Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2003 J. Micromech. Microeng. 13 568 (http://iopscience.iop.org/0960-1317/13/5/307) Download details: IP Address: 146.201.208.22 The article was downloaded on 16/05/2013 at 08:11 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Liquid flow in microchannels: experimental observations and computational analyses of
microfluidics effects
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2003 J. Micromech. Microeng. 13 568
(http://iopscience.iop.org/0960-1317/13/5/307)
Download details:
IP Address: 146.201.208.22
The article was downloaded on 16/05/2013 at 08:11
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Received 23 December 2002, in final form 21 March 2003Published 14 May 2003Online at stacks.iop.org/JMM/13/568
AbstractExperimental observations of liquid microchannel flows are reviewed andresults of computer experiments concerning channel entrance, wall slip,non-Newtonian fluid, surface roughness, viscous dissipation and turbulenceeffects on the friction factor are discussed. The experimental findings areclassified into three groups. Group I emphasizes ‘flow instabilities’ andgroup II points out ‘viscosity changes’ as the causes of deviations from theconventional flow theory for macrochannels. Group III caters to studies thatdid not detect any measurable differences between micro- and macroscalefluid flow behaviors. Based on numerical friction factor analyses, theentrance effect should be taken into account for any microfluidic system.It is a function of channel length, aspect ratio and the Reynolds number.Non-Newtonian fluid flow effects are expected to be important forpolymeric liquids and particle suspension flows. The wall slip effect isnegligible for liquid flows in microconduits. Significant surface roughnesseffects are a function of the Darcy number, the Reynolds number andcross-sectional configurations. For relatively low Reynolds numbers,Re < 2000, onset to turbulence has to be considered important because ofpossible geometric non-uniformities, e.g., a contraction and/or bend at theinlet to the microchannel. Channel-size effect on viscous dissipation turnsout to be important for conduits with Dh < 100 µm.
1. Introduction
The compactness and high surface-to-volume ratios ofmicroscale fluid devices make them attractive alternatives toconventional flow systems for heat transfer augmentation,chemical reactor or combustor miniaturization, aerospacetechnology implementations and biomedical applications,such as drug delivery, DNA sequencing and bio-MEMS, toname a few.
While experimental evidence indicates that fluid flow inmicrochannels, especially in terms of wall friction, differs frommacrochannel flow behavior, laboratory observations are ofteninconsistent and contradictory. Thus, theoretical microfluidics
investigations are necessary; however, for any computationalanalysis of microchannel flow, some pertinent questions haveto be considered (Gad-el-Hak 1999, Pfhaler et al 1990): is thecontinuum assumption still valid? For example, the gas flowKnudsen number, Kn = λ/L, should be very small, i.e.,Kn < 0.1, for the Navier–Stokes equations to hold.Kleinstreuer (2003) discussed alternative Knudsen numbersfor liquid and two-phase flows. Are entrance effects persuasivein light of the typically short channel length? Microchannelinlet configurations, flow inlet conditions, flow developmentsand viscous dissipation have to be considered as ‘entranceeffects’. Is the onset of instabilities affected by the smallsize of the conduits? For example, complex inlet geometries,
Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects
significant wall roughness as well as interactions of largefluid molecules and microparticle suspensions may trigger ormodulate local turbulence.
Theories have been well developed for gas flow inmicrochannels, and they have shown good agreementswith experimental observations (Gravesen et al 1993,Beskok and Karniadakis 1994, 1999, Harley et al 1995,Beskok et al 1996, Chen et al 1998). For example, Gravesenet al (1993) highlighted wall slip and compressibilityphenomena in microchannels as important causes fordeviations from macroscale gas flows. For liquid flows,Gravesen et al (1993) emphasized the importance of existingbubbles in a channel, the independence of viscosity from thecharacteristic dimensions, and the channel length effect. Ifbubbles are present, they increase the pressure drop, due tothe surface tension effect. Bubbles are hard to eliminatecompletely; they may entrain when first filling a channel.Gravesen et al (1993) found no microfluidic device that worksin the fully-developed (turbulent) regime, and many devicesare so short that the pressure drop is dominated by inertiallosses. Elwenspoek et al (1994) mentioned the entrance effectas a possible cause for the deviation of published experimentalresults from the conventional theory applied to macroscalesystems. They observed that contamination of liquids byparticles occurs very often, and their origin is quite unclear.Without filtering, they would have never obtained reproducibleresults, and if they reproduced, they were in accordance withtheoretical expectations. They also emphasized the effect ofbubbles trapped in the microchannels and the importance ofmixing in microscale devices. Ho and Tai (1998) pointed outthe necessity to re-examine the surface forces in the momentumequation. It was found that the frictional force between theirrotor and the substrate is a function of the contact area. Thisresult departs from the traditional friction law, which states thatthe friction force is linearly proportional to the normal forceonly. They considered this as an example of the importance ofsurface forces in microscale devices such as the van der Waalsforce, electrostatic force and steric force. They concluded thatan apparent viscosity would represent the integral effects ofthese surface forces.
In this paper, recent experimental liquid flows inmicrochannels are reviewed, and results of computerexperiments are discussed, focusing on entrance effects, wallslip, non-Newtonian fluid flow behavior, surface roughness,turbulence and viscous dissipation effects.
2. Experimental observations
The model configurations, laboratory conditions andexperimental findings for liquid flow in microchannels aresummarized in table 1.
The first two groups in table 1 list contributors whomeasured microfluidics deviations when compared withconventional, or macro-scale, theory. Group I emphasizespossible occurrence of ‘flow instabilities’ and group IIpoints out ‘viscosity changes’ as the causes of discrepanciesin friction factor values when compared to conventionaltheory. Group III caters to studies that did not detectany measurable differences between micro- and macroscalefluid flow behaviors. Specifically, the experimental findings
discussed by the individual research teams of groups I to IIIare as follows:
• Peng and Peterson (1996), Peng et al (1994) andXu et al (1999) observed increase/decrease of frictionfactor/pressure gradient compared to conventional theory,and they attributed this to an early onset of laminar-to-turbulent flow transition.
• Pfhaler et al (1990, 1991), Urbanek et al (1993), Mala andLi (1999), Qu et al (2000), Papautsky et al (1999), Renet al (2001) and Guo and Li (2003) assumed the deviationwould originate from surface phenomena such as surfaceroughness, electrokinetic forces, temperature effects andmicrocirculation near the wall. Most results indicate thatthe friction factor/pressure gradient would increase dueto effects of surface phenomena.
• Tuckerman and Pease (1981), Xu et al (2000), Sharp(2001) and Judy et al (2002) insisted that the frictionfactor/pressure gradient results are the same as the valuespredicted by conventional theory. They claimed thatthe deviations which other researchers had observed (seegroups I and II) might have originated from measurementerrors of channel size, experimental uncertainties andbecause of overlooking the entrance effect.
Thus, the use of the Navier–Stokes equations appears tobe appropriate for microchannel flows of liquids as long aslsystem � 0.1 µm for conduits filled with, say, water understandard conditions. This condition is based on the globalKnudsen number for liquids, Kn (=λIM/lsystem), which is verysmall, where the intermolecular length for water molecules isλIM = 3 A (Kleinstreuer 2003). Nevertheless, as the systemsize decreases, surface phenomena, such as near-wall forcesand relative roughness, become more and more important, andhence the Navier–Stokes equations have to be augmented asshown in the next sections.
If the water temperature rises from 300 to 310 K, the waterkinematic-viscosity decreases by 20%, which will result in a25% increase of the Reynolds number. Thus, fluid temperaturechanges in microchannels may affect the friction factor. Forexample, Pfhaler et al (1991) measured the friction factor ofisopropanol flow in microchannels ranging between 0.48 and40 µm in diameter. They observed a friction factor decrease,maximum of 25%, with channel diameter, attributing it toviscous dissipation. Contrary to such observations, Urbaneket al (1993) reported a friction factor increase with temperaturerise. Toh et al (2002) investigated numerically heat transferphenomena inside heated microchannels. They found thatthe heat input lowers the frictional losses, particularly at lowReynolds numbers (LRN). Judy et al (2002) observed liquidflow temperature rises (e.g., 6.2 K for isopropanol in a longsquare channel of 74.1 µm diameter for Re ≈ 300), and theyrelated this to viscous dissipation. They suggested that theviscosity change due to temperature changes should be takeninto account to estimate the friction factor.
3. Theory
3.1. Macro-scale friction factor equations
Based on the continuum mechanics assumption, the continuityequation and equation of motion (or the Navier–Stokes
569
JK
ooand
CK
leinstreuer
Table 1. Classification of experimental observations.
Group Reference Microchannel configurations Laboratory conditions Observed microeffects
I Peng et al (1994), Peng and Peterson Dh : 0.48–367 µm Liquid: water Flow Friction factor(1996), Mala and Li (1999), ReDh : 50–4000 Material: stainless/aluminum steel instability increaseXu et al (1999), Qu et al (2000) Rectangular, trapezoidal, substrate, fiberglass plastic cover effects or decrease
circular conduitsEnhanced surface
II Pfhaler et al (1990, 1991), Dh : 5–254 µm Liquid: iso-propanol, n-propanol, phenomena due Apparent FricitionYang and Li (1998), ReDh : 50–2500 1-, 2-propanol, 1-, 3-pentanol, KCl to increased viscosity factor increaseMala and Li (1999), Li (2001), Rectangular, trapezoidal, solution in water surface-to-volume ratio effectsPapautsky et al (1999), Qu et al (2000), circular, triangular conduits Material: stainless/Aluminum steelRen et al (2001), substrate, fiberglass plastic coverUrbanek et al (1993),Guo and Li (2003)
III Tuckerman and Pease (1981), Dh : 15–344 µm Liquid: deionized water, No deviation from → Same asElwenspoek et al (1994), ReDh : 8–4000 1-propanol, 20% solution of glycerol conventional theory macrochannelXu et al (2000), Rectangular, circular Material: fused silica, stainless flow dataSharp (2001), Judy et al (2002) conduits steel, silicon substrate, pylex cover
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Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects
equations for constant fluid property cases) are being solvedin the conventional theory approach. With the resultinginformation, the friction factors for various internal andexternal flow systems can be calculated.
The friction factor is defined as the ratio between the wallshear stress and the dynamic pressure, i.e.,
f = τw12ρu2
(1a)
where u is the mean fluid velocity. The friction factor definedin equation (1a) is called the Fanning friction factor, or skin-friction coefficient. There is another commonly used frictionfactor called the ‘Darcy friction factor’, which is four timesgreater than the Fanning friction factor (Kleinstreuer 1997):
fD = 8τw
ρu2= 4f. (1b)
Friction factor deviations can be expressed as ‘percentagechange’
�f ∗ = fobserved − fbase
fbase× 100% (1c)
where fbase (see equation (1a), (2), (3a) or (4)) is thetheoretical friction factor value and fobserved is the experimentalor computational microchannel value.
Friction factor relationships for turbulent fully-developedflow through macrochannels are typically expressed as afunction of the Reynolds number and relative roughnessheight. For example, Colebrook combined the surfaceroughness and Reynolds number effects in an interpolationformula, expressed as (White 1999)
1
f1/2D,turb
= −2.0 log
(ε/d
3.7+
2.51
ReDhf1/2D,turb
),
where ReDh = UDh
v(2)
where Dh is the hydraulic diameter.For laminar pipe flow, ReD < 2300, the friction factor is givenby
fD,lam = Cf
ReD
(3a)
or
fD,lamReD = Cf (3b)
where Cf = 64 for circular tubes, and for rectangular channels,Cf will have different values that are determined by the aspectratio b/a, knowing the hydraulic diameter (White 1991).
The friction factor in the entrance region of any conduit isalways larger than that for fully-developed flow; the differenceincreases with the Reynolds number for a given channel lengthand decreases as the channel length increases. Shah andLondon (1978) suggested an experimentally validated formulafor the laminar friction factor, which is valid within ±2.4% formany macro-scale duct shapes (White 1991):
fappRe ≈ 3.44√ζ
+fpRe + K∞/4ζ − 3.44/
√ζ
1 + c/ζ 2(4a)
with
Re = U0Dh
v(4b)
where fpRe = 14.23,K∞ = 1.43, c = 0.000 29 and ζ is aGraetz-type variable defined as ζ = (x/Dh)/ReDh forchannels, where x is the distance from the inlet.
3.2. Transport equations
The governing equation for a general flow model, which is anextension of the Navier–Stokes equations, can be written ingeneric form as (Kleinstreuer 2003)
∂
∂t(αρ) + ∇ · (ρ
��K · �u) − ∇ · (��K · ∇) = αS (5a)
where α is the porosity, ��K is an area porosity tensor, �u is thevelocity vector, ρ is the fluid density, is the fluid diffusivity(viscosity), S is a source term and is an arbitrary transportquantity.
For example, the corresponding continuity andmomentum equations, i.e., = 1, S = 0 and = �u, S =�S, respectively, are
∂
∂t(αρ) + ∇ · (ρ
��K · �u) = 0 (continuity equation) (5b)
∂
∂t(αρ �u) + ∇ · (ρ(
��K · �u) ⊗ �u) − ∇ · (η��K · (∇�u + ∇ �uT ))
= −α �S − α∇p (momentum equation). (5c)
For steady, incompressible and constant fluid property flowsin an isotropic medium
where p is the pressure, η is the effective viscosity and ve is theeffective kinematic viscosity, which is the sum of molecularkinematic viscosity and turbulent viscosity, and �S is the sourceterm which becomes the resistance vector for flow in a porousmedium layer (see section 4.4).
On the left-hand side of equation (5c), the first termis the time derivative or local time variation of momentum,the second term represents the spatial momentum convectionand the third term stands for viscous diffusion of momentum.On the right-hand side, the first term captures the augmentedsurface forces induced by the scale-down of the system, andthe last term is the pressure force acting on the fluid.
The low-Reynolds number k–ω model (Wilcox 1998,Zhang and Kleinstreuer 2003) is used to investigate turbulentflow effects. The governing equations are the following. LRNk–ω model:
ui
∂k
∂xj
= τij
∂ui
∂xj
− β∗k� +∂
∂xj
[(v + σkvT)
∂k
∂xj
](7a)
ui
∂ω
∂xj
= αω
kτij
∂ui
∂xj
− βω2 +∂
∂xj
[(v + σωvT)
∂ω
∂xj
](7b)
where the turbulent viscosity is given as vT = cµfµk/ω, andthe function fµ is defined as fµ = exp[−3.4/(1 + RT/50)2]with RT = ρκ/(µω).The other coefficients in the above equations are
Cµ = 0.09, α = 0.511, β = 0.8333,
β∗ = 1, σκ = σω = 0.5. (8)
The energy equation for steady, axisymmetric flow is given as
Figure 1. Comparison of friction factor deviations in a channelentrance (h/w = 1.0) based on experimental observations(equations (4)) and computer simulation results for variousReynolds numbers.
where ρ is the fluid density, Cp is the fluid specific heat, vi
are the velocity components, T is the fluid temperature, κ isthe fluid heat conductivity, µ is the fluid viscosity and q is thesource heat transfer rate; is defined as
={
2
[(∂vr
∂r
)2
+
(∂vx
∂x
)2]
+
(∂vr
∂x+
∂vx
∂r
)2}
. (10)
The equations are discretized using a central differencescheme except for the convective terms, which are discretizedby using a higher-order upwind scheme, and are integratedby means of a commercial code (AEA Technology 2001)enhanced with our Fortran programs for pre- and post-processing. For a simplified microchannel flow system,reduced ordinary differential equations (section 4.4) areintegrated using the modules provided in MATLAB 6(MathWorks 2002). In order to integrate the reduced partialdifferential equations (section 4.6), a finite-difference codewas developed. Results from computer experiments focusingon different effects potentially important in microchannel floware discussed in the next sections.
4. Analyses and results
4.1. Entrance effects
The microchannel effects influencing the friction factor andultimately the liquid flow behavior analyzed and discussedinclude channel entrance, fluid power law, wall slip, channelroughness and onset to turbulence. Starting with the frictionfactor in macrochannels, figure 1 shows a comparison betweenequation (4) and computer simulation results, i.e., a solutionto the reduced momentum equation (6a):
(�u∗ · ∇∗)�u∗ = −∇∗p∗ +1
ReDh
∇∗2 �u∗ (11)
where p∗ = p
ρU 20, �u∗ = �u
U0, x∗ = x
Dh, y∗ = y
Dh, ReDh
= ρU0Dh
µ,
and U0 is the inlet or average flow velocity. The boundary
L/Dh
f
100 200 300 400 5000.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
0.034
h/w = 0.0h/w = 0.2h/w = 0.5h/w = 1.0
Figure 2. Effect of height-to-width ratio on apparent friction factorfor Re = 1000.
conditions are uniform velocity at the inlet and constantpressure at the channel exit.
As the height-to-width ratio of a ‘channel’ changes froma square (1.0) to a parallel plate (0.0), the friction factorincreases for any given Reynolds number (figure 2). Thisimplies that square channels have the lowest friction factoramong the rectangular channel family for a given Reynoldsnumber. The reason is that the maximum centerline velocitydecreases with decreasing height-to-width ratio and hence thepressure gradient needed to accelerate the inviscid core isless.
4.2. Non-Newtonian fluid effects
In order to evaluate a non-Newtonian fluid effect on thefriction factor, equation (5c) is solved where τij = ηγij
with η = η0(γij )n−1 under the assumption of steady,
laminar, incompressible flow. The effect depends highlyon the exponent n of the power law where n = 1 impliesNewtonian fluid. By varying it within reasonable bounds, i.e.,0.9 � n � 1.1, the friction factor decreases by 50% andincreases by around 100% as shown in figure 3. Relativelylow-Reynolds-number flows are more affected than high-Reynolds-number flows.
Despite the dramatic impact of n, ordinary liquids inmicrochannels do not turn suddenly into non-Newtonianfluids. Sharp (2001) brought the possibility of the effect ofhigh shear rate on the change of viscosity into question, sayingthat high shear rates, as high as 10 000 s−1, do not cause waterto behave like a non-Newtonian fluid. The flow may only shownon-Newtonian behavior when there exist long chain polymersor when fine particle suspension are considered. However, ifliquids obey the power law, it greatly affects the friction factor.
4.3. Wall slip effects
From the numerical experiments for a 200 × 200 µm2 squarechannel and using Maxwell’s equation, uslip = l
(dudy
)y=0,
where l = 3 A for water, the maximum possible slip velocity
572
Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects
Exponent n of "Power Law"
f*=
f ob
serv
ed/f
bas
e
0.9 0.95 1 1.05 1.1
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
ReDh = 500ReDh = 1000ReDh = 2000
Figure 3. Effect of non-Newtonian fluid behavior on the frictionfactor.
is found to be about 14 µm s−1 where the average velocity is105 µm s−1, i.e., it is only 0.0014% of the average velocity.
Thus, in general, the effect of wall slip on the frictionfactor is negligible for liquid flows in microchannels.
4.4. Surface roughness effects
The effect of surface roughness on flow characteristics wasinvestigated by Mala and Li (1999) and Qu (2000) et al, whoproposed a ‘roughness-viscous’ model (RVM). It assumes thatthe roughness–viscosity is a function of y, the distance fromthe wall. The viscosity introduced by surface roughness has afinite value at the wall and is zero at the channel center, and isproportional to the Reynolds number. It should be noted thatthe RVM requires an experimentally determined coefficient tomatch system specific friction factors. Sharp (2001) claimedthat the RVM does not fully explain friction factor trends athigh Reynolds numbers referring to their experimental results.
In this study, surface roughness effects are investigatedwith a thin porous medium layer (PML) on the walls forwhich the Brinkman–Forchheimer extended Darcy equationholds. Thus, the effects of surface roughness on the frictionfactor, pressure gradient and flow structures in microchannelflow are elucidated. Generally, the PML results show thatsurface roughness may increase or decrease the friction factor,depending upon the roughness-equal-PML permeability, i.e.,if the Darcy number, Da = κ/D2
h, approaches infinity, thenthe PML resistance vanishes and when Da → 0, the PML actslike a solid coating, reducing the cross-sectional flow area.
The PML, or surface roughness, effects on the pressuregradient and friction factor are analyzed using a steady fully-developed flow model. Typical relative surface roughnessvalues, which depend on the wall material and machiningprocess, are listed in table 2. Generally, micromachiningresults in higher roughness than etching processes, andmicrochannels made of stainless steel have higher values ofsurface roughness than those made of silicon. A typicalroughness value for etching processes was reported to be20 nm (Xu et al 2000), which is negligible in most
Table 2. Typical values of relative surface roughness (h/Dh ×100%).
Relative surfaceAuthor roughness (%) Material
Pfhaler et al (1991) ∼1% SiliconPeng et al (1994) ∼0.6–1% SiliconMala and Li (1999) ∼3.5% Stainless steel/
fused silicaPapautsky et al (1999) ∼2% SiliconXu et al (1999, 2000) ∼1–1.7% Aluminum, silicon
microchannels where Dh > 10 µm. However, considering flowin channels with Dh � 10 µm, the effect of surface roughnessshould be taken into account.
The source term in the momentum equations (5c)simulating the resistance in the porous medium is the resistancevector, �R, which can be represented as (Bear 1972)
�R = (RC + RF|�u|β)�u (12)
where RC is a resistance constant, RF is the resistance speedfactor and β is a ‘resistance speed power’ (usually 1.0). Thequadratic drag term (∼RFu2) in equation (12) is dominantfor relatively high-Reynolds-number flows (Nield and Bejan1992).
For steady (∂/∂t = 0), 2D (∂/∂z = 0) and fully-developedflow (∂/∂x = 0) in an open channel as well as through anisotropic medium, equation (5c) reduces to the Brinkman–Forchheimer extended Darcy equation, which can be readilysolved using MATLAB. Specially, for the channel (or parallel-plate) case
0 = −dp∗
dx∗ +4
ReDh
d2u∗
dy∗2+
{− 4
DaH/2ReDh
− CF
Da1/2H/2
u∗2
}PML
(13)
where p∗ = p
ρU 20, u∗ = u
U0, x∗ = x
H/2 , y∗ = y
H/2 ,
DaH/2 = κ(H/2)2 , ReDh
= ρU02H
µ, H is the channel height,
and CF (≈0.55) is drag coefficient.For the tubular case:
0 = −dp∗
dx∗ +2
ReD
(d2u∗
dr∗2+
1
r∗du∗
dr∗
)
+
{− 2
DaRReD
− CF
Da1/2R
u∗2
}PML
(14)
where x∗ = xR
, r∗ = rR
,DaR = κR2 , ReD = ρU0D
µ, R is the tube
radius, and U0 is the average velocity. The relative roughnesslayer thickness was calculated based on the pipe diameter,i.e., D = 2R. The terms in braces {}PML are effective only inthe PML, i.e., the roughness region. Typical values of modelparameters are listed in table 3.
The boundary conditions are no-slip velocity (u = 0)at the wall, and velocity gradient at the interface, r∗ = ξ ,between open and porous regions, i.e., du∗
dr∗∣∣r∗=ξ
= ReD
2dp∗dx∗ ξ .
The interface velocity gradients should be obtained byiteration.
Figure 4 shows the Darcy number effect on the axialvelocity profile. When decreasing the Darcy number, themaximum velocity at the pipe center increases, while the
Fluid diffusivity For laminar flow, ≡ µRC Resistance constant RC = f (Da, Re)RF Resistance speed factor RF = f (Da, |�u|)β Resistance speed power β ≈ 1.0
velocity profiles near the wall flatten. As expected, the velocityprofile in the PML is a function of the Darcy number. It isfound that the effect of the Forchheimer term on the frictionfactor is much more significant in tubes when compared tothe parallel-plate case (see figure 5). This can be explainedby the fact that the surface area increases with the radius,attaining a maximum at the shell where the roughness layerlies. Increasing surface area causes an increase in the effects ofsurface phenomena, e.g., surface roughness effect in this study,while it is maintained to be constant for the parallel-plate case.
574
Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects
(a)
log10(Da)
Cha
nge
infr
ictio
nfa
cto
r(%
)
-7 -6 -5 -4 -3 -2 -1 0-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
Re=500Re=1000Re=2000
SR = 2.0 %
SR = 1.0 %
SR = 0.5 %
(b)
log10(DaR)
Cha
nge
infr
ictio
nfa
cto
r(%
)
-8 -6 -4 -2 0 2 4-4
-2
0
2
4
6
8
10
12 Without the Forchheimer termReD = 500ReD = 1000ReD = 2000
SR = 0.5 %
SR = 2.0 %
SR = 1.0 %
Figure 5. Effects of Reynolds number, surface roughness, Darcynumber and Forchheimer drag term on friction factor for(a) parallel-plate case and (b) circular tube case.
Figure 6 shows comparisons between the PML modelpredictions and selected experimental results. Specifically,the experimental results of Mala and Li (1999), which fallinto the region predicted by the PML model, indicate a strongf ∗(ReD) dependence (figure 6(a)). The experimental data setsof Guo and Li (2003) are well matched with those of thePML model (figure 6(b)). Clearly, roughness elements of the179.8 µm diameter tube have a higher Darcy number whencompared to the 128.8 µm diameter tube. The f ∗(ReD) datafor the 179.8 µm diameter tube seem to indicate the effect oflaminar-to-turbulent flow transition, when ReD > 1700. Theyclaimed that ‘the form drag resulting from the roughness isone reason leading to the increased friction factor’; the formdrag is captured by the Forchheimer term in the PML model.
4.5. Turbulence effects
In general, when designing flow systems, operation in thetransitional laminar-to-turbulent regions should be avoided
ReD
f* =f o
bse
rve/f
bas
ef* =
f ob
serv
e/f
bas
e
750 1000 1250 1500 17501.000
1.025
1.050
1.075
1.100
1.125
1.150
1.175
1.200
1.225
1.250
1.275
1.300
1.325
1.350
1.375
1.400
Mala & Li (1999) - 50 µm diameter casePML with Da=2x10-2
(a)
(b)
ReD
500 1000 1500 20000.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Guo & Li (2003) - D=179.8 µm caseGuo & Li (2003) - D=128.76 µm casePML with Da = 10-4
PML with Da = 10-2
Figure 6. Comparisons of porous medium layer (PML) modelpredictions with experimental data: (a) Mala and Li (1999) and(b) Guo and Li (2003).
because of highly non-linear flow instabilities and the difficultyof predicting flow characteristics.
In the 1990s (see table 1), different researchersobserved an increase in the friction factor for microchannels,and tried to explain the phenomenon by virtue of earlylaminar-to-turbulent flow transition (cf Peng et al 1994,Peng and Peterson 1996, Mala and Li 1999, Xu et al 1999,Qu et al 2000). Sharp (2001) performed bulk-flow resistanceexperiments and micro-PIV measurements to investigatethe validity of this ‘early laminar-to-turbulent transition’hypothesis. She did not detect any deviation from theconventional theory and concluded that the laminar-to-turbulent transition theory is invalid. However, the graphsin her dissertation, showing centerline velocities for variousReynolds numbers and microtube diameters, seem to indicateturbulence effects for Reynolds numbers around 1800. Onthe other hand, Judy et al (2002) measured the frictionfactor in microtubes and rectangular channels, and confirmedthat the friction factor can be predicted using laminartheory as long as ReDh � 2300. In contrast, Xu et al
575
J Koo and C Kleinstreuer
D0D1=2⋅D0
Flow inlet
Flow exit
Contraction
L/D0=510 L/D0=100
x*
Figure 7. Schematic diagram of the geometry used in computingthe effect of sudden contraction on laminar-to-turbulent flowtransition.
(2001) developed a model to predict friction factors forturbulent flows in microtubes using a one-equation model,which improved the accuracy for low-Reynolds-number flows(Wilcox 1998). They modified a constant, which controls themixing length increase with distance from the wall, to fit theirexperimental data. They claimed that microtubes smaller than130 µm show turbulent flow characteristics different frommacrochannels. By adjusting a constant in their turbulencemodel equation, they obtained turbulent velocity profiles formicrochannels smaller than 130 µm. However, their resultsshow a somewhat strange behavior in terms of the turbulencekinetic energy (TKE) distribution in radial direction (see theirfigures 9–14).
Zhang and Kleinstreuer (2003) performed a comparisonstudy to select a LRN turbulence model suitable for thehuman oral airways, which are composed of unique areaconstrictions and expansions. For medium inhalation rates(Q ≈ 30 l min−1), they observed a sudden TKE increase afterthe major constriction. They concluded that a modified LRNk–ω model (Wilcox 1998) is suitable for simulating laminar–transitional–turbulent flows in constricted tubes, such as thehuman upper airways, stenosed blood vessels, obstructedpipes or engine inlet ducts (cf Kleinstreuer and Zhang 2003).Similar non-uniform geometries can be found in manymicrochannels which, for example, have ‘sumps’ or 90◦
bends before their actual inlets. Such sudden contractionsand bends cause the flow to have strong radial velocitycomponents, which may increase the TKE and inducelaminar-to-turbulent transition at relatively low Reynoldsnumbers.
Considering a typical step contraction as given in figure 7,it is assumed that the incompressible Newtonian fluid flowis steady and axisymmetric values for a uniform velocity,uin, TKE, k, and pseudo-vorticity, ω, were prescribed atthe inlet. Specially, at the inlet (AEA Technology 2001,Zhang and Kleinstreuer 2003):
k = 1.5(I × uin)2 and ω = k0.5
0.3D(15)
where uin is the mean velocity (ReD = 1000), I is the turbulenceintensity, usually taken as 0.037, and D is the diameter of theinlet tube. The length of the intake part was determined forthe flow to be fully developed.
Table 4. Magnitude comparisons between convective, diffusive andviscous dissipation terms (µ = 8.67 × 10−4 (kg−1 ms−1), ρ =996 (kg m−3), κ = 0.611 (W−1 m K−1) and Cp = 4178 (J kg K)).
Diffusive DissipationTube radius (m) Convective term term term
Figure 8 shows cross-sectional views of various flowcharacteristics at different locations. The axial velocityprofile (figure 8(a)) is laminar and fully developed before thesudden contraction, and changes dramatically in the smallermicrotube where it eventually fully develops to a turbulentvelocity profile. The associated radial velocity component(figure 8(b)) is zero before the contraction and becomessignificant, especially near the wall at and after the contraction.The turbulence kinetic energy, which is set to be uniform at theinlet, is damped out and hence negligible before the contraction(figure 8(c)). It has a peak near the contraction and evolvesinto a near-uniform profile with a maximum between the flowcenter and the wall. The pressure gradient increases greatlyafter the contraction (figure 8(d )). The recirculating zone nearthe contraction causes a bump just after the contraction. The‘entrance’ length, starting at the contraction, is much shorterthan that for the laminar flow case.
4.6. Viscous dissipation effect
As is discussed in section 2, the fluid temperature effect on theflow field has been investigated mainly in conjunction with heattransfer problems. No detailed information has been reportedconcerning system-size effects on viscous dissipation and thefriction factor in microchannels.
If constant-property flow is hydrodynamically fullydeveloped in the x-direction and has no heat source, the non-dimensionalized energy equation reduces to
CpReDµ
RL(1 − r∗2)
∂[T ∗(Tm − Tw)]
∂x∗ = κ
R2r∗ (Tm − Tw)
× ∂
∂r∗
(r∗ ∂T ∗
∂r∗
)+ 4
µ3Re2D
ρ2
r∗2
R6(16)
where R is the radius of a tube, r∗ = rR
, ReD = ρUmeanD
µ,
T ∗ = T −TwTm−Tw
, Tw is the wall temperature, and Tm is the meantemperature. For a given fluid and flow condition, the viscousdissipation term changes inversely proportional to R6 while thediffusion and convection terms vary inversely proportional toR2 and R, respectively. In table 4, the relative magnitude ofeach term is listed for the case of water at 300 K and ReD =2000. The relative magnitude of the dissipation term increasesdrastically with a decrease in channel size. This can also beexplained with the Brinkman number, which is the ratio ofdissipation and heat diffusion.
576
Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects
(a)
r* (= r/D0)
r* (= r/D0)
u*(=
u/U
ave)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
x* = 500x* = 510x* = 510 - 522x* = 610
(b)
u* r(=
ur/U
ave)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
x* = 500x* = 510x* = 510 - 511x* = 610
(c)
r* (= r/D0)
TK
E/U
02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.002
0.004
0.006
0.008
0.01
0.012
0.014x* = 500x* = 510x* = 510 - 511x* = 610
(d)
x* (= x/D0)
P*(=
(P-P
atm
)/ρU
02)
450 550500 6000
0.5
1
1.5
2
2.5
3
3.5
Before contractionAfter Contraction
Contraction
Figure 8. Evolution of velocity components, turbulent kinetic energy, apparent viscosity and pressure in a microchannel (50% contraction,Re = 2000): (a) evolution of axial velocity profile, (b) evolution of radial velocity profile, (c) evolution of turbulent kinetic energy and(d ) pressure profile in axial direction.
Br = V 2µ
κ�T= µ3Re2
D
ρ2R2κ�T, (17a)
V = µReD
ρR(17b)
where V is a representative velocity and �T is a representativetemperature difference.
Adding the axial heat diffusion term back intoequation (16), the size effect on viscous dissipation in termsof a possible temperature rise in a microtube was investigated.The length of the tube was selected to be 800D, where thePrandtl number for water is Pr = 5.9. To pinpoint the lower andupper temperature limits due to viscous dissipation, isothermalwall as well as adiabatic wall conditions were considered.Specifically,
T = 300 K at x = 0 (18a)
∂T
∂r= 0 at r = 0 (18b)
T = 300 K at r = R
(for the constant wall temperature case) (18c)
∂T
∂r= 0 at r = R (for the adiabatic wall case) (18d)
∂
∂x
(T − Tw
Tm − Tw
)= 0
at the exit (for thermally fully-developed flows). (18e)
Figure 9 shows the tube-size effect on temperature rise for theRe = 2000 case. Clearly, temperature changes depend greatlyon the conduit size and the particular thermal wall condition.For example, in a 200 µm diameter tube, the temperature riseis negligible for both wall conditions. In a tube of diameterDh = 100 µm, the viscous dissipation effect on the frictionfactor is clearly measurable. Indeed, Pfhaler et al (1991)showed that the friction factor decreases for isopropanol flowin smaller channels. Regarding the size of their conduits (0.48–40 µm in diameter), the data seem to be a typical example ofthe influence of the viscous dissipation effect.
577
J Koo and C Kleinstreuer
X [m]
T[K
]
0.05 0.1 0.15299
300
301
302
303
304
305
20 µm40 µm74 µm200 µm
Tube diameter:
Note:
Tube lengths L = 800D
(a)
X [m]
T[K
]
0.05 0.1 0.15
300
305
310
315
320
325
20µm40 µm74 µm200 µm
Note:
Tube lengths L = 800D
Tube diameter:(b)
Figure 9. Effect of tube size on temperature change, i.e., viscous dissipation (Re = 2000): (a) constant wall temperature case and(b) adiabatic wall condition case.
5. Conclusions
In this paper, experimental observations concerning thefriction factor and flow behavior in microconduits are reviewedand classified into three groups. Group I emphasizes ‘flowinstabilities’ and group II points out ‘viscosity changes’ asthe causes of these discrepancies. Group III caters to studiesthat did not detect any measurable differences between micro-and macroscale fluid flow behaviors. Next, the various effectsassociated with microfluidics systems are analyzed by virtueof computer experiments.
The entrance effect should be taken into consideration formicrofluidic systems; it is a function of channel length, aspectratio and the Reynolds number. It becomes more importantfor the case of flow in short channels with high aspect ratiosat high–Reynolds-number conditions. Non-Newtonian fluideffects are expected to be important for polymeric liquids andparticle suspension flows. Wall slip effects are negligible forliquid flows in microconduits. Surface roughness effects area function of the Darcy number, the Reynolds number andcross-sectional configurations. Turbulence effects becomevery important for Reynolds numbers above 1000, mainly dueto (upstream) geometric non-uniformities. For example, theturbulence kinetic energy has a maximum value right after atypical contraction and may induce transition to turbulent flow.The peak value is a function of the geometry of the contraction(especially, the diameter ratio). Viscous dissipation effectson the friction factor are not negligible in a microconduit,especially for conduits with Dh < 100 µm.
In summary, conventional theory, selectively augmentedwith the LRN k–ω turbulence and/or our PML model, can
readily predict benchmark experimental results for liquid flowsin microconduits.
Acknowledgments
The authors want to express their thanks to Dr JosephP Archie and Sarah Archie for the endowment of theMcDonald-Kleinstreuer fellowship for Junemo Koo. Useof the software package CFX4 from AEA TechnologyEngineering Software, Inc. (Pittsburgh, PA) and access tothe SGI Origin 2400 at the North Carolina SupercomputingCenter (RTP, NC) are gratefully acknowledged.
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