannelKohl et al. [25] explored and proposed the optical lever methodfor pressure measurement. The microchannel presents a numberof micrometric pressure taps connected to silicon membraneswhich deform according to the local pressure. The deformationwas measured recording the change in deflection angle of a fixedincident laser targeting the membrane surface. The change indeflection angle was measured by a photodiode sensor which canbe precisely moved and positioned. Based on this principle, an inte-grated pressure sensor can be produced. The sensitivity of thisintegrated pressure sensor can be easily adjusted by changingthe spatial resolution (the distance between the membrane andthe photodiode sensor). The uncertainty on the measured pressureranged from ±2.4% to ±13.3%.

Since a very precise optical system to test quantitatively theslight slope of the pressurized membrane is involved, it requiresgreat labor to adjust this system. Also such optical system, whichincludes a laser, a lens and several adjustable stands for precisepositioning, is expensive. In addition, the range of the measurablepressure is limited by the mechanical strength of the membranematerial and the channel must be customized for use with thistechnique.

Even microstrain gauges membranes can be employed to sensethe pressure induced at the tap from the microchannel. The defor-mation of the membrane results in the variation of the electricalresistance of the material making up the membrane, thereforeinformation on pressure can be transformed into an electrical sig-nal, which is usually magnified for better accuracy via a Wheat-stone bridge on the membrane, as in the work by Zohar et al.[17] and Bavière and Ayela [8]. After careful calibration, a correla-tion between pressure and voltage in the circuit can be given. Thispressure sensor can be used to test local pressure along the micro-channel and has very low temperature sensitivity (±0.011% �C�1),as reported by Bavière and Ayela [8], who found no effects whichmight actually be the result of inaccurate measurements at thesmall scale. The constant coefficient which links pressure and elec-trical voltage varies from one membrane to another, and eachstrain gauge needs to be calibrated over a certain pressure rangebefore measurement. Even in this case, the microchannel and themembranes must be specifically designed in order to use this tech-nique. For liquids through very small microchannel, Celata et al.[16] demonstrated that the measurement of the differential pres-sure along the channel can be linked to the fluid temperature risebetween the inlet and the outlet of the channel due to the viscousdissipation. This effect is more evident for fluids having a low spe-cific heat and a large viscosity (i.e. iso-propanol). For gases, thetemperature rise due to the viscous dissipation is compensatedby the cooling due to the gas acceleration along the channel. Forthis reason, this effect is not useful for the determination of the dif-ferential pressure along a microchannel for gas flows.


3.3. Measurement of temperature

The measurement of the bulk fluid temperature along a micro-channel cannot be made directly without large disturbance on thefluid flow. Suspended micro temperature sensors within micro-channels have been proposed in the past, as reported in an inter-esting review paper of Nguyen [34]. The first prototypes ofsuspended sensors made by Petersen et al. [35] and Lammerinket al. [36] demonstrated that full integration between this kindof temperature sensors and the microchannels was possible buttheir impact on fluid flow was not negligible at all. Nowadaysmicrofabrication offers a large variety of devices to achieve localtemperature measurements. Among the existing devices it is pos-sible to distinguish three groups: (i) thin film resistance tempera-ture detector (lRTD), (ii) thin film thermocouple (lTFTC) and (iii)semiconducting sensors (SC).

In microchannels, to avoid disturbances of the fluid flow, inmany cases only the wall temperature along the channel is deter-mined by means of integrated temperature sensors and the directdetermination of the fluid bulk temperature can only be obtainedat the inlet and outlet sections of the channel by using specific ple-nums. Sometimes, the thermal inertia of the temperature sensorsplaced at the walls of the microchannel is larger or comparableto that of the channel walls. In this case, the presence of the sen-sors can modify the wall temperature distribution. The problemis how to calculate the fluid temperature when the wall tempera-ture is known. In adiabatic flows this is simple if one can assumethat thermal equilibrium exists between the walls and the fluid.This is not the case when the flow is heated.

In the following a summary of the most interesting temperaturemicrosensors developed in the last years is given with the indica-tion of the main advantages and drawbacks of each one. In Fig. 3a schematic representation of these three different kind of temper-ature microsensors together with their characteristic calibrationcurves is given in order to highlight the main differences amonglRTD, lTFTC and SC due to their different working principle.

The first kind of temperature microsensors which can be inte-grated in microchannels are the Microresistance TemperatureDetectors (lRTDs). This type of temperature sensors is widely used

(a) µRTD

Typical sensors layout

CharacteristicCurve

T: TemperatureR: ResistanceV: VoltageI: Current

Metal aMetal a

Tmeas

R

Fig. 3. Schematic representation of the most common temperature micro-sensors withdetector (lRTD); (b) thin film thermocouple (lTFTC) and (c) semiconducting sensors (S

in microtechnology devices firstly because it is easy to make andsecondly because it generally is characterized by a linear response.The functioning principle of a lRTD is the same as that of the clas-sical RTD in which the temperature-dependent electrical resistivityof a material is used in order to estimate the temperature (Fig. 3a).

Among the several materials used for the realization of thelRTDs the most common are Pt, poly-Si, Al, Ni, W, Au or yet Ag[37–43]. The fabrication of the sensitive metal elements for lRTDsis well controlled by using sputtering or evaporation CVD pro-cesses. Platinum is commonly used for different applications inmicrotechnology for its catalytic property in combination withsome gases or, at other times, for its inert properties in combina-tion with a large variety of fluids. However, platinum films canpresent some drawbacks: the first one is the need of an adhesionlayer on almost all the substrates. In many cases the depositionof the platinum film generates an alloy with different thermalproperties than the pure material and the characteristic curve ofthe electrical resistance vs temperature of the lRTD can changecompletely. Moreover, platinum films can be damaged if the tem-perature becomes larger than 550 �C. Other drawbacks of the plat-inum films is their cost and their CMOS incompatibility [39]. Forthese reasons, some microfluidics technical applications of lRTDuse polysilicon (poly-Si) [40].

Thin Film Thermocouples (lTFTCs) are active elements whichuse the Seebeck effect to measure the temperature of a two-metaljunction (Fig. 3b). Microfabrication of lTFTCs can be more or lesscomplex depending on the existing material constraints for thespecific microdevice needed. Zhang et al. [44] proposed a chromelthin film thermocouple embedded on a Ni substrate and demon-strated that their behavior is similar to the standard K-Type ther-mocouples. In fact, the embedded TFTCs of Zhang et al. [44] onNi substrates have shown that their thermal sensitivity is not af-fected by the junction size between 25 lm and 80 lm. This ther-mal sensitivity has been precisely measured for a chromel film of100 nm and a junction of (60 lm � 60 lm) up to 900 �C; the See-beck coefficient of this lTFTCs was 40.6 lV �C�1 in agreement withthe standard value of this coefficient for the classical K-Type ther-mocouples. The results obtained by Zhang et al. [44] were con-firmed by Choi and Li [45] who showed as a K-lTFTCs with a

(b) µTFTC (c) SC

Metal b

Tref

PN Junction

dV

IV

V

I

the indication of their characteristic curve; (a) thin film resistance temperatureC).


junction of 25 lm � 25 lm and a film thickness of 150 nm wereable to provide a good sensitivity (40.4 lV �C�1) up to 800 �C anda fast response time (28 ns).

However, the authors underline that a microfabrication troublefor this kind of microcomponents is linked to the surface quality(especially roughness) of the thin film layers used in these lTFTCswhich can strongly influence the stability of these temperaturemicrosensors.

With the aim to take advantage of the CMOS compatibility pro-cess (integration, low cost) an alternative for the temperature mic-rosensors are the Semiconducting sensors (SCs) which usePolysilicon active elements (Fig. 3c). Bianchi et al. [46] present astate of the art of integrated smart temperature SCs sensors likeMOS transistors, bipolar transistors and diodes. The use of MOStransistors as temperature sensors needs the use of the transistorsin weak inversion in order to obtain a linear relation between tem-perature and their electrical parameters even if some limitationsdue to leakage currents at high temperature can exist [47,48].Bipolar transistors and diodes present directly a junction voltageproportional to the absolute temperature. Generally, at room tem-perature, silicon p–n junctions have a forward voltage drop of0.7 V, and this voltage decrease by 2 mV for every degree of in-crease [49]. Typical temperature range of these sensors could be�55 to 175 �C with a typical accuracy between ±0.1 and ±3 �C[46]. Transistors with CMOS or bipolar IC technologies enable therealization of all the needed circuits in order to integrate directlythe temperature compensation or calibration functions to the sen-sors. Filanovsky and Lee [48] realized two temperature sensorswith signal-conditioning amplifiers to extract directly from aMOS transistor the linear temperature dependence of VT. UsingBiCMOS technology, the first solution chosen was a thresholdextractor circuit which gave a linear temperature characteristic be-tween �40 and 150 �C but showed some disparities linked to fab-rication. The second one was a resistive Wheatstone bridge whichgave a non-linear temperature response but with a good repeat-ability in the same range of temperature.

Finally, planar diffused silicon diodes seem to be a simpler wayto measure temperature when employing SCs sensors. Guha et al.[39] showed a high temperature gas sensor consisting of anembedded diode able to give a linear response up to 260 �C accord-ing to the calibration with a sensitivity of �1.2 ± 0.005 mV/�C. In apH sensitive ISFET chip, Chin et al. [50] have integrated a p-n diodewhich presented good linearity with a sensitivity of �1.51 mV/�C(in the range 0–50 �C) and which also have been used with a spe-cific compensation circuit dedicated to the application.

Even if, up to now, CMOS technologies are not so frequentlyused in Microfluidics applications, in the future SC sensors couldhave a wide range of applications in this field due to their flexibil-ity and to their high scale of integration that enables to expand thesingle sensor to an array of sensors. For example, Han and Kim [51]

Table 1Characteristics of the micro-sensors for temperature measuremen

Authors Material Thermal

RTDs Schöler et al. [38] PlatinumWu et al. [42] Silver paintWu et al. [40] SiliconChoi et al. [43] Gold

TFTCs Zhang et al. [44] Chromel 40.6 lVChoi and Li [45] Chromel 40.4 lVKim and Kim [61] Chromel

SCs Guha et al. [39] Silicon �1.20 mBianchi et al. [46] SiliconChin et al. [50] Silicon �1.51 mHan and Kim [51] Silicon

have developed a 32 � 32 diodes (1024 diodes) array for measur-ing the temperature distribution on a small surface (8 mm �8 mm). This kind of sensor reduce the 2048 interconnections padsneeded for 32x32 RTD or TFTC sensor array to only 64 interconnec-tions pads that involve a tiny chip and the simplest design. In Hanand Kim’s work [49], calibration experiments evidenced the linearoutput temperature signal of this array in the range 0–100 �C withan accuracy of ± 0.5 �C. In compensation of this flexibility at thefabrication level, array sensors need a multiplexer to read themeasurements.

In Table 1 a selected list of microsensors for temperature mea-surement is shown with the indication of the main technical char-acteristics (when available) of these sensors (thermal sensitivity,accuracy and range of application).

In many of the published scientifical works temperature mea-surements are still obtained using traditional sensors like thermo-couples or Platinum RTD. Bavière et al. [52] measured thetemperature along a rectangular channel by arranging four ther-mocouples uniformly in line with the channel axis. The thermo-couples were planted in blind holes 500 lm from the channelwall surface. To predict the temperature of the wall surface a cor-relation was presented based on calibration experiments and thetemperature measured by thermocouples a constant distance awayfrom the wall surface. This approach provides an indirect way toobtain the local wall surface temperature along the whole channel,which makes it possible to analyze the local heat transfer charac-teristics. However, the existence of these holes near the channelwalls may alter the temperature distribution inside because theyintroduce a heterogeneous thermal conductivity. Demsis et al.[21] investigated the convective heat transfer coefficient in a coun-ter flow, tube-in-tube heat exchanger. Two ports perpendicular tothe axis of the microchannel were machined and connected to theinlet section of the inner tube 35 mm before the heated section,one for the measurement of temperature and the other for pres-sure. The measurement of the outlet pressure and temperaturewas proposed in the same way through two ports 35 mm afterthe heated section. The risk of this technique is linked to the erro-neous evaluation of the fluid outlet temperature: if the part of tubebetween the end of the heated section and the outlet side port isnot thermally insulated with care, the fluid temperature decreasesalong this stretch and the temperature recorded at the exit sideport differs from the fluid temperature at the end of the heatingsection. Also, the side ports may disturb the main stream insidethe tube. This kind of side ports can be readily applied to non-circular microchannels or circular microtubes with relatively largeinner diameter; however, if the inner diameter is very small, below0.3 mm, it becomes difficult to manufacture them.

Besides thermocouples or diode arrays which can provide singleor multipoint measurement of temperature, some specificoptical techniques have been used for the measurement of a

t.

sensitivity Accuracy Temperature range

20–80 �C45–105 �C

0.05 �C

�C�1 20–900 �C�C�1 20–800 �C

V �C�1 0.005 mV �C�1 20–260 �C±0.1–3 �C �55–175 �C

V �C�1 10–50 �C±0.5 �C 0–100 �C


two-dimensional temperature field on a solid surface at micromet-ric scales, such as liquid crystal thermography (Muwanga andHassan [53]), infrared thermography (Mosyak et al. [54]) andtemperature sensitive paints (Liu and Sullivan [28]). The mostimportant constrain of these optical techniques is that thesemethods can be successfully applied only in the cases in whichthe external surface of the test section is not thermally insulated.

3.4. Measurement of local velocity data

There is a crucial need of local experimental gas velocity datathrough microdevices but all the velocimetry techniques proposedfor liquid flows in microchannels present strong limitations if ap-plied to the analysis of gas flows through microchannels. For thedetermination of the local velocity of a gas in a microchannel threeoptical velocimetry techniques have been proposed in the open lit-erature up to now: (i) the Laser-Doppler Anemometry (LDA); (ii)the microParticle Image Velocimetry (lPIV) and (iii) the Molecu-lar-Tagging Velocimetry (MTV).

The Laser-Doppler Anemometry (LDA) has been preliminaryused by Ladewig et al. (cited in [55]) in order to measure gas-phasedata in an operational U-shaped mini-fuel cell with channels of2 mm � 2 mm rectangular cross-section. The flow was seeded withwater droplets, but the velocity data obtained were not yet com-plete enough to fully characterize the flow inside the minichannels.Furthermore, LDA is a point-measurement technique; in order toobtain a more representative visualization of the whole velocityfield in a microdevice a field-measurement technique as the micro-Particle Image Velocimetry can be more useful. For the problemsrelated to the systematic use of the LDA technique for the recon-struction of the velocity profiles in microchannels the LDA tech-nique can be considered useless in measuring the velocity withinchannels having an inner hydraulic diameter below 1 mm. Onthe contrary, for the investigation of the velocity field in micro-channels having inner diameters below 1 mm the microParticleImage Velocimetry is nowadays a consolidated technique but onlyfor liquid flows. An interesting and complete review of the progressof this technique in the recent years is due to Lindken et al. [4]. Upto now, few works have described the application of this techniquefor the analysis of gas flows through microchannels.

One reference paper in this field is due to Yoon et al. [55]; in thispaper the micro PIV technique has been used to directly measuregas-phase velocities in situ in an operational fuel cell with chan-nels having characteristic dimensions of 1–2 mm. For PIV measure-ments, in macro-scale gas flows, the gas is often seeded with oliveoil particles generated by an atomizer. To this aim, high pressuresand a high flow rates are needed for the generation of the oil seedparticles. On the contrary for microscale gas flow applications, asmicrofuel cell, the gas flow rate is relatively small (Reynolds num-bers of the order of hundreds to a few thousands) and this can cre-ate problems in the generation of appropriate tracer particles.Since PIV is based on imaging of particulates introduced into theflow, the accuracy of the gas-phase flow velocity measurement isstrongly dependent on the ability of the tracer particles to followthe stream. In the work of Yoon et al. [55], water droplets havebeen used as seed particles for measuring the flow in a straightminichannel. This choice was motivated for fuel cells with theopportunity to introduce the water particles in the gas flow usingthe humidifier section without contaminating the system. The re-sults of Yoon et al. have demonstrated that certain limitationsare encountered when trying to use such tracer in flows with sig-nificant decelerations, such as in a 180� switchback turn. In partic-ular, the experimental results of Yoon et al. have evidenced thattracer particles smaller than 1 lm are required to accurately followthe flow in such regions. This aspect is one of the most importantconstraints in the application of the lPIV technique for the analysis

of gas flows in microchannels. Sugii and Okamoto [56] have ap-plied the PIV technique in order to investigate the velocity distri-bution in a fuel cell with 1 mm � 0.5 mm rectangular polymermicrochannels. Even in this paper, it is stressed how the mainproblem for the application of this technique to the investigationof micro gas flow is the generation of proper seeding. In this workfluorescent oil particles having a diameter between 0.5 and 2 lmwere used as tracer particles in a nitrogen flow for low Reynoldsnumbers between 26 and 130.

Molecular-Tagging Velocimetry (MTV) is a whole-field opticaltechnique that relies on molecules that can be turned into long life-time tracers upon excitation by photons of appropriate wavelength[57]. These molecules can be either premixed or naturally presentin the flowing medium (unseeded applications). Typically, a pulsedlaser is used to ‘‘tag’’ the regions of interest, and those tagged re-gions are interrogated at two successive times within the lifetimeof the tracer.

Lagrangian displacement vector provides the estimate of thevelocity vectors. This technique can be thought of as essentially amolecular counterpart of Particle Image Velocimetry (PIV), and itcan offer advantages compared to particle-based techniques wherethe use of seed particles is difficult, or may lead to complications asfor gas flows in microdevices. However, this technique has beenonly used for the analysis of liquid flows in microchannels[58,59]. On the contrary, up to now the application of thistechnique for the analysis of gas flows in microchannels is notcompletely consolidated. A specific European research project(GASMEMS project) devoted to the development of velocimetrytechniques for gas flows in microchannels has been launched inorder, among others, to demonstrate the possibility of using lPIVand MTV for the analysis of the gas flows in channels having innerdimensions below 1 mm [60].

It is possible to conclude that for the measurement of the localvelocity data in microchannels much work is still needed in orderto adapt the velocimetry techniques proposed for liquid flowsthrough microdevices to the analysis of gas flows; this field is ofstrategic importance especially for the optimization of microfluidiccomponents as, for example, the micro- and mini-fuel cells.

3.5. Measurement of flow rate

The flow rate through a microchannel can be measured directlyby means of appropriate instruments (mass or volumetric flow me-ters) or in an indirect way by checking the value taken by othermeasurable quantities, like pressure, forces, weight, volume, tem-perature or a combination thereof. As a rule of thumb, it is possibleto use a commercial mass or volumetric flow meter for gas flowsonly if the gas mass flow rate through a microchannel is larger than10�8 kg/s (0.1 Nml/min). On the contrary, for very low mass flowrates (<10�8 kg/s) indirect methods can be considered more reli-able in order to determine the flow rate with low values ofuncertainty.

The commercial flow meters for gas flows can be divided in twogroups: the volumetric flow meters and the mass flow meters. Thevolumetric flow meters give an indication of the flow rate regard-less of the fluid tested; the mass flow meters, on the contrary, canbe used only with the fluid for which they have been calibrated.

The sensors of the commercial gas flow meters can be divided innon-thermal flow sensors and thermal flow sensors. The first kindis classified according to the mechanical working principle bymeans of which the flow rate can be measured indirectly:

(1) by the drag force (i.e. using silicon cantilevers);(2) by pressure measurements (i.e. using capacitive and/or

piezoresistive pressure sensors);(3) by using the Coriolis principle.


The main disadvantage of the non-thermal flow sensors islinked to the dependence of the force, pressure difference and Cori-olis force on the density of the working fluid. Since the gas densitydepends on the gas temperature, a temperature compensation is acompelling need for this kind of sensors.

As indicated by Nguyen [34], the thermal flow sensors em-ployed in the commercial flow meters can be classified as follows:

(1) thermal mass flow sensors which measure the effect of theflowing fluid on a hot body (micro hot-wire and micro hot-film sensors);

(2) thermal mass flow sensors which measure the asymmetry oftemperature profile around a micro heater which is modu-lated by the fluid flow (calorimetric sensors);

(3) thermal mass flow sensors which measure the delay of aheat pulse over a known distance (time-of-flight sensors).

The most popular sensors among the commercial flow metersare the calorimetric sensors: this kind of flow meters are able topredict the mass flow rate through the sensor by means of the fol-lowing balance equation:

_m ¼ Q th

cpðTb;out � Tb;inÞð15Þ

which puts in evidence how the gas mass flow rate can be deter-mined by means of two local measures of temperature (Tb,out andTb,in) and the knowledge of the heat power (Qth) transferred to thegas by the heater.

By using microfabrication, this kind of thermal mass flow sensorcan be directly integrated into a microchannel. Schöler et al. [38]have shown as two classic Pt RTDs and a microheater made on aglass substrate can be directly integrated on a SU-8 epoxy resistmicrochannel (50 lm � 15 lm � 8000 lm). As demonstrated byKim and Kim [61] by using a capacitive mass flow sensor usingtwo K-Type TFTC (150 lm � 150 lm) associated to a PDMS micro-channel (800 lm � 800 lm � 4 cm), the accuracy of this kind ofmass flow meters can be strongly influenced by the position andthe thermal insulation of the microsensor.

When the gas mass flow rates that one wants to measurethrough microchannels are very low (<10�8 kg/s) the use of com-mercial devices becomes unsuitable because of the high level ofuncertainty afflicting the measurements. In addition, accuratemeasurements of mass flow rate in microchannels are challengingfor gas flows, which have a density very sensitive to the room pres-sure and temperature fluctuations.

Low gas flow rates can be determined using the general expres-sion of the mass flow rate as the change of mass over time:

_m ¼ dmdt

ð16Þ

If the gas can be considered as an ideal gas, the mass flow rate canalways be expressed as follows:

_m ¼ pRT

dVdtþ V

RTdpdt� pV

RT2

dTdt

ð17Þ

It can be noted from Eq. (17) that if two of the three variables (vol-ume, pressure and temperature of the gas) are kept constant whilethe third one is monitored during the experiment, the mass flowrate of the gas can be determined. This provides the basic insightsinto gas mass flow rate measurements. In practice, it is impossibleto keep two of the three variables strictly unchanged, so great ef-forts are made to minimize the fluctuations of the nominally un-changed variables to determine the mass flow rate as accuratelyas possible. These fluctuations are usually the main source of errorsin low mass flow rate measurements for gas microflows.

Based on Eq. (17) several techniques have been proposed for thedetermination of the mass flow rate below 10�8 kg/s.

In the droplet tracking technique the gas volumetric flow rate ismeasured by introducing the outflow gas into a pipette, in whichthe gas pushes a droplet forward. By recording the position ofthe droplet versus time under the assumption of constant pressureand temperature, the velocity of the droplet can be determined andknowing the diameter of the pipette the mass flow rate of gas canbe calculated. In this case, Eq. (17) can be re-written as:

_m ¼ pRT

dVdt¼ p

RTpd2

pip

4DlDt

ð18Þ

where dpip is the diameter of the pipette and Dl is the distance thedroplet moves during a certain time interval (Dt). This techniquedoes not require expensive meters and can be performed by usingmultiple microchannels. The lowest mass flow rate that can be mea-sured with this technique depends on the specific setups and on themaximum number of microchannels tested in parallel. As rule ofthumb, the lowest value of the mass flow rate that can be measuredby using the droplet tracking technique is of the order of 10�10 kg/s.However, it is possible to obtain lower mass flow rates if the mea-surement is made using hundreds of identical microchannels inparallel in a single test section. This is the case of Shih et al. [62]who measured the mass flow rate of helium through microchannelsin the order of 10�12 kg/s, which is the same order measured byEwart et al. [63] with nitrogen. Colin et al. [64] obtained the massflow rate down to the order of 10�13 kg/s for helium, while Celataet al. [16] reached flow rates of the order of 10�8 kg/s for the sametype of gas with this technique. However, it is technically difficult tomaintain a constant moving speed for the droplet. Another problemis the accurate measurement of the pipette inner diameter, whichweighs most in determining the mass flow rate, as shown in Eq.(18), and, in the case of parallel channels, their dimensions mayvary significantly and thus the value obtained is only an average.

If gas flow is accumulated in a tank for a certain time interval,there will be a pressure rise in the tank during the flow. If the pres-sure in the tank is under reasonable control and not very high, theexpansion of the tank or the increase in its volume can be safelyneglected. In this case Eq. (17) becomes:

_m ¼ VRT

dpdt� pV

RT2

dTdt

ð19Þ

If the second term in the right-hand side of Eq. (19) is very smallcompared to the first one (i.e. <1%) it can also be dropped and bycareful measurement and recording of pressure rise inside the tankversus time the mass flow rate can be determined. This is usuallyreferred to as constant volume technique. Based on this principle,Ewart et al. [65] built an experimental setup in which the outletof the microchannel is connected to a large tank. By recording thepressure versus time inside the tank, the mass flow rate can bedetermined. The lowest mass flow rate measured with this tech-nique was of the order of 10�13 kg/s. The volume of the tank shouldbe carefully chosen so that the pressure change is detectable by apressure sensor for mass flow rate calculation and at the same timenegligible to remain a nearly constant outlet pressure. Great careshould also be paid to the temperature fluctuation of the gas as itis compressed into the tank, so that the temperature change canbe neglected, which greatly simplifies the calculation of the massflow rate.

Arkilic et al. [66] used a two-tank, modified constant-volumeaccumulation technique to measure the mass flow rate of gas, asthe single-tank procedure would not ensure the thermal stabilityneeded for their experiment. In this case, Eq. (19) is not sufficientto calculate the mass flow rate, as both upstream and downstreampressures vary during the experiment. Instead, the pressure


difference between the two tanks (both located downstream of themicrochannel) is measured. During the test one tank is used toaccumulate the gas flowing out of the microchannel and the otherone remains in steady state which provides a pressure reference.The pressure difference between the two tanks is measured insteadof the absolute pressure for the determination of mass flow rate, asgiven by:

_m ¼ VRT

dðDpÞdt� DpV

RT2

dTdt

ð20Þ

where Dp is the pressure difference between the reference tank andthe flow tank. As the two tanks are connected before the flow starts,this quantity is 0 at the beginning. Due to the relatively small massflow rate of gas through the microchannels, when the flow finishesthe pressure difference can be very small, quite several orders ofmagnitude smaller than the absolute pressure in Eq. (19). Therefore,the sensitivity of mass flow rate to the tank temperature fluctuationis reduced by several orders, and the tank for gas accumulation canbe reasonably regarded as isothermal during the experiment. Inaddition, this technique requires that the two tanks have identicaltemperature and undergo the same thermal fluctuations, whichcan be realized by a good design, arrangement and thermal insula-tion of the tanks. For this reason, in the work by Pitakarnnop et al.[67] the whole setup was thermally regulated with Peltier modulesto maintain a constant temperature.

This method provided data for mass flow rates of an order of10�11 kg/s and a sensitivity as low as 7 � 10�15 kg/s (Arkilic et al.[68]) was reported; Pitakarnnop et al. [67] developed a new setupfor gas microflows in which the constant volume method for massflow rate measurement was implied to both the upstream anddownstream flow. This offers a double-check providing the possi-bility to compare the measured results for the same flow. Theirmeasured values of mass flow rates can be as small as to7.1 � 10�14 kg/s.

Table 2Summary of the experimental techniques proposed for the analysis of the gas flows in mi

Physicalquantity

Technique Range

Mass flowrate

Flow meters 50000–0.05 Nml/min

Droplet tracking �10�8 kg/s�10�10–10�13 kg/s (hundredsparallel microchannels)

Single tank constant volume �10�13 kg/s

Two-tank constant volume �10�11 kg/sMass spectrometry method 4 � 10�17 kg/s–4 � 10�8 kg/s

Diameter Accumulation of heavy liquid 150 lmSEM 0.5–40 lm

133–730 lm10–300 lm30–254 lm

Temperature Thermocouples (directly, at inlet andoutlet)

0–200 �C

Thermocouples (indirectly, alongmicrochannel)

–

Thermocouples (directly, after inletand before outlet)

–

Infrared thermography 50–130 �CLiquid crystal thermography 43–50 �CTemperature sensitive paint 10–100 �C

Pressure Pressure transducer 0–1.5 MPa

Tube cuttingPressure-Sensitive Paints From near vacuum to 2 atm.Pressure-sensitive molecular film 10�2–104 PaMicrostrain gauge membrane 0.1–0.4 MPaOptical lever 0–1.4 MPa

To achieve a constant mass flow rate with this technique, thepressure difference between the two tanks should increase linearlywith time during the experiment. As the outlet of the microchannelis directly connected to the tank, the outlet pressure keeps chang-ing and is uncontrolled. The inlet pressure should be very carefullyadjusted with the passage of time to achieve a linear increase ofpressure difference between the two tanks. This is very difficultto achieve in practice. Thus, the process of gas flow in the micro-channel becomes time-dependent with this method.

In order to complete this review, it is possible to highlight thatthe most sensitive technique for the gas detection and for the mea-surement of very low mass flow rates in leaks is based on massspectrometry and it has been presented by Tison in 1993 [69].Tison developed a specific setup to measure gaseous capillary leakrates ranging from 10�6 mol/s to 10�14 mol/s. A series of complexoperation steps are involved in the test so that the final determina-tion of flow rate is dependent on the ratio of measured quantitiesinstead of their absolute values, which reduces the sensitivity ofpossible fluctuations (temperature, pressure and so on) and ex-pands the measurement to very low range (pressures between10�4 and 10�8 Pa). In this work, however, the focus is on the deter-mination of leak rates of a specific gas, helium, which, at such pres-sures, is actually part of a mixture, so that it is the partial pressurewhich has to be determined, and either ion gauges or mass spec-trometers are used to this aim. It can be highlighted that Tison’stechnique is time-consuming (6–8 h for a single test) with respectthe other techniques reviewed and its use in microfluidics can beconsidered useful only for specific tests in which the gas mass flowrates are very low (<10�15 kg/s).

To summarize the main observations made in section for eachtechnique, Table 2 shows the most important methods proposedfor the fluid-dynamic investigation in microchannels.

For each technique, the typical ranges of values for which theyhave been applied together with the typical values of uncertainty

crofluidic applications.

Typical uncertainty References

0.6%, 1%, 2% Morini et al. [18], Celata et al. [16],Tang et al. [15]

9.64% (calculated) Celata et al. [16]of 3–4% Shih et al. [62], Ewart et al. [63],

Colin et al. [64]4.5% Ewart et al. [65], Pitakarnnop et al.

[67]– Arkilic et al. [66,68]1–8% Tison [69]

0.13% Asako et al. [7]1.25–2% Zohar et al. [17]2% Morini et al. [18]0.3–2.98% Tang et al. [15]2.48–3.67% Celata et al. [16]

0.17%, 0.25% Tang et al. [15], Morini et al. [18]

Systematic error reduced byaround 0.8 K

Baviere et al. [52]

Inlet and outlet effectsminimized

Demsis et al. [21]

Sensitivity: 0.1 �C Mosyak et al. [54]1.1–1.5 �C Muwanga et al. [53]Sensitivity. 1% �C�1 Liu et al. [28]

0.25%, 0.5% Tang et al. [15], Morini et al. [18],Asako et al. [7]

Minor losses minimized Asako et al. [7], Celata et al. [16]1 mbar Huang et al. [27], Liu et al. [28]– Matsuda et al. [32,33]1% Zohar et al. [17], Baviere et al. [8]2.4–13.3% Kohl et al. [25]


are shown. The data in Table 2 may be employed during the designof a new test rig in order to select the best measurement tech-niques with the goal to minimize the global uncertainty on the de-rived quantities, as shown in the next section.

4. Uncertainty analysis

Eq. (13) highlights that the friction factor is a function ofmany parameters. Eqs. (4) and (5) link the friction factor and themeasurable quantities. The friction factor is a general functionf ¼ f ðx1; x2; . . . ; xnÞ in which the single measurable variable xi ischaracterized by a known absolute uncertainty dxi which corre-sponds to the relative uncertainty dxi

xi. Based on the theory of

uncertainty propagation, the absolute uncertainty on the frictionfactor is given by [70]:

df ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

@f@xi

� �2

ðdxiÞ2 þ 2Xn

i¼1

Xn

j¼iþ1

@f@xi

@f@xj

covðxi; xjÞ

sð21Þ

where covðxi; xjÞ is the covariance associated with xi and xj. If xi andxj are independent, their covariance is 0; if the two variables arecorrelated, their covariance will not be 0. However, in experimentthe singly measured variables are independent of each other, whichgreatly reduces the complexity of calculation. Even with such sim-plification, the measurable variables might be correlated by theinstrumentation used in the experimental setup (i.e. the use ofthe same pressure sensor or temperature sensor and so on) and/or by the room conditions (via data acquisition system). However,this kind of correlation is relatively weak (i.e. Agilent 34420A digitalmultimeter for data acquisition presents a slight sensitivity of themagnitude of 0.0008% with regard to room temperature) and canbe either negative or positive, which tends to balance out the totalcontribution to the overall uncertainty of the function f. For this rea-son, in the following discussion the measurable variables will beconsidered independent to each to other, and Eq. (21) can be simplyexpressed as follows for the total relative uncertainty of f:

dff¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@f@x1

x1

f

� �2 dx1

x1

� �2

þ @f@x2

x2

f

� �2 dx2

x2

� �2

� � �þ @f@xn

xn

f

� �2 dxn

xn

� �2s

ð22Þ

For flow in microtubes, by considering all the directly measurablequantities (inner diameter, channel length, temperature, pressuredrop, outlet pressure and mass flow rate) in the determination ofthe value of friction factor, the relative uncertainty of friction factoris given by:

dff¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1dDh

Dh

� �2

þk22

dLL

� �2

þk23

dTT

� �2

þk24

dDpp

� �2

þk25

dpout

pout

� �2

þk26

d _m_m

� �2s

ð23Þ

Table 3Sensitivity coefficients of operative parameters in the calculation of friction factor.

Sensitivitycoefficients

In-compressibleflow

Compressible flow(negligible aa)

Compressible floStrategy #1

k1 (diameter) 5 5 5 + 4b ln(1 + a)k2 (length) 1 1 1k3 (temperature) 0 1 1 + b ln(1 + a)k4 (pressure

drop)1 1 1þ a

2þa� b a1þaþ

k5 (outletpressure)

0 1 11þa

2þ b 2

2þa lnð1þh

k6 (mass flowrate)

2 2 2 + 2b ln (1 + a)

a a = Dp/pout; b = 2Dh/(fL).

where the sensitivity coefficients, k1, k2,. . ., k6, which show how theuncertainty of each measurable parameter contributes quantita-tively to the uncertainty of friction factor, are listed in Table 3 forboth incompressible and compressible flow. The analytical expres-sion of these coefficients, reported in Table 3, can be calculated byEqs. (22) and (23) for which:

ki ¼@f@xi

xi

fð24Þ

and depends on the flow conditions in the experimental test as wellon the measurement strategies adopted (i.e. the use of two absolutepressure sensors at the inlet and at the outlet of the microchannelor the use of one differential pressure sensor with one absolutepressure sensor and so on).

Eq. (23) highlights that the relative uncertainty of the frictionfactor is more sensitive to the measured variables which are asso-ciated to larger sensitivity coefficients and the largest one domi-nates the total relative uncertainty. Specifically, the accuracy ofmicrochannel diameter measurement has the greatest weight inthe determination of the friction factor uncertainty, followed bymass flow rate and pressure (for compressible flows) measure-ment, while the accuracy of length and temperature measurementis comparatively less important. This analysis suggests that foraccurate calculations of the friction factor the measurement ofthe channel diameter is of crucial importance: even a small errormay induce a large error in the calculated quantity.

When compressibility effects become important and for verylow mass flow rates (<10�8 kg/s), uncertainty in flow rate measure-ments becomes large; as explained in the previous section, in thesecases the flow rate is not directly measured but it is calculated withEqs. (17)–(20) as a function of temperature, pressure and volumeof the system: this increases the influence of such quantities mea-surement on the total uncertainty of the friction factor.

It can also be noted from Table 3 that, for incompressible flowsand compressible flows with low pressure drop, the uncertainty offriction factor can be directly calculated from the error of eachmeasurement, without knowing the absolute values of the mea-sured parameters. On the contrary, for compressible flows withcomparatively large pressure drops (which is usually the case inmicrochannels), the friction factor uncertainty depends on theabsolute values of the following experimental parameters:

a ¼ Dppout

; b ¼ 2Dh

fLð25Þ

where Dp is the pressure drop along the whole microchannel, pout isthe outlet pressure, f is the friction factor and L in the total length ofthe microchannel.

For compressible flows different strategies can be used in orderto determine the friction factor depending on the type of pressuresensors involved in the measurement.

w (non-negligible aa) Compressible flow (inlet and outlet pressuremeasured) Strategy #2

5 + 4b ln(1 + a)11 + b ln(1 + a)

b 1þ a2þa

� �lnð1þ aÞ 2þ 2

a2þ2a� bþ 2b 1þ 1a2þ2a

� �lnð1þ aÞ (for pin instead

of Dp)

aÞ þ a1þa

i2

a2þ2a ½1þ b lnð1þ aÞ� � b

2 + 2b ln (1 + a)

0 500 1000 1500 2000 25000

5

10

15

20

25

30

Re

unce

rtain

ty o

f fric

tion

fact

or (%

)

total uncertaintypressure drop, FS ±0.5%mass flow rate, FS ±0.5%

Fig. 5. Contribution of the differential pressure sensor and of the mass flow ratesensor having an uncertainty equal to ±0.5% of FS (Re = 2300) on friction factoruncertainty.


For example, one can measure:

� the total pressure drop due to the microtube by using a differ-ential pressure sensor together with the determination of theabsolute value of the outlet pressure (indicated as Strategy #1in Table 3) or� the absolute value of the pressure at the inlet and at the outlet

of the microtube (indicated as Strategy #2 in Table 3).

Of course, other combinations of pressure sensors are possible.Each different strategy causes some difference in the expression

of friction factor and thus changes the sensitivity coefficients ofeach measurement (Eq. (24)), as shown in Table 3.

In order to give some numerical examples, the influence ofsensitivity coefficients of the six measurable quantities (inner diam-eter, microtube length, temperature, total pressure drop, outletpressure, mass flow rate) on the uncertainty of the friction factoris investigated quantitatively with a set of error-free data generatedfrom gas flow in a 100 lm microtube with a length of 30 mm. Theflow is assumed to be compressible, isothermal and in the laminarregime, and the gas is discharged to the atmosphere. The typicalvalue of uncertainty for each measurement can be chosen by usingthe data in Table 2; we have assumed an uncertainty of ±2% for innerdiameter measurement, ±0.3% for length measurement, ±0.25% fortemperature measurement, ±0.5% for pressure measurement and±2% for mass flow rate measurement.

Fig. 4 shows the influence of each measurement on the uncer-tainty of friction factor. It can be seen that the accuracy of frictionfactor is most sensitive to the measurement of inner diameter,which results in an uncertainty of more than 10% on the frictionfactor. This contribution grows gradually with the increase ofReynolds number. The importance of mass flow rate measurementtakes the second place and a 2% deviation results in an error of4–5.5% in friction factor. Compared with diameter and mass flowrate, the influence of other quantities (pressure drop, outlet pres-sure, length and temperature) on the accuracy of friction factor issmall.

In practice, the uncertainty evaluation is more complicated thanthe previous simulation, which based on assumption of constantvalues for the relative uncertainty of each measured quantity. Dur-ing a specific test round for isothermal flow, four out of the sixquantities to be measured (inner diameter, microtube length, tem-perature, total pressure drop, outlet pressure, mass flow rate) al-most remain constant when the imposed Reynolds number ischanged, namely inner diameter, length, temperature and outlet

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

Re

unce

rtain

ty o

f fric

tion

fact

or (%

)

diameter, ±2%length, ±0.3%temperature, ±0.25%pressure drop, ±0.5%outlet pressure, ±0.5%mass flow rate, ±2%

Fig. 4. Influence of each measured parameter with typical uncertainty for amicrotube having an inner diameter of 100 lm and a length of 30 mm.

pressure (here it is assumed that the gas is discharged directly intothe atmosphere). On the other hand the value of the pressure dropand the mass flow rate increase greatly in microchannels when theReynolds number increases. This causes a change in the relativeuncertainty of these two quantities, which tends to be larger whenthe pressure drop and mass flow rates are farther from the fullscale (FS) value of the instrument.

To further explore this effect, we assume the relative uncer-tainty for pressure drop and mass flow rate is ±0.5% of the full scale(FS) reached when the Reynolds number is 2300.

Fig. 5 shows the single contribution of the uncertainty of thesetwo parameters to the uncertainty on friction factor. The total fric-tion factor uncertainty is calculated based on the changing uncer-tainty of pressure drop and mass flow rate, as well as the constantrelative uncertainty of inner diameter (±2%), length (±0.3%), outletpressure (±0.5%) and temperature (±0.25%). It can be seen that forlow Reynolds number (<500), or more generally when the mea-sured values are far from the full scale, the uncertainty of frictionfactor becomes very large and sometimes this leads to unreason-able results in Microfluidics experiments.

In order to control the uncertainty during experimental testsover a wide range of Reynolds numbers it becomes mandatory touse a series of devices with different full scale (FS) values.

As shown in Fig. 6, if two pressure meters and two flow rate me-ters are used having their FS at Re = 500 and at Re = 2300 when the

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

Re

unce

rtain

ty o

f fric

tion

fact

or (%

)

total uncertaintypressure drop, FS ±0.5%mass flow rate, FS ±0.5%

Fig. 6. Effect of the use of two differential pressure sensors and mass flow ratesensors having different FS (Re = 500 and Re = 2300) on friction factor uncertainty.


Reynolds number becomes lower than 500 the measurement canbe switched to the set of devices having their FS at Re = 500 andthis can greatly reduce the overall uncertainty of friction factorat lower Reynolds number.

By comparing the data of Fig. 6 with those of Fig. 5 it is evidentthat using more devices having different full scale values it is pos-sible to expand the range of the Reynolds number in which the fric-tion factor uncertainty stays below ±14%.

0 500 1000 1500 2000 2500-1

0

1

2

3

4

5

6

Re

sens

itivi

ty c

oeffi

cien

t

pressure dropoutlet pressure (corresponds to pressure drop)inlet pressureoutlet pressure (corresponds to inlet pressure)

Fig. 7. Comparison of the influence of different strategies for the pressuremeasurement on the sensitivity coefficients of the inlet, outlet and differentialpressure (sensors with a fixed uncertainty (±0.5%)).

Table 4Declared uncertainties of quantities measured in experiments with liquid flows in percen

Authors Geometry

d (%) (circular) h, w (%) (rectangular) L (%)

Li et al. [73] 2 – 0.1Liu and Garimella [74] – – 3.9Wu and Cheng [75] 1.83 – 0.37Mala and Li [76] 2 – 0.2Judy et al. [77] 2.5 2.5–5 –Chen et al. [78] – – –Celata et al. [16] – – –Peng et al. [79,80] – – –

Table 5Declared uncertainties of quantities measured in experiment with gas flows in percentage

Authors Geometry Pressure(%)

d (%) (circular) h, w (%)(rectangular)

L (%)

Ewart et al. [63] 1.39 – – 1.5

Ewart et al. [65] – h: 2.13; w: 0.20 1.06 0.5Celata et al. [16] 2.48–3.67 – – –

Asako et al. [7] 0.13 – 0.003 0.42Zohar et al. [17] – h: 2; w: 1.25 – <1Hsieh et al. [22] – h: 1; w: 1 – 0.7Maurer et al.

[81]– h: 2; w: 1 1 2

Morini et al. [18] 2 – 0.3 0.5Araki et al. [82] triangle and trapezoid

Dh: 3.230.2 5.13 –

Turner et al. [83] – – – –Tang et al. [15] 0.3–2.98 0.02–0.07 0.25 0.17Colin et al. [64] – h: 2.23–18.5 w:

0.58–1.40.2 –

Pitakarnnopet al. [67]

– h: 5.3 w: 1.4 0.2 0.5

As shown in Table 3, using pressure sensors of identical uncer-tainty but different strategies of pressure measurement may resultin a different accuracy for friction factor too. In this sense, an ‘‘apriori’’ uncertainty analysis can become an important design toolfor microfluidics experiences as underlined in [71].

Fig. 7 indicates that if the inlet and outlet pressure are mea-sured for the determination of the friction factor for a compressiblegas microflow (Strategy #2), the sensitivity coefficients for frictionfactor will be very large at lower Reynolds number compared withthe measurement of pressure drop and outlet pressure. It is evidentthat for Reynolds number larger than 500, it is more advisable tochoose an absolute pressure sensor with a higher accuracy forthe inlet section in order to reduce the total uncertainty on frictionfactor.

On the contrary, if the pressure drop and outlet pressure aremeasured (Strategy #1), it is more beneficial to use a more accu-rate differential pressure transducer in order to obtain a low totaluncertainty.

Uncertainty analysis can also be used to check the validity ofthe results published in the literature: sometimes discrepanciesfrom the predictions of the theory can be traced back to the useof inaccurate instrumentation for the measurements. A list ofexperimental papers has been examined in order to check the levelof uncertainty associated to the measured values of the friction fac-tors for liquids and gases.

In Table 4 the declared uncertainties on the friction factors re-ported by Ferguson et al. [72] for experiments with liquid flowsare summarized in order to have a reference for a comparison withthe values described in this paper for gas flows. It is interesting to

tage (from Ferguson et al. [72]).

Pressure (%) Mass flow rate (%) Re (%) Friction factor (%)

1.5 2 – –0.25 1.01 10.5 11.80.68 1.69 – –2 2 3 9.20.25 – – –2 5 10.5 5.4– 7 5 71.5 2.5 8 10

.

Temperature(%)

Mass flow rate (%) Re(%)

Friction factor (%)

0.02 4.5 (const. volume); 4.2 (droptracking)

– –

0.02 4.5 – –– – – 19 (flow meter);27

(pipette)– 0.98 – 12.5<1 <8.5 – –0.04 4.0 1.46 1.82– 2 – –

0.25 0.5–0.6 3 102.2 – 10.9

– 4.9 4.82 4.5 5.9– – – –

0.2 4 (const. volume); 3.1 (droptracking)

– –

Table 6Typical values of parameters in experiments with gas flows.

Authors Geometry and size (lm) Length (mm) Pressure (kPa) T (K) Mass flow rate (kg/s) Re

Inlet Outlet

Ewart et al. [63] Circ. d: 25.2 lm 53 1.22–12.11 0.25–2.47 296.5 0.02–2 � 10�10 0.0018–2.5Ewart et al. [65] Rect. h: 9.38; w: 492 9.4 0.060–115.47 0.012–32.65 – 0.0049–22.5 � 10�10 –Celata et al. [16] Circ. d: 30–254 50–91 Up to 1000 100 – – 0.8–500Asako et al. [7] Circ. d: 150 30–50 300 190 300 4 � 10�6 1508–2188Zohar et al. [17] Rect. h: 0.5 and 1; w: 40 4 Up to 400 100 – 1.5–6 � 10�9 –Hsieh et al. [22] Rect. h: 50; w: 200 24 2.7–64.63 0.68–13.19 300 0.88 � 10�8–40.9 � 10�8 2.6–89.4Maurer et al. [81] Rect. h: 1.14; w: 200 10 140–500 48–100 296 6 � 10�12�5 � 10�10 0.001–0.07Morini et al. [18] Circ. d: 133–730 200–1000 – – – – 100–5000Araki et al. [82] Triangular and trapezoidal Dh: 3.92–10.3 15 and 25 – 100 295 0.2 � 10�10�1.0 � 10�10 0.042–4.19Tang et al. [15] Circular and rectangular Dh: 10–300 27.5–100 – 100 295 – 3–6200Colin et al. [64] Rect. h: 0.5–4.5; w: 20, 50 5 – 65–200 294 4 � 10�13�2 � 10�9 –Pitakarnnop et al. [67] Rect. h: 1.88; w: 21.2 5 Up to 300 2–50 298.5 8 � 10�14�2 � 10�11 –


note that the typical values of the friction factor uncertainties varyfrom ±5.4% to ±11.8% and, in many papers, the most importantuncertainty, related to the inner diameter, was not declared. Inthese works the flow was adiabatic and in many cases the temper-ature of the test flow was not checked.

Table 5 reports a collection of declared uncertainties for exper-iments with gases. Comparing the data of Table 4 with those of Ta-ble 5 it is seen that for gas flows the uncertainty on the frictionfactor can be larger than ±12% due to the smaller dimensions ofthe tested microchannels (less than 150 lm generally) and to thelarger uncertainties on mass flow rate measurement. In fact, verylow flow rates and low values of pressure are used in these exper-iments in order to obtain large values of the Knudsen number(large rarefaction effects).

By observing the total uncertainties of the friction factor of Ta-ble 5 it is interesting to note that in many experimental works thedata about the uncertainty analysis are incomplete and it is impos-sible to know the real value of the total uncertainty on the exper-imental friction factors.

The typical values of temperature, pressure and mass flow ratefor these works are given in Table 6, together with the range ofReynolds numbers investigated.

It is interesting to highlight that in the experimental works inwhich the tested Reynolds numbers were larger than 10 only com-pressibility effects were evidenced and analysed; on the contrary,in order to study the rarefaction effects on the gas microflowsthe Reynolds number must be less than 10 (very low mass flowrates). In fact, in order to study the gas rarefaction effects smallchannels (usually less than 20 lm) and low values of pressure (lessthan 0.1 bar) must be used. Under these conditions the values ofthe mass flow rate are very low (10�6�10�14 kg/s) and specifictechniques like the droplet tracking method or the gas accumula-tion method have to be employed to calculate the Reynolds num-ber accurately. In the gas accumulation technique experimentsmust be devised so as to keep the temperature of the tank fixed,which may lead to difficulties in the design of the test rig.

5. Conclusions

The need for the development of specific measurement tech-niques for the field of Microfluidics is nowadays very high. In thispaper a critical review of different approaches for the measure-ment of the main operative parameters useful to determine thefriction factors and pressure drops through microchannels (suchas microchannel sizes, pressure, temperature and gas flow rate),is presented highlighting both the advantages and possible prob-lems one may encounter with each specific measurement ap-proach. This offers advice for both researchers and engineers tochoose a suitable method based on the accuracy to be achieved,

ranges of parameters to be measured, temperature under opera-tion, materials and sizes of microchannel, type of working fluid,etc. It has been shown in the paper how the propagation of uncer-tainty can be used as a powerful design tool for microfluidicexperiences.

It has been demonstrated that the uncertainty of each quantitydirectly measured in experiments has a quite different influence indetermining the uncertainty of friction factor: accordingly, oneshould pay greatest care to the accuracy of diameter measurement,followed by that of mass flow rate and pressure measurement. Inthe paper the uncertainties and typical values of various parame-ters in the work of different researchers are compared, which pro-vides useful references for future research in the field of theanalysis of the gas flow through microdevices. Of course, these datacan be useful also for the design of new experiment test rigs andfor the selection of the more appropriate devices in order to obtaina controlled uncertainty of the data and meaningful results.

As result of this critical review the following main conclusionscan be drawn:

� About the estimation of the average inner diameter of a micro-channel: the method of accumulation of heavy liquid proposedby Asako et al. [7] seems to be a good one in order to reduce theuncertainty accompanying this measurement (<0.2%). Thistechnique is based on the assumption that for liquid flows inlaminar regime the Poiseuille law holds also in microchannels.However, for very small hydraulic diameters this techniquerequires very long time for the liquid to accumulate; whenwater is used as test fluid, care must be used in order to esti-mate the mass lost due to evaporation during the experimentaltests, which strongly depends on the environmental conditions.� About the estimation of the flow rate: commercial thermal mass

flow sensors can be used down to 10�8 kg/s (0.1 Nml/min).When the mass flow rate is very low (<10�8 kg/s) the bestmethod is the single-tank constant volume method proposedby Ewart et al. [65] and, more recently, the same method withdouble measurement and precise temperature regulation byPitakarnnop et al. [67]. During the measurement, the environ-mental conditions must be accurately checked in order to main-tain constant temperature.� About the determination of the pressure in a microchannel: it is

possible to reconstruct the pressure distribution along a micro-channel but this requires specific techniques and sensors (Pres-sure-Sensitive Paints, microstrain gauge membrane, opticallever) which need a specific calibration and sometimes a spe-cific test section. If one is only interested in the total pressuredrop along a microtube, usual pressure sensors can be used.These devices can be employed with low levels of relativeuncertainty (<±0.5%) by placing them in the plenums at the inlet


and outlet of the microchannel. However, in this case the cor-rect evaluation of the minor losses at the ends of the microtubebecomes a crucial point. In addition, for compressible andincompressible flows, different strategies of pressure measure-ments (i.e. the combinations of absolute and differential pres-sure sensors) may result in a different accuracy of the frictionfactor.� About the temperature: thermocouples, thermal liquid crystals

or infrared thermography can be used to estimate the surfacetemperature of the microdevices but, in general, the use of clas-sical Pt-RTD sensors and thermocouples is usually recom-mended in order to determine the temperature in microfluidicdevices. However, it has been shown in the paper that theimprovement in the microfabrication techniques has conductedto the realization of specific lRTD, TFTC and SC microsensorsand their integration in microchannels has been demonstratedto be very promising for the analysis of gas dynamics inmicrochannels.� About the determination of the local velocity distribution: the

velocimetry techniques proposed for liquid flows in microchan-nels present strong limitations if applied to the analysis of gasflows. For the determination of the local velocity of a gas in amicrochannel two techniques have been proposed: the micro-Particle Image Velocimetry (lPIV) and the Molecular-TaggingVelocimetry (MTV). For lPIV, the accuracy of the gas-phase flowvelocity measurement is strongly dependent on the characteris-tics of the tracer particles; it has been demonstrated that for gasflows through microchannels the seeding (water or fluorescentoil) must be characterized by diameters below 1 lm in order toclosely follow the flow especially in regions of rapid decelera-tion (bends). This fact can determine strong problems for thegeneration of the tracer particles especially for low gas flowrates. On the other hand, the application of MTV to the analysisof gas flows through microchannels can be considered only inits preliminary phase. The determination of the gas local veloc-ity in microdevices still remains an important open question.

Acknowledgement

The research leading to these results has received funding fromthe European Community’s Seventh Framework Programme (ITN –FP7/2007-2013) under grant agreement no. 215504.

References

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Morini 2011 Experimental Thermal and Fluid Science

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