microfluidics (colin/microfluidics) || single-phase heat transfer

Chapter 5

Single-phase Heat Transfer

5.1. Introduction

The computer industry increasingly needs efficient thermal management of electronic components and devices: as processor speeds increase circuit board power density increases as well by a factor two to three per decade. Thermal management is also a limiting factor in the development of efficient devices used in power electronics. Nowadays, conventional cooling devices have reached their maximum performances with regard to demand and new technologies require specific thermal systems to be designed that are appropriate for the scale of microdevices.

The idea of using micro heat exchangers advanced three decades ago. A large amount of heat can be evacuated through forced convection in microchannels micromachined directly over silicon wafers containing the microcircuit. Figure 5.1 shows such a micromachine. Today it is possible to use microchannels to evacuate power densities going from 200 to 400 W/cm2. The design and optimization of microscale heat exchangers requires us to understand the flow dynamics and heat transfer processes in very small channels. A vast amount of work has been done on this topic and reported in the literature, and the aim of this chapter is to provide a summary from a somewhat critical point of view.

We provide a short summary of scalar transport processes in conventional laminar and turbulent wall flows at the beginning of this chapter. Macroeffects have been confused with microeffects in some studies, at least in some pioneering investigations. We discuss this particular point in detail in section 5.3. A special

Chapter written by Sedat TARDU.

Microfluidics Edited by Stéphane Colin © 2010 ISTE Ltd. Published 2010 by ISTE Ltd.

196 Mic

section iliquid flo

5.2. Hea

5.2.1. Tr

This facilitatetextbookmechaniequation

ρ

where,

material where [5.1]. In density represen

∇

t

crofluidics

is devoted to ows are discus

Figure 5.1.

at transfer in

ransport of th

section briefle the reading ks such as: [Aism in a contn of enthalpy

ˆD 1ˆˆD

Ht RePr

ρ =

is the vecto

derivative tha stands for tin the rest of and conducti

nts the dissip

H

∇

t

gas microflossed in section

Microchannels(image cou

channels of c

e enthalpy

ly reviews claof the rest

ARP 84, BEJ 8tinuous flow f

that reads:

ˆ ˆ Eck Tr Re

∇ ⋅ ∇ +

or gradient op

at regroups ththe time. The f the chapter, ivity of the ation that re

H

ows (section 5ns 5.5 and 5.6

s obtained by silurtesy of IMEP

conventional

assic heat tranof the chapte

84, ECK 72, Hfield is relate

ˆDˆˆD

c pEce t

Φ +

erator, and

he inertial termnon-dimensio

is the presfluid, and sults from th

DD

pT

5.4). Some sp6.

licon DRIE micP-Grenoble)

sizes

nsport phenomer (details canHIN 75, SCH ed to the non-

ms related to onal quantitiesure, and is the tem

he work don

DD

Ut t

∂∂

= + ⋅∇

ρT

pecific scale e

cromachining

mena in this sn be found i79]). The hea-dimensional

the flow fieldes are denoted

are respectmperature. Fun

e against the

iUt∂∂ ∂

∇= +

k

effects in

section to in classic at transfer

transport

[5.1]

is the

d , and d by in tively the nction e viscous

ix∂∂

Uq

Φ

Single-phase Heat Transfer 197

stresses. It is given by in tensorial notations for a Newtonian fluid

with dynamic viscosity .

Important non-dimensional numbers appear in [5.1], namely:

– Reynolds number: Uref ref

ref

LRe =

ν;

– Prandtl number: ref

ref

Prνα

= ;

– Eckert number: .

The index “ref” correspond to reference values used in the scaling of the corresponding quantities, such as the reference kinematic viscosity and diffusivity . The reference temperature is usually taken as a difference (for example between the wall temperature and bulk temperature in an internal flow or temperature T∞ at infinity in a boundary layer). The left-hand side of the transport equation [5.1] is the classic inertia term, while the first term on the right-hand side is the molecular diffusion. The compressibility effects represented by the last term of [5.1] are directly proportional to Ec, while large Ec (important compressibility effects) and/or small Re (slow viscous Stokes-like flows) may result in preponderant viscous dissipation that consequently cannot be negligible. The Eckert number is directly related to the Mach number that is the ratio between the

reference velocity and reference sound velocity , i.e.: . We can,

indeed show that:

[5.2]

for an ideal gas. Parameter γ stands for the ratio between the specific heat coefficients under constant pressure cp and constant volume cv. We will use these relationships later in the section devoted to gas flows in microchannels. The transport equation [5.1] reduces to:

ji

j i

UUx x

∂∂μ∂ ∂

Φ=

μ

ref

ref

UEc

H=

refν

refα refTΔ

,s refU,

ref

s ref

UMa

U=

( ) 21ref

ref ref

U TEc Mah T

γ ∞= = −Δ

198 Microfluidics

2ˆ ˆD ˆ ˆˆDT EcTt RePr Re

α= ∇ + Φ [5.3]

for an incompressible laminar flow, where we made use of . The non-dimensional diffusivity is denoted by in the last equation.

5.2.2. Channel entry problem: hydraulic and thermal development

The entry zone of an internal channel flow is under the effect of two boundary layers related to the velocity and temperature that develop on both sides and coincide at and , respectively, above which the flow is dynamically and thermally developed (see Figure 5.2). The ratio of these development lengths is a

function of the Prandtl number and we typically have ~dt

d

xPr

x. The hydraulic

development length depends on the Reynolds number through dd

x C Rea

= , where

the constant in laminar flows, and a is the half-width of the 2D channel. The Reynolds number here is based on a and the bulk velocity, which is the velocity averaged over the cross-section. The hydraulic development length is four to six times shorter in turbulent flows and strongly depends upon the entry conditions (such as triggering roughness elements). The thermal development length is related

to the Péclet number , with . The transition coefficient is

for a laminar flow subject to constant temperature at the wall, and is only slightly different when the heat flux is constant [PRI 51]. The thermal development process is more complex in turbulent flows, and the reader is referred to Chapter 11 of [BUR 83] for further details.

The heat transfer process in the entry zone is governed by the Graetz number: aGz RePrx

= . One of the most popular correlations used in constant wall

temperature laminar flows is ( ), where the Nusselt number

is based on the bulk temperature (defined as the enthalpy averaged

over the cross-section (see equation [5.5] in section 5.2.3) and stands for the heat exchange coefficient. More precisely, denoting the heat flux at the wall and the wall temperature by ( )0 0

/y

q k T y=

= − ∂ ∂ , coefficient is defined through

pDH c DT=

α

dx dtx

0.12dC ≈

Pe RePr= dtdt

x C Re Pra

=

0.2dtC ≈

1/30.94Nu Gz= 10Gz ≥haNu k= VT

h0T

h


in this specific case. The ensemble of these relationships is valid

when the axial diffusion is negligible, which is the case when the Péclet

number is large enough, 1Pe RePr= (typically larger than 50).

Heat transfer in the entry zone is governed by the equation:

( )

( )

1

1 1

t

tt

T TU yPrx y y

Tv yPr Pry y

α+ +

+ + ++ + +

++ +

+ +

⎡ ⎤∂ ∂ ∂⎛ ⎞= +⎢ ⎥⎜ ⎟∂ ∂ ∂⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞∂ ∂= +⎢ ⎥⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

[5.4]

in the turbulent regime. The “+” in this equation refers to the scaling by the viscosity v and the shear velocity related to the shear at the wall through

. The temperature solution depends upon the distributions of

turbulent diffusivity and eddy viscosity or turbulent viscosity .

Generally the former is related to the latter by the turbulent Prandtl number , which is constant and approximately 0.9.

Figure 5.2. Schematic view of the hydraulic and thermal development in the entry zone of a channel at Pr >>1

5.2.3. Fully developed laminar or turbulent internal flows

Flow is dynamically developed when the two boundary layers in the entry zone coincide at the channel centerline, and the velocity becomes independent of the streamwise direction . For heat transfer, in return, the dimensional temperature distribution still depends upon x, but the non-dimensional temperature

( )0 0Vq h T T= −2

2

Tx

∂α∂

( )0y

u U yτ ν ∂ ∂=

=

( )t yα + + ( )t yν + +

t t tPr ν α=

U 0 , T 0

ay

x

xdxdt ~ xd Pr

U y( )

x( ),T x y

200 Microfluidics

defined by becomes independent on the axial direction in the fully

developed thermal regime. In the following the index w will refer to the wall. The bulk temperature results from the averaged enthalpy and is given by:

[5.5]

For a 2D turbulent channel flow, the convection equation reduces to:

( )ˆ ˆ1 1wt

t

dT dTv v Pr TU vdx dx Pr y Pr y

θ θ +⎡ ⎤⎡ ⎤ ⎛ ⎞∂ ∂− + = +⎢ ⎥⎜ ⎟⎢ ⎥ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦

[5.6]

The Nusselt number in a fully-developed 2D turbulent flow is given by the integral:

Nu =U

U V

1

1 +Pr

Prtν t

+

U

U Vdη" dη '

0

η '

∫η

1

∫

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

dη0

1

∫

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

−1

[5.7]

Solutions in the laminar regime are of course obtained by taking .

The closure problem is of a fundamental nature in modeling turbulent flows. We are not aware on the scale effects on the fine structure of transfer mechanisms (such as the eddy viscosity) in minichannels. This is because of important difficulties rising in the development of microsensors on one hand, and the flow is essentially laminar in microchannels on the other. We will consequently not insist on these particular aspects in this chapter. The temperature distribution in a fully-developed laminar 2D channel flow subject to a constant wall heat flux is:

ˆ w

w V

T TT T

θ−

=−

VT

1V S

V

T U T dSU S

= ∫

( ) 0t yν+ + =

( )2 435ˆ 5 6136

w

w V

T TT T

θ η η−

= = − +−


where (see Figure 5.2). The Nusselt number based on the hydraulic diameter of the channel is under these conditions and

in the case of constant temperature at the wall. Following a similar procedure for flow in a pipe of diameter , we get Nu = 4.36 and Nu = 3.66 for constant flux and temperature at the wall respectively.

5.3. “Macroeffects” in microchannels: single-phase liquid flows

The early version of this chapter was written before 2000. Huge progress in heat transfer in single-phase micro- and microchannels has been achieved since. The main conclusion we had a decade ago still holds: except in very specific configurations dealing with channels of size below 1 µm (as will be discussed later), there are no specific non-continium effects in liquid flows. Thus, wall transfer phenomena are still governed by Navier-Stokes continium and enthalpy equations. The differences observed and labeled as microeffects are mostly due to a misinterpretation of the data, as we will show hereafter.

5.3.1. Geometrical effects

We discuss some early investigations dealing with heat transfer in microchannels in this section. The differences observed between experiments in microchannels and channels of conventional sizes are most often due to the geometrical shape effects that are not taken into account when comparing experimental data with conventional correlations. Shape factor Z, is defined as the ratio between the channel height and its spanwise width. The streamwise extension of the channel that determines whether the flow is hydraulically and thermally developed or not and the structure of the wall (roughness) are factors that we will classify as macroeffects. To be precise, here we identify microeffects as the influences related to the scales delimiting the validity of continuum media together with the non-slip boundary conditions and continuity of temperature distributions. The molecular interactions at solid–fluid interfaces, such as electrostatic and Van der Waals surface forces, are also part of the microeffects category.

The first pioneer investigations showing the efficiency of micro heat exchangers, wherein the cooling fluid is a liquid, were carried out three decades ago [TUC 81, TUC 82]. Using a simplified analysis, these authors have shown the possibility of evacuating a power as great as 1,300 W/cm2 through a network of parallel microchannels of 50 µm height with a temperature difference of 50°C between the inlet and outlet. This configuration is actually optimum for a silicon substrate of 1x1 cm2 dimensions. The experiences did not reveal any particular

y aη =

2hD a= 4.12hNu hD k= =3.77Nu =

hD D=

202 Microfluidics

microeffect on the flow or heat transfer characteristics in this configuration (see the first line of Table 5.1).

[PFA 90] and [PFA 91] investigated the flow characteristics in rectangular microchannels of heights as small as 0.8 µm. They reported that the drag coefficient

(where stands for the wall shear stress) is three times larger in the microchannel of dimensions 0.8x100 µm2 (wherein the hydraulic diameter is Dh = 1.59 µm), than in the 1.7x100 µm2 case. An important modification is observed in the ( )fC Re distribution when Dh = 3 µm. It has to be emphasized however that there is a difference of 75% between the drag coefficient they measure in a larger microchannel of 53x135 µm2 and shape factor 0.4 and in the fully developed regime compared to the classic correlations of [SHA 78]. They conclude microeffects for microchannels Dh = 80 µm.

It is well established now that this limit is overestimated. Among several factors, the experimental uncertainties in determining the channel dimensions (especially the height that is difficult to characterize when the surface is rough) are at the origin of these discrepancies. To be more precise, consider a Poiseuille flow. The drag coefficient based on the bulk velocity is where the Reynolds number is based on the bulk velocity and hydraulic diameter of the channel. This is in Hagen Poiseuille flow, i.e. in the case of a round tube

. The pressure drop over a length L is and the

flow rate is . Combining the results in , the drag coefficient is experimentally deduced from the measurements of pressure drop and flow rate. It is clearly seen that 10% of the error in results in 50% of the error in Cf.

We have to respect the necessary conditions of the geometrical parameters and in particular the shape factor before determining a plausible microscale effect. We cannot compare the drag coefficient or Nusselt number obtained in a microchannel with a shape factor Z > 0.02 with the classic correlations that are valid for 2D channels (Z→0). This point is crucial not only in the entry zone, but also in the fully developed turbulent flow in minichannels. The 3D induced by the large Z results in flow structures at the corners that might not only affect the transition, but may also influence the wall transfer in a non-negligible manner.

The investgation by [PEN 94] is a nice example that illustrates this remark well. The microchannels used by these authors have a shape factor , as shown in Table 5.1. The 3D effects can thereore not be neglected in this specific case. This

22f w VC Uτ ρ= wτ

C f = 2τ w ρUV2 = 24 ReDh

16hf DC Re= 24 1

2f Vh

Lp C UD

ρ⎛ ⎞⎟⎜Δ = ⎟⎜ ⎟⎜⎝ ⎠

2 4V hm U Dρ π= 2 532 f hp C Lm DρΔ =

hD

0.5Z ≥


transitional Reynolds number based on the bulk velocity and hydraulic diameter

they indicate is ,

which is significantly smaller than

2,300trRe = of a conventional Poiseuille flow. The first plausible explanation is the non-homogenity created by corner vortices.1

This scenario may be similar to the stability mechanism governing 3D boundary layers [SCH 79, p. 535]. This argument is of course hypothetical, and there are no more detailed investigations in the literature on the mechanism ensuring the channel stability of large shape factors (for example in [DRA 81]). Another plausible explanation is the combined 3D effects with roughness in the entry zone. Details concerning the roughness are not provided in [PEN 94] and we cannot therefore comment further on this point.

[PEN 94] also report large differences in the Nusselt number they determine in both laminar and turbulent regimes, compared with the conventional 2D channel correlations. They indicate that 0.62 1/3Nu Re Pr∝ in a laminar regime2. They

compare their results with the relationship 1/3

1/3 hDNu Gz Re PrL

⎛ ⎞∝ ∝⎜ ⎟⎝ ⎠

that is valid

in the entry zone, as we already pointed out in section 5.2.

Once more, this procedure is open to discussion. Their results are well regrouped by a correlation of Dittus-Boelter type 4/5 1/3Nu Re Pr∝ in the turbulent regime, as in conventionally-size channels. The proportionality coefficients and in the laminar ( ) and turbulent ( ) regimes depend upon the geometry (G) and shape factor Z (see Table 5.1). To conclude, the effects observed by [PEN 93] can obviously not be interpreted as microeffects. Navier-Stokes equations with appropriate boundary conditions should perfectly model their cases. This remark is also valid for other results published by this group in parallel microchannel network configurations (see the last line of Table 5.1).

1 The critical Reynolds number based on the bulk velocity and half the channel height is 3,543 according to linear stability analysis [LIN 46]. It is not convenient to compare the critical Reynolds number in a rectangular channel and the “critical” Reynolds number of an axisymetric Hagen Poiseuille flow. The latter is linearly stable, while the Poiseuille flow in a 2D channel is not. We opt for this presentation just to conform with the one used by [PEN 93]. 2 This Nusselt number is based on the hydraulic diameter. It is usual to express the heat exchange coefficient h on the temperature difference between the wall and bulk flow. [PEN 93], however, use the inlet temperature to define h. This makes the comparisons difficult.

700V htr

U DReν

= ≈

GlC GtCl t

204 Mic

Referen

[TUC 81[TUC 82

[PFA 90[PFA 91

[PEN 94[PEN 95

[PEN 95[PEN 96[WAN 9

T

crofluidics

nce Characte

1] 2]

Parallel m

Deionized

0] 1]

Single mic

H = 135 µW = 100 µµm

N-propanisopropyl

4] 5a]

Single mic

Water

5b] 6] 94]

Parallel m

Water, meFour para

Table 5.1. Char

W Wc=

300

minmax

2

50

H

Z

WDh WL m

=

=

=

=

300

0.33

34

45

H

Z

DhL m

=

=

=

=

eristics

microchannel n

d water

crochannel

W = µm µm, H = 1.7 an

nol, silicone oil,alcohol

crochannel

microchannel n

ethanol allel channels

racteristics of linvestiga

50 3m Hμ= =

( )( )

100

n ,0.33

x ,

343 1

.

m

H WH W

WHH

mm

μ−

=

= −+

200

3 1.

3 133

.

m

m

mm

μ

μ

÷

÷

÷

Corre

etwork

_

53 µm,

nd 0.8

_

Lamin

Nu =

Turbu

Nu =

etwork Lamin

Gl

Nu

C

=

=

Turbu

Gt

Nu

C

=

=

liquid flows accations in micro

300 mμ

3 1.

33 mμ

−

elations

______

______

nar: 0.62 1

GlC Re Pr

ulent: 4/5 1/

GtC Re Pr

nar:

(0.62 1/3

,

GlC Re Pr

f D W Hh c=

ulent:

(4/5 1/

,

GtC Re Pr

f D W Zh c=

cording to the fiochannels

Remark

There armicroeff Fully deflow configur

Plausiblemicroeff

1/3

/3

Transitio

c 70Re ≈

Geometreffects

)3

W

)/3

Z

Geometreffects

first experimenta

10

0.8

1.h

W

H

D

=

=

=

ks

re no fects

eveloped

ration

e fects for:

on 00

rical

rical

al

00

8

59 .

m

m

m

μ

μ

μ


The main conclusion we reach in this section is that we cannot observe microeffects for liquid flows in microchannels with hydraulic diameters larger than at least 20 µm. The differences observed in the literature and misleadingly classified as microeffects (especially the early results before 2000) are essentially due to the fact that the comparisons are generally made with respect to correlations that are valid for 2D channels. For the microchannel network configurations, on the other hand, carefull analysis of the eventual effects of heat conduction in the substrate and conjugal heat transfer effects is also necessary. The numerical simulations performed by [AMB 00] perfectly illustrate this particular point. These authors studied the experiments of [HAR 97] through complete 3D Navier Stokes and convection equations (without taking into account the viscous dissipation terms). They used a closure in the turbulent regime. The conduction equation in the substrate is coupled with the appropriate boundary conditions at the interface for a microchannel network. They obtained a perfect agreement between the measurements and their numerical model. Figure 5.3 shows the results.

Figure 5.3. Comparison of the numerical simulations conducted by [AMB 00] and the experiments of [HAR 97] at x/L=0.75 in a 68 microchannel network. The classic macroscale

correlation is shown by a continuous line

The experiments conducted by [HAR 97] have been performed in a network of 68 identical channels, each having a hydraulic diameter Dh = 404 µm and 25 mm length. The measurements are indicated by open circles in Figure 5.3. The numerical results are in good agreement with the measurements, except some differences at high Reynolds numbers that do not exceed 10%. We opted for the Bejan

k ε−

206 Microfluidics

representation [BEJ 84] of the Nusselt number versus the square root of the Graetz

number. This results in h

1/ 21/ 2h

DDxxNu Re Pr Gz⎛ ⎞∝ ∝⎜ ⎟

⎝ ⎠ in the entry zone (and not

the classic correlation , see [BEJ 84, pp.95-102]. The continuous line in Figure 5.3 shows the results of [SHA 78] related to a channel flow subject to constant wall flux under canonical conditions. There are clearly important macroeffects induced by the geometry of the microchannel network that lead to the Nusselt number distribution deviating from the conventional theory, both in developing and developed regimes. [WEI 00] have clearly shown that the pressure gradient distribution versus Reynolds number is linear for Re ,≤ 1 500 and

hD mμ≤ 50 . They indicated that important effects are induced by roughness. This particular point will be discussed in the next session.

[BAV 05] reported well-controlled experiments in microchannels of half-widths varying from 4.6- 20.5 µm. The pressure gradient has been measured in situ by MEMS-type sensors. Figure 5.4 shows the distribution of the Poiseuille number

versus the Reynolds number obtained in the experiments conducted

by this group. No significant microeffects are observed in these investigations (at least concerning the drag coefficient) and the classic laminar number represents the measurements in the laminar regime well.

Figure 5.4. Poiseuille number versus Reynolds number based on hydraulic diameter in mini- and microchannels according to [GAO 02] and [BAV 05]

1/3xNu Gz∝ ∝

hf DPo C Re=

24Po =


The Nusselt number distribution versus the half-width of the channels shown in Figure 5.5 comes from the same group and shows that Nu differs only slightly from the theoretical Nusselt number in microchannels up to a = 50 µm.

Figure 5.5. Ratio of the measured to theoretical Nusselt number in microchannels versus the channel half-width according to [GAO 02]

5.3.2. Axial conduction and conjugate heat transfer effects

In a laminar macroflow subject to constant wall flux, longitudinal variations of the wall and bulk temperatures are equal and constant, i.e. we have:

where stands for the wall flux, i.e. (see for example [FAV 09]). This relation neglects the effect of heat conduction into the substrate (wall). It implies that there is a temperature gradient between the inlet and outlet that subsequently induces longitudinal diffusion in the substrate. The conjugate conduction-convection heat transfer at the solid/fluid interface alternates the longitudinal gradient of the fluid temperature that stops being constant. The conjugate heat transfer effect is generally negligible in conventional macrochannels since the channel walls are thin with respect to half-width . In microchannels or in

theoNu

V w w

V

dT dT qTdx dx x caU

∂∂ ρ

= = =

wq ( )w wq k T y∂ ∂=−

a

208 Microfluidics

microtubes, however, the thickness of the wall (or capillary) can become comparable to the cross-sectional flow scale. If this is the case, the bulk temperature no longer varies linearly in the strteamwise direction but will be convex according to the numerical simulations performed by [MAR 04]. This effect is discussed, for example, in [CEL 06a] for microtubes. The axial conduction effect induced by conjugate heat transfer can be quantified through the parameter:

where is the conductivity of the substrate. and are respectively the outer and inner hydraulic diameters. The conjugate heat transfer that cannot be classified as a microeffect becomes important when .

5.3.3. Viscous dissipation

The viscous dissipation term appearing in transport equation [5.1] is not a microeffect either. It is proportional to , and may become significant in microchannels (or microcapillaries). It can be shown, for instance, by integrating the temperature equation it can be shown that the cross-sectional non-dimensional dissipation is proportional to the Poiseuille number . Furthermore, in a flow subject to constant wall flux, the longitudinal variation of the bulk temperature is modified by the viscous dissipation term and, in non-dimensional form, becomes:

where and are respectively the cross-section area of the channel and its axial length scaled by the hydraulic diameter [CEL 06b]. In macrochannels the contribution to the bulk temperature gradient of the dissipation term is entirely negligible. The viscous heating may increase the bulk temperature by several degrees K in microchannels of hydraulic diameters typically smaller than 100 µm. The dissipation can also modify the viscosity and may cause errors in determining the Reynolds number.

5.3.4. Roughness

Bypass transition from a laminar to turbulent regime is sensitive to roughness. Well controlled experiments allow us to hinder transition up to Reynolds numbers as

2 2 1s ho hiC

hi

k D DP

k D L RePr

⎛ ⎞− ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠

sk hoD hiD

210CP −≥

Ec Re

fPo C Re=

ˆ ˆˆ ˆˆˆ

V wdT q L Ec L Podx RePeS

= +

S L


large as 100,000 [HIN 75, p. 707]! Roughness, however, can accelerate the transition, particularly in the entry zone. Thus, Hagen Poiseuille flow that is linearly stable is subject to boundary layer-type linear instability in the region of hydraulic development, which strongly depends upon roughness [SCH 79]. The lower limit of the Reynolds number, based on the half-width of the channel for nonlinear transition in Poiseuille flow, is 1,000. Thus macrochannel flows are subject to transition between 1,000 and the critical Reynolds number inferred from linear stability analysis, which is 6,000 depending on several parameters (especially roughness).

[MAL 99] conducted well-controlled experiments in microtubes whose diameter varied from 50-254 µm. They used deionized water and reached Reynolds numbers of up to 2,500. Two series of experiments with 3 and 6 cm length channels were performed to analyze the entry effect. The distribution of drag coefficient versus Reynolds number revealed flow characteristics modification for Re > −300 900 compared with macro Hagen-Poiseuille flow. This was first interpreted as early transition and the establishment of a fully developed turbulent regime in the rangeRe , ,> −1 000 1 500. They subsequently used a roughness model to interpret their data, which were also well regrouped through this model. The roughness is modeled using a virtual viscosity that can be expressed as:

2

k 1 expr kr

Rey a y aA Rek Re k

μμ

⎡ ⎤⎛ ⎞− −= − −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

[5.8]

for a bidimensional rectangular channel. The half-width of the channel is a, y is the distance to the wall, and is a constant depending upon the Reynolds number and roughness. The similarity between this approach and the eddy viscosity formulation of wall turbulence has to be noted. The roughness Reynolds number kRe is defined

as ( )

22

2kkRe kuτν

+= = , where k stands for mean roughness height and the shear

velocity is as usual3. The streamwise momentum equation is

written as: . The streamwise velocity distribution is:

3 We use a notation similar to turbulent wall flows in which (+) indicates scale quantities of the viscosity and shear velocity.

rA

0y

uuyτ

∂ν

∂=

⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠

( )rdp dudx y dy

∂μ μ

∂

⎡ ⎤⎢ ⎥= +⎢ ⎥⎣ ⎦

210 Microfluidics

[5.9]

The roughness function is expressed as:

[5.10]

in its non-dimensional compact form. The wall shear stress is straightforward. It is given by:

[5.11]

This relation clearly suggests that roughness increases shear at the wall in laminar flow. [MAL 99] adapted coeficient from a least-squares analysis of the experimental data and obtained good agreement between the model and their measurements.

This methodology is open to discussion but it is interesting because it draws attention to an eventual roughness effect in laminar flows. Results similar to [MAL 99] have also been reported [WEI 00] for trapezoidal microchannels. In situ measurements performed by MEMS-type microprobes can provide clear answers on the effect of roughness on microflows. Use of wall MEMS probes, such as pressure or wall shear stress sensors, can also be useful to experimentally determine the transitional Reynolds numbers by sensing the pressure/shear stress fluctuations. One of the open questions is whether or not a similar approach can be used to determine the roughness effect on near-wall heat transfer using roughness diffusivity

and the transport equation . The Nu number

measurements reported, for example, in [PEN 94] can be confronted to this model to bring some preliminary answers to this question. Recent numerical simulation results of [GAM 08] and [GAM 09] show that roughness increases the drag coefficient more than the heat transfer coefficient for a pattern of parallelepipedic elements of height k. The actual challenge is to achieve the opposite by optimizing the shape and distribution of the roughness elements.

( )( )0

1 y

r

dp au y ddx F

ηη

μ η−

= ∫

( ) ( )2

1 1 expr rV

k aF A k aU a

ηη η

+ + ++ + +

+ +

⎡ ⎤⎛ ⎞− ⎟⎜⎢ ⎥⎟= + − − −⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

( ),

2, 0

1 10

1 1 exp

w k

w k r

rV

F kA k aU

τ

τ += + ++

= =⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥⎟− − ⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

rA

( )r yα

( )( )rT Tu yx y y

∂ ∂ ∂α α

∂ ∂ ∂

⎡ ⎤⎢ ⎥= +⎢ ⎥⎣ ⎦


Systematic studies on roughness effects on the flow and transfer characteristics in turbulent minichannels are missing in the literature. In wall-turbulent macroflows, the roughness increases turbulence activity in the viscous sublayer when it is confined to the buffer layer, extending typically from 10-30 units from the wall (see for example [ROT 62]). The drag coefficient and Nusselt number become independent of Re and Pr when the mean roughness is higher than 30. To be brief, let us summarize by indicating that the Nusselt number behaves according to the Taylor-Prandtl-Colburn analogy over rough walls. It is therefore proportional to the drag coefficient, i.e.: ( ), ,f fNu C f Re Pr C= .

5.4. Gas microflows: rarefaction and compressibility

5.4.1. Knudsen number and compressibility effects

Both the flow physics and transfer process are relatively well understood in gas flows in microsystems through the effect of the Knudsen number, ,

defined as the ratio of the mean free-path of the molecules:

[5.12]

and a typical macro length scale L of the flow. The quantities defining are the Boltzmann constant , the temperature T, the pressure p, the diameter o of the molecules modeled as the spheres, and the ideal gas constant R. Using the relation between the viscosity v and the speed of sound and introducing the Mach number Ma (that is the ratio of velocity to the speed of sound), we can relate the Knudsen

number to the rarefaction and compressibility effects by 2

MaKnRe

π γ= where

stands for the ratio of the specific heats under constant pressure and constant volume [ECK 72, GAD 99].

Continuum regime models and the related boundary conditions of the no-slip and temperature-jump type are valid for small Knudsen numbers, such as . The local equilibrium is then respected and the velocity probability distributions are of Maxwell type. For large Knudsen numbers, , the continuum model no longer makes sense; the flow is highly rarefied and is in the molecular regime. The media is moderately rarefied in the transition regime . It is no longer

Kn L=

2 22Bk T

RTpπ

νπ ο

= =

Bk

γ

310Kn −≤

10Kn≥

110 10Kn− ≤ ≤

212 Microfluidics

valid to model the slightly rarefied regime corresponding to the range , by making use of the Navier-Stokes equations coupled except for

the no-slip boundary. This regime is commonly labeled as slip-flows, wherein the wall boundary conditions are formulated by a slip velocity and temperature jump. The latter were formulated by [MAX 79] and [SMO 98] for monoatomic gases over a century ago. Some molecules lose their momentum when exchange the wall shear stress after contact and some of them do not. This results in wall slip. The ratio of the number of molecules that exchange their momentum and the molecules that conserve it is defined as the accommodation coefficient . The latter depends upon the nature of the wall, the fluid and roughness. The accommodation coefficient is small in the case of a smooth wall. The slip velocity is expressed by:

2 ˆˆ2 3 1ˆ ˆˆ ˆ1 2gas w

w w

Kn u Tu u Re KnbKn y x

ν

ν

σ ∂ γ ∂σ ∂ π γ ∂

⎛ ⎞⎛ ⎞− −− = + ⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠ [5.13]

where it is recalled that subindex w refers to the wall. We scaled the temperature through:

[5.14]

which is generally used for adiabatic walls. We can also rewrite equation [5.13] by

making use of the Eckert number as:

2 ˆˆ2 3 1ˆ ˆˆ ˆ1 2gas w

w w

Kn u Re Kn Tu ubKn y Ec x

ν

ν

σ ∂ γ ∂σ ∂ π γ ∂

⎛ ⎞⎛ ⎞− −− = + ⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠ [5.15]

The last term of this equation represents the thermal creep that regenerates a flow in the direction of the streamwise gradient of the wall temperature. The order of magnitude of the thermal creep term is Kn2 , showing that it can be significantly large towards the beginning of a moderate rarefaction regime. Coefficient b, introduced by [BES 94], extends the validity of slip velocity to the second order. A

3 110 10Kn− −≤ ≤

νσ

2ˆ ref

ref

p

T TT

Uc

−=⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

2ref

p ref

UEc

c T=

Δ


Taylor series analysis shows that b is indeed related to the ratio of first and second derivative of the velocity at the wall, i.e.:

[5.16]

The classic Maxwell theory leads to b = 0. Using these relationships, one we easily obtain the velocity profile in the slightly rarefied regime in a fully developed 2D laminar channel flow subject to constant wall temperature. The velocity profile can be expressed in a compact form by:

[5.17]

where the first term on the right is the canonical Poiseuille velocity profile

in the coordinate system given in Figure 5.2. We notice

that the wall slip velocity increases the velocity distribution uniformly across the channel. Thermal creep has to be taken into account under non-isothermal conditions and if the Knudsen number is relatively large. Then, the velocity profile is expressed as:

[5.18]

The Reynolds and Eckert numbers are based on: . The

length scale is half the channel width.

The longitudinal pressure distribution can also be easily determined in the fully developed isothermal flow, assuming that the ideal gas relationship still holds in the rarefaction regime [ARK 94, BES 96]. The continuity equation is used to compute the mass flux from which is deduced. It goes without saying that the compressibility results in a nonlinear pressure distribution, and the gradient of the pressure squared is constant when the Kn effect is not taken into account. The procedure is similar when the rarefaction effects are not negligible. According to [BES 96], the pressure distribution is given by:

2 2

2w

U ybU y

∂ ∂∂ ∂

⎡ ⎤⎢ ⎥= ⎢ ⎥⎣ ⎦

( ) 02

2ˆ ˆ 212 12

Knref

U U KnU UU dp dx a Kn

ν

ν

σ

σμ =

−= = = +

− +

22

0ˆ ˆ1 1Kn

yU ya=

⎛ ⎞⎟⎜= − = −⎟⎜ ⎟⎜⎝ ⎠

2

0

ˆ2 3 1ˆ ˆ 21 ˆ212

Kn

w

Kn Re Kn TU UEc xKn

ν

ν

σ γ ∂σ π γ ∂=

⎛ ⎞− − ⎟⎜ ⎟⎜= = + + ⎟⎜ ⎟⎟⎜⎝ ⎠+

( )2 2refU dp dx a μ=−

p RTρ=

( )p x

2dp dx

214 Microfluidics

[5.19]

where is the non-dimensional pressure distribution scaled by the

pressure at the exit (E) of the channel of streamwise length L. The constant A appearing in the last equation is related to the conditions at the channel entry. Equation [5.19] is valid for second-order the Knudsen numbers.

In the preceding analysis it is assumed that the pressure is uniformly distributed across the channel and that the thermal effects are negligible. These hypotheses are valid for small Mach numbers but are otherwise not appropriate. Indeed, the direct numerical simulation results of Navier-Stokes equations coupled with slip-boundary conditions conducted by [BES 96], and the direct Monte Carlo simulations of [PIE 96] differ by less than 15%. There is also a good agreement between the theoretical predictions we discussed before and the majority of the experiments, at least in the slip-flow regime. We recapitulate and comment on some experimental investigations on gas flows in microsystems in Table 5.2. The agreement between measurements and theory is globally satisfactory for isothermal flows, the predicted quantities having been obtained by using an accommodation coefficient roughly equal to one. For instance, the experiments performed at CALTECH (California Institute of Technology), wherein the pressure distribution has been determined in situ by MEMS-type4 sensors (last line of Table 5.2) are in global agreement with a second-order slip model and .

The early experiments conducted by [WU 83] constitute one of the rare disagreements with the theory. These authors measured the drag coefficient and Nusselt number in silicon microchannels. They reported an increase in and a drastic decrease in the critical Reynolds number ( Recr = 400 ). Such large effects cannot be attributed to the Knudsen number effects. The Knudsen number was not mentioned in their papers but we estimate that , which is within the slip regime. It is interesting to point out that the work reported in [WU 83] used to be cited as a situation in which microeffects are predominant. Yet, rarefaction influences cannot explain such large discrepencies, which should rather result from roughness and/or geometrical effects. Here we maintain this view in the case of the 17% decrease in drag coefficient reported in round microtubes experiments [CHO 91], in which small Knudsen numbers exclude rarefaction microeffects.

4 Microelectro-mechanical-systems.

( ) ( ) ( )2 22 21 12 1 12 lnE EKn Kn A L xν ν

ν ν

σ σ

σ σ

− −−Π + −Π + Π = −

( )( )E

p xxp

Π =

0.1Ma ≤

1σν =

fC

fC

0.001Kn≤


Reference Configuration Fluid *Characteristics and observations

*Concordance with slip models *Remarks

[SRE 68] Round tube Diameter: 5 cm

Rarefied gas

* *Large pressure gradient

Good agreement with first order slip models for

[WU 83] Silicon

microchannels Hydraulic diameter: 134-164 µm

* (estimated) *Increase in drag coefficient *More rapid transition *Larger Nusselt number in turbulent regime

*Negligible rarefaction effects *Geometrical effects and roughness

[CHO 91] Round tube Diameter: < 10 µm

* (estimated) *Drag coefficient is 17% larger

Negligible rarefaction effects

[PFA 91] Silicon microchannels Hydraulic diameter: 8 µm Streamwise length: 11 mm

*

* The exit Mach number is 0.7

Good agreement with a first-order slip model using σ ν = 1

[TIS 93] Round tube Diameter: 2 mm Streamwise length: 400 mm

*

Good agreement with a first-order slip model at using σ ν = 1

[ARK 94a] [ARK 94b]

Silicon microchannels 0.11x6.66 µm2

1.04x31.14 µm2

* * The pressure between the exit and inlet sections varies between 1.2 and 4.2

* Excellent agreement with a second-order slip model at

with σ ν = 1

*Slight difference (5%) with respect to direct Monte Carlo simulations for [PIE 96]

[HAR 95] Silicon microchannels Depth: 0.5/19.79 µm

Air

Good agreement with first-order slip models using

[LIU 93] [PON 94] [LIU 95] [SHI 96]

Silicon microchannels Depth: 1.2 and 1.33 µm

* * In situ measurements with MEMS

Good agreement with a second-order slip model using

Table 5.2. Some measurements and their characteristics in gas microflows

0.265Kn ≤

0.13Kn ≤

2N 0.001Kn ≤

2N 0.005Kn≈

2N

He0.001 0.363Kn≤ ≤

2N

HeAr

200Kn ≤0.6Kn ≤

2N

He

0.44Kn ≤

0.1Kn ≤

0.44Kn =

2N

He

0.363Kn ≤

1σν =

He2N

0.156Kn ≤

1σν ≈

216 Microfluidics

The slip models become less efficient in the transition regime , as expected. [BES 96] and [PIE 96] show that the rarefaction effects become opposed to the compressibility effects as Kn increases. Recall that the pressure varies as

in an isotherme compressible microchannel flow, where . However, numerical simulations have shown that under the Kn number effect, towards the end of the slip-flow regime. Thus, the compressibility effects are opposed to rarefaction and the flow behaves as if it was incompressible. Some experimental results, in particular those reported by [ARK 94b] and [PON 94] at slightly disagree with these predictions. The differences between the global measurements and the model are smaller than 5%, which is within the margin of experimental errors.

5.4.2. Effects on heat transfer

The temperature jump at the wall is modeled in a similar way to the slip velocity:

ˆ2 2ˆ ˆˆ1 Pr

Tgaz p

T p

Kn TT Ty

σ γ ∂σ γ ∂

⎛ ⎞−− = ⎜ ⎟+ ⎝ ⎠ [5.20]

where stands for the thermal accommodation coefficient. It is easy to obtain analytical solutions of the temperature distribution in a fully developed 2D laminar, incompressible flow subject to a constant wall heat flux. The formulation of the problem is based on the non-dimensional transport equation:

22

2

ˆ ˆ ˆ1ˆˆ ˆPr Re Reˆ

T T Ec UUx yy

∂ ∂ ∂∂ ∂∂

⎛ ⎞= + ⎜ ⎟

⎝ ⎠ [5.21]

that takes viscous dissipation into account. The hypothesis of thermaly developed

flow leads to: . The temperature profile can consequently be put as a

sum of four distributions that can be individually analyzed. Namely:

[5.22]

0.1 3Kn≤ ≤

2p x∝ 0Kn →p x∝

0.5Kn ≈

Tσ

ˆ ˆ

ˆ ˆw

T Tx x

∂ ∂∂ ∂

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

0ˆ ˆ ˆ ˆ ˆ ˆ

w Kn JU JT CT T T T T T=− = + + +


The first term on the right of this equation is the classic temperature profile without rarefaction effects:

2 44

0ˆ ˆ ˆ 5 Prˆ ˆRePr (1 )ˆ 2 12 12 3Kn

p

T y y EcT yx

∂∂=

⎛ ⎞⎛ ⎞= − − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

[5.23]

where in the second group on the right represents the viscous dissipation effects. The distribution takes into account the velocity (U) and jump (J):

[5.24]

The effect of the temperature jump at the wall is represented through:

[5.25]

Finally the thermal creep effects (C) are given by:

[5.26]

Other solutions with different boundary conditions can also be analytically obtained in a slip regime (see for example [BES 94] for profiles corresponding to isolated lower and upper walls subject to constant flux).

Thermal creep can dominate the heat transfer process when the Knudsen number is large. The flow rate significantly increases with respect to macroflows in this case. A negative longitudinal temperature gradient has the opposite effect. The viscous dissipation, on the other hand, plays a dominant role when the Knudsen number is large. The longitudinal temperature gradient can become positive in a in a cooled microchannel because of viscous dissipation and the flow rate can subsequently increase. These effects are proper to the slip regime and can sometimes modify the heat transfer process in an unexpected way, resulting in large differences with respect to conventional continuum macrochannles.

It can be interesting to analyze the effect of velocity and temperature jumps on the Nusselt number for academic reasons. The thermal creep effect is negligible in

ˆJUT

( )2ˆ 2ˆ ˆ 1

1ˆ 12

JU

w

T KnT Re Pr yx Kn

ν

ν

σ∂∂ σ

⎛ ⎞ −⎟⎜ ⎟⎜= −⎟⎜ ⎟⎟⎜⎝ ⎠ +

2 2ˆ ˆ1

TJT

T

KnT qPr

σ γσ γ−

=−+

( )22 2

2ˆ3 1ˆ ˆ 1ˆ4C

w

Pr Re Kn TT yEc x

γ ∂π γ ∂

⎛ ⎞− ⎟⎜ ⎟⎜= −⎟⎜ ⎟⎟⎜⎝ ⎠

218 Microfluidics

the slightly rarefied regime if ReEcKn << . The viscous

dissipation is also negligible provided that 1ReEc << . The parameters that affect the

heat transfer are and , respectively, under these conditions. They are defined by:

2 2 22 1 1 Pr12

TU T

T

Kn KnS SKn

ν

ν

σ σ γσ σ γ− −= =

++ [5.27]

The Nusselt number based on the bulk temperature and channel half-width in a flow subject to constant wall flux can easily be computed, leading to:

[5.28]

Figure 5.6 shows the variation in Nusselt number given by [5.28] versus the Knudsen number. The Nusselt number increases as the rarefaction effects become significant in the slip regime. We must recall, however, that this is a significantly simplified, idealized model that does not take the dissipation and thermal creep into account.

Figure 5.6. Nusselt number variation versus Knudsen number in a flow subject to constant wall flux and in which thermal creep and viscous dissipation effects are negligible. The

thermal and velocity accommodation coefficients are both equal to one

0.001 0.1Kn≤ ≤

US TS

2

2 2

23

868 2315 15 3 3

U

U UT U

SNu

S SS S

⎛ ⎞⎟⎜ + ⎟⎜ ⎟⎜⎝ ⎠=

⎛ ⎞⎟⎜+ + − + ⎟⎜ ⎟⎜⎝ ⎠


The compressibility may profoundly change the transfer process. The problem becomes complex, which may explain why there are only few analytical contributions concerning this aspect in the literature. Consider the enthalphy equation for a compressible fluid in a 2D channel:

[5.29]

in its simpliest form, wherein we neglected the axial diffusion and dissipation. [PRU 86] introduced a separation of variables procedure that considerably simplifies the problem. They use the state equation of an ideal gas and suppose that the velocity distribution may be written as:

[5.30]

They subsequently obtain the longitudinal distribution of the cross-averaged temperature. We checked that the same procedure leads to the expression of bulk temperature. Using the expression for specific heat under constant pressure in an

ideal gas and defining the non-dimensionalized bulk temperature by

, the integral over the cross-section of the first term in the

enthalpy equation gives , where is the mass-flow rate.

Decomposition [5.30] allows us to obtain the surface integral of the second term:

. The surface integral of the term on the right-hand side of [5.29] leads

to the wall flux. After scaling, we obtain:

[5.31]

where the Reynolds number is based on the bulk velocity, the Nusselt number is defined with respect to the channel half-width and is the ratio of the streamwise length of the channel to its half-width. The last term of equation [5.31] is clearly due to compressibility effects. This term is proportional to , since without rarefaction the streamwise gradient of the pressure square is constant. The

2

2pT dp Tc U U kx dx y

∂ ∂ρ

∂ ∂− =

( )( )( )

( )( )

,f y RTU x y f y

x p xρ= =

52pc R=

( )( )ˆ V p

Vp

T x TT x

T−

=

ˆ52

Vd TR q

dxq

RT dp qp dx

−

ˆ ˆ2ˆˆ ˆ ˆPr Re 5V

V

d T Nu d pTdx p dx

β= +

β

21 p

220 Microfluidics

compressibility may therefore seriously effect the heat transfer mechanism for short microchannels (small ) subject to a small pressure being imposed.

Analytical solutions of temperature distribution that include rarefaction, compressibility, dissipation and thermal creep are difficult. Solutions can only be obtained by numerical simulations. The continuum models of [DIL 98] and [CHE 00] (there are many others) have to be extended to broader ranges of Knudsen number, including the transition regime. The formulation and analysis of the entry problem is particularly difficult in these circumstances.

5.4.3. External flows in a rarefied regime

The boundary conditions of slip and temperature jumps do not allow the obtention of self-similar solutions in external flows. We can indeed show that the only class of Falkner-Skan flows that admits self-similar solutions of the dynamic and thermal boundary layers is the flow where external velocity varies linearly with steamwise direction . Local self-similarity exists and iterative procedures may be formulated. The reader is referred to [TAR 02] for further details.

5.5. Molecular effects of liquid flows in microchannels

The Knudsen number cannot be used in liquid flows, because of its strong incompressibility with respect to gas flows. In liquid flows, the slip regime and accompanied slip velocity and temperature jump at solid/liquid interface have to be validated by a dynamic model adequately taking intermolecular interactions into account [GAD 99]. [THO 97] use the Lennard Jones model, in which the intermolecular forces are given by:

[5.32]

where r is the intermolecular distance, and are characteristic energy and length scales. The interaction coefficients and depend upon the nature of the solid and fluid at the interface.

The Lennard Jones model is criticizable for complex molecular structures, such as water. Yet, we will still use [THO 97] results to estimate the liquid/solid interfacel effect on a Poiseuille flow in a microchannel. Let’s define the non-dimensional slip length by:

β

ˆ ˆU x∞ ∝

( )13 748

2ij

ij ij

dr rF r c− −⎡ ⎤⎛ ⎞ ⎛ ⎞Ξ ⎢ ⎥⎟ ⎟⎜ ⎜= −⎟ ⎟⎜ ⎜⎢ ⎥⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠Λ Λ Λ⎢ ⎥⎣ ⎦

Ξ Λijc ijd


[5.33]

in a channel of half-width . As in rarefied flows, it is easy to show that the wall singularity induces a supplementary term in the velocity distribution that reads:

[5.34]

The shear is unaffected by the wall singularity in a pressure-driven channel

flow, contrary to Couette flow [GAD 99]. This fact excludes the molecular liquid/wall interaction as a factor triggering transition in channel flows. Consequently, the early transition reported by [PEN 94] and [WAN 95] can hardly be explained by some molecular effects, as we have indicated before.

The slip length is related to:

[5.35]

where is the characteristic length scale of the interaction between the wall and the liquid, is the energy scale and is the ratio of the wall and fluid densities. Denoting the asymptotic limit by , at which the conventional no-slip and no-jump conditions are valid, we can write:

[5.36]

according to the molecular dynamic simulations. Relation [5.36] is reminiscent of a catastrophic increase in slip length scale when the shear becomes close to the critical limit . The critical limit is related to the molecular time scale

where m stands for the molecular mass. The critical shear for water is:

[GAD 99]. The half-width of a channel for which the wall

ˆ ˆˆˆ ˆˆ ˆ

sliquid w s

w w

L u uu u La y y

∂ ∂∂ ∂

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟− = =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

aˆ

SU

( )2ˆ ˆ2

2S S

S sref

U UU L

U dp dx a μ= = =

−

ˆˆuy

∂∂

( ), ,pf pfs s p fL L σ ε ρ ρ=

pfσpfε w fρ ρ

0sL

( )( )

1 2

0 1s scr

u yL L

u y∂ ∂∂ ∂

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

( )cru y∂ ∂

1/ 22

mmt σε

⎛ ⎞⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜⎝ ⎠

( ) 11 110cr

u y s∂ ∂ −=

222 Microfluidics

shear would be critical is ( )hD3 34 cr

a Re nmu y

ν∂ ∂

= = at D 1h

Re = , which is a

nanochannel!

Let us determine the slip velocity in a microchannel of hydraulic diameter 100 µm by supposing that the Reynolds number is 2,000.

hDRe = The asymptotic

slip length is , corresponding to a ratio of fluid-to-wall densities of four ( for water). We obtain:

( )h

1 20 4

D 23ˆ 2 1 Re 4 104

sSS

ref cr

LUUaU a u y

ν∂ ∂

−−

⎡ ⎤⎢ ⎥= = − = ×⎢ ⎥⎣ ⎦

[5.37]

which is only 0.06% of the bulk velocity. The conclusion is clear: molecular expects are totally excluded in micro- (and a fortiori mini-) channels.

5.6. Electrostatic effects: interfacial electrostatic double layer

5.6.1. General

The electrostatic charges at the solid/fluid interface exist in all wall-bounded flows, since the majority of surfaces have electrostatic charges. Their effects are negligible at macroscales and become somewhat significant only in microfluidics. Several authors, such as [MAL 97], have investigated on an analytical basis that this effect implies some plausible modifications in transfer processes and hydrodynamic stability.

The electrostatic double layer results from the disequilibrium between the co-ions and counterions in the flow. The surface electrostatic charges attract the counterions and establish an electrical field. The ionic concentration near the wall becomes more important compared to the bulk flow. In a thin layer near the wall of thickness, typically 0.5 nm the ions are strongly attracted by the wall and become immobile. In the adjacent diffusive double layer, they are mobile and convected by an imposed pressure gradient. This results in a charge in density charge that is a measure of the excess co-ions. The mobile ions are transported by the imposed pressure gradient leading to an electric potential and subsequent electric conduction current in the opposite sense. [MAL 97] show that the velocity profile may be decomposed as , where is the classic

Poiseuille profile and is the component induced by the electrical field

0 20sL σ≈102.89 10 mσ −= ×

( )yϕ

dp dx

( ) ( ) ( )P EDLU y U y U y= + ( )PU y

( )EDLU y xE


leading to the external force in the streamwise momentum equation. The component induced by the diffusive double layer is:

[5.38]

with and being respectively the dielectric constant of the medium and the permittivity of vacuum. is the zeta potential at the edge of the diffusive layer. is the ratio of the channel half-width to the thickness of the electrostatic double layer (EDL):

[5.39]

The inverse of the denominator of this equation is the Debye-Huckel parameter. This parameter is related to the ionic concentration , balence of positive or negative ions , electron charge , absolute temperature and to the Boltzmann constant . The EDL component can also be put in the form:

[5.40]

where stands for the ratio of the EDL forces to viscous forces,

, and is the non-dimensional potential. The non-dimensional

potential results from the equilibrium between the current induced by the transport of charges across the pressure gradient, and the conduction current generated in the opposite direction by . It can easily be seen that the EDL effect is maximum at the centerline , and slows the flow. The ratio between the EDL and Poiseuille components at the centerline is approximately:

xE ( )yϕ

( )( )( )

0sinh

1sinhEDL x

ya

U y E

κε ε

ξμ κ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪=− −⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

ε 0εξ κ

12

02 2

02B a

a

k Tn z e

κεε

=⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

0nz e aT

Bk

( ) ( )( )2

sinhˆˆ21

sinhEDL s

EDL

yaEyU a

κξ

κκ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪Γ ⎪ ⎪⎪ ⎪=− −⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

20

EDLn z e a

lξ

μΓ =

ˆB a

z ek Tξ

ξ= ˆ ss

EE ξ=

sE0y =

224 Microfluidics

[5.41]

at . This equation is valid for sufficiently large ( ). The exact relationships can be found in [MAL 97]. We have slightly simplified the equations here. The second parameter appearing in equation [5.41] is the ratio of current induced by and the conduction current is:

[5.42]

where is the fluid conductivity. For a given , the EDL

effect is inversely proportional to , or in other words proportional to the fourth power of the EDL thickness. This shows why EDL is negligible in macroscale canonical flows. The differences between the velocity profiles and drag coefficient under EDL effect compared to Poiseuille flow is also proportional to the zeta potential but the effect hardly exceeds a few percent.

It should be usefull at this step to discuss the EDL effect through a practical exemple and provide some typical values. We deal with a typical case here, which is similar to that in [MAL 97]. We consider a microchannel 25 μm with and cm long (streamwise) submitted to a pressure gradient of 3.107 N/m3. The Reynolds number of the corresponding Poiseuille flow is only 21, with Up(y = 0) = Up0 = 2.5 m/s. We deal with a KCl solution whose ionic concentration is n0 = 6.1020 m-3. The

parameters we introduced read and . We

consider a zeta potential of ξ = 80 mV. The EDL thickness is 0.3 µm and . Figure 5.7 shows the velocity profiles with and without EDL effect under these conditions. There is no spectacular modification of the velocity profile except a decrease at the centerline. The analysis of profiles near the wall, however, show some subtle differences that play a role in transition. This point will be discussed in the next section.

We can define an apparent viscosity by identifying bulk velocity under the EDL effect and setting equality between the Reynolds numbers of EDL and Poiseuille flows. The EDL increases the apparent viscosity by a factor of three at small values and the effect on becomes negligible for . [MIG 87]

( )( ) ( )

2

3 2

ˆ0 8ˆ0 2

EDL EDL c

P EDL c

U yr

U yξ

κ κ ξ

= Γ Γ= ≅−

= + Γ Γ

0y = κ 10κ≥

cΓ

sE

0

0c

n z e lλ ξ

Γ =

0λ( )2

0

0EDL c

n z e aG

λ μΓ Γ = =

4κ

1l=

5

03.10EDL

PU−Γ = 8

0 4.10c PUΓ =

41κ=

aμ

2κ≈ aμ 10κ≥


indicates that the apparent viscosity increases for round microtubes of diameters smaller than 1 µm. The results reported in [PFA 91] are contradictory however and indicate an opposite tendency, i.e. a decrease in apparent viscosity with decreasing tube diameter. Neither EDL nor roughness effects can explain these discrepancies.

Figure 5.7. Velocity profile under EDL effect through the parameters indicated in the text

5.6.2. Effect on transition

The velocity profiles under EDL have an inflexional point near the wall. We have:

[5.43]

where . This relation shows the existence of an inflexional point at:

[5.44]

( )( )

2 2

2

ˆˆ2sinh

ˆ sinhEDL sEU a dp y

dxyξ∂

κμ κ∂

Γ=− +

ˆ 0yy a= ≥

( )

( )

2

2

1ˆ arcsin h - sinhˆˆ2

1 2arcsin h - sinh

EDL s

a dpydxE

r

κκ μ ξ

κκ κ

⎧ ⎫⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪Γ⎪ ⎪⎩ ⎭⎧ ⎫⎪ ⎪⎪ ⎪≈ ⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

226 Microfluidics

Recall that is the ratio of the EDL and Poiseuille velocity components defined in equation [5.41]. Near the wall and for a sufficiently large , the inflexional point is situated at:

[5.45]

according to the first terms of the Taylor series of equations [5.44]. The retardant effect of EDL renders the inviscid flow unstable according to the Fjortoft criteria, because there is an inflexional point I, together with a region in the flow wherein

. Detailed analysis of the linear stability characteristics has been performed by [TAR 04]. The neutral curves deduced from the hydrodynamic stability analysis are summarized in Figure 5.8. It is clearly seen that the critical Reynolds number decreases by a factor nearly equal to two under the EDL effect at

.The critical wave and Reynolds numbers of the microflow are respectively and , compared with and from

conventional Poiseuille flow. This effect is better appreciated if we recall that at the friction factor only increases by some 10%. It is clear that one of the

most significant effects of EDL is the decrease in critical Reynolds number, rather than the increase in friction coefficient or apparent viscosity. More details concerning the linear stability analysis of EDL flow can be found in [TAR 04].

Figure 5.8. Neutral curves of the EDL flow compared with the Poiseuille flow. The open circles correspond to Poiseuille flow with . Bold circles correspond to =41,

G = 12,720 and ξ = 2.1254 . The rest of the results are obtained by changing the microchannel height and keeping the rest of the parameters constant.

The triangle is obtained for =164. The neutral curves for =8, 16 and 41 are compared with the macroscale flow

0r ≤κ

2

1 2ˆ 1 lnyrκ κ

⎛ ⎞⎟⎜ ⎟≈ + −⎜ ⎟⎜ ⎟⎜⎝ ⎠

2 2ˆ ( ) 0IU y U U∂ ∂ − <

41κ=1.10cα = 3190cRe = 1.02cα = 5772cRe =

41κ=

κ=∞ κ

κκ


[TAR 10] investigated the effect of the EDL on the bypass transition mechanism through direct numerical simulations. An initial perturbation velocity field was introduced in Poiseuille and EDL flows and the time-space evolution of the perturbed field was analyzed for short times. The electrokinetic effects rapidly destabilize the flow when the local disturbance and/or the Reynolds number are respectively strong and large enough to overcome the transient growth regime. It was found that a small perturbation quickly leads to transition through the bypass and nonlinear interactions under the EDL effect. A larger disturbance an order of magnitude larger is incapable of destabilizing the macroscale flow. The EDL develops some new transitional wall structures during the bypass process.

The practical implication of the present study is clear. The electrostatic effects can be efficiently used to enhance mixing and heat transfer in microchannels. Consider a channel flow at a given Reynolds number. Note by the Stanton number corresponding to the developed turbulent regime, and by the Stanton

number in the laminar flow configuration. We can easily show that .

Thus, a decrease in the transitional Reynolds number by some factor leads to an increase in the Nusselt number by approximately the same amount. This requires the use of a convenient fluid with a small enough Debye Hückel parameter, i.e. with a diffuse layer thicknesses of greater than 5 µm or so. For smaller thicknesses, the effect on transition is not significant, since large Reynolds numbers cannot be achieved in microchannels. Advances in colloidal and interface science are crucial concerning these aspects, and recent research in this area is promising.

5.6.3. EDL effect on heat transfer

The decrease in bulk velocity under EDL effect is accompanied by a decrease Nusselt number, as expected. It is perfectly feasible to analyze the EDL effect on Nu in a simple way, assuming that:

– the flow is fully developed, both hydraulically and thermally (the heat

coefficient is independent of the longitudinalcoordinate );

– it is subjected to constant wall heat flux;

– the Péclet number Pe = Re Pr is large, so the axial diffusion can be negligible;

– the ratio Ec Re is small, therefore the dissipation can be ignored.

The Nusselt number can easily be determined through equation [5.7], wherein the eddy viscosity is set to zero (the flow is laminar) under these circumstances. The

tSt

lSt

3/4t

l

St ReSt

∝

0hx

∂∂ =

228 Microfluidics

analytical solution, although not difficult to obtain, is long and difficult to interpret. It is, however, possible to give an approximate relationship that is quite clear and valid for . Indeed, the wall normal gradient of component is confined to a thin layer of near the wall, and is approximately constant in 80% of the cross-section at . A rough estimate consists of taking the EDL component into account as a bulk flow, and as the approximation often made for

metal liquid flows. Thus, taking as a constant bulk

flow in equation [5.7] leads to:

[5.46]

where subindex refers to the wall as usual and is the ratio given in equation [5.41]. For the numerical example we are dealing with, ratio r is r = -0.80, which is only 5% smaller than Nu = 2.06 of the macroflow without EDL effect. (Recall that the Nusselt number here is based on the channel half-width; it is four times smaller than Nu based on the hydraulic diameter.) In Figure 5.9 we show the ratio with and without the EDL effect versus . These results were

obtained by keeping the physical characteristics of the fluid that we introduced in the example in 5.6.1 constant and modifying the channel width. It is clearly seen that the wall transfer process can only be significantly affected by EDL at , at which limit the Nusselt number decreases by 35% with respect to macroflow. Parameter , related to some experiments found in the literature, is also indicated in Figure 5.9 by assuming that the EDL thickness is roughly 0.3 μm, as in the example given here. Parameter in these investigations is systematically larger than 100, suggesting that there is no plausible effect of EDL. In some other experiments, wherein a decrease as large as 50% has been reported [PEN 93], º 2,000 is extremely large, and EDL certainly cannot explain such differences.

A similar conclusion is reached for microflows subject to constant wall temperature, although the analytical solution cannot be given in this particular case, which requires an iterative procedure. The EDL effect in the channel entry has not been investigated in the literature. [MAL 97] combine EDL with axial conduction and dissipation in a flow subject to constant wall temperature. They base the Nusselt number on the difference between the wall and inlet temperatures. This makes it difficult to compare analytical solutions. Their results indicate a moderate effect of EDL on wall transfer of less than 15% for .

10κ≥ ( )EDLU y2 κ

20κ≥

( ) 2

ˆˆ2 EDL sEDL

EyU aξ

κΓ

=−

( )( )2

2

1 3 217 6 335 5 4

w

w Vw

rT Th aNuk y a T T rr

∂∂

+−= = =

−+ +

w 0r ≤

0

NuNu ξ=

κ

20κ≤

κ

κ

κ

40κ≥


Figure 5.9. Effect of the EDL on Nusselt number in a fully developed flow subject to a constant wall flux. See section 5.6.3 for the hypothesis and typical case we chose.

The thickness of the EDL is 0.3 μm

5.7. Conclusion

The micromolecular effects are unexpected in liquid flows in microsystems, except when the size of the latter becomes significantly smaller than one micron (which would make it a nanochannel). Most of the effects that are misleadingly classified as microeffects in liquid flows in the literature are macroeffects that become preponderant because of the system scales. Gas flows in the rarefied slip-flow regime are well modeled through first- and second-order slip and temperature jump formulations. The Debye length has to be several microns, in order for the electrokinetic effects to become really significant, which is not the case for conventional fluids such as water.

To conclude, as my colleague Gian Piero Celata would say “a channel is a channel”. 5.8. Acknowledgment

The author thanks Prof. Gian Piero Celata for interesting discussions and having provided some material that helped the redaction of this chapter.

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230 Microfluidics

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microfluidics (colin/microfluidics) || single-phase heat transfer

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