Microfluidics and Nanofluidics

, Volume 1, Issue 2, pp 190–196

A criterion for experimental validation of slip-flow models for incompressible rarefied gases through microchannels

Authors

    • DIENCAUniversità di Bologna
  • Marco Lorenzini
    • DIENCAUniversità di Bologna
  • Marco Spiga
    • Dipartimento di Ingegneria Industriale
Research Paper

DOI: 10.1007/s10404-004-0028-1

Cite this article as:
Morini, G.L., Lorenzini, M. & Spiga, M. Microfluid Nanofluid (2005) 1: 190. doi:10.1007/s10404-004-0028-1

Abstract

This paper is devoted to analysing the friction factor for incompressible rarefied gas flow through microchannels. A theoretical investigation is conducted in order to underline the conditions for experimentally evidencing rarefaction effects on the pressure drop. It is demonstrated that for a fixed geometry of the microchannel cross-section, it is possible to calculate the minimum value of the Knudsen number for which the rarefaction effects can be observed experimentally, taking into account the uncertainties related to evaluation of the friction factor.

Keywords

MicrochannelsRarefied gasSlip-flowPressure dropExperimental uncertainty

1 Introduction

Several experimental works in the open literature are devoted to determining the pressure drop through microchannels. Recent reviews of these works are presented by Morini (2004) and Kandlikar and Grande (2003); yet, the experimental results on friction factor in microchannels are not univocal. The following points summarise the main conclusions reported on the friction factor in micro-fluidic devices highlighting their peculiarities with respect to the conventional macro-channels. As remarked by different authors:
  • The value of the friction factor for laminar fully developed flow is found to be lower than that for conventional gas flow and higher for liquid flow.

  • The Poiseuille number (fRe) for laminar fully developed flow depends on the Reynolds number.

  • The friction factor for gaseous laminar fully developed flow decreases with the Knudsen number (rarefaction effects).

  • The friction factor for gaseous laminar fully developed flow with Mach numbers higher than 0.3 increases with the Mach number (compressibility effects).

  • The friction factor depends on the material of the microchannel walls (metals, semi-conductors, and so on) and/or on the test fluid (polar fluid or non-polar), thus evidencing the importance of electro-osmotic phenomena at the microscale level.

  • The friction factor depends strongly on the relative roughness of the walls of the microchannels, also in the laminar regime.

From these results one can conclude that, for a gas flowing through a microchannel, deviations of the friction factor from classical theory can be explained in terms of several effects. This paper will focus on the effects of gas rarefaction and compressibility on the Poiseuille number.

2 Rarefaction and compressibility effects

The rarefaction effect becomes important when the molecular mean free path of the flowing medium has the same order of magnitude as the channel size; in this case, the continuum model is not suitable for the fluid. If the Knudsen number (Kn=λ/h, where λ is the molecular mean free path and h is the channel depth) ranges between 0.001 and 0.1, the flow can still be approached by the slip-flow model based on the Navier–Stokes equations with slip conditions at the walls. The rarefaction effects could become important in microchannels with hydraulic diameters lower than 100 μm in the presence of a gas flowing at low pressure. The gas rarefaction leads to a reduction of the friction factor for increasing Knudsen numbers, as demonstrated theoretically by Ebert and Sparrow (1965), Sreekant (1968), Harley et al. (1995) and Morini and Spiga (1998).

The compressibility effects can be significant when a gas flows through a microchannel because the small hydraulic diameters allow high velocities and high Mach numbers to be reached even at low Reynolds numbers. Indeed the pressure drop per unit length through a microchannel is large and can determine a relevant variation of the gas density along the microchannel, which induces an acceleration in the gas and a change of the velocity profile not only quantitatively but also in shape. As a consequence, the friction factor increases with the inlet Mach number. It is important to remark that, for developing flows in microchannels, the Mach number could change strongly between the inlet and the outlet of the microchannel, as showed by Guo and Wu (1997). Li et al. (2000) demonstrated experimentally that the effect of the compressibility can be neglected for an average Mach number (between inlet and outlet) lower than 0.3 and if the pressure drop is less than about 5% of the initial static pressure.

In general, for gas flows in a microchannel, the effects of rarefaction and compressibility are coupled and tend to contrast each other. For gas flows, the Reynolds number (Re = hW/ν), the Knudsen number (Kn) and the Mach number (Ma = W/c, where c is the speed of sound) can be related by using the results of the classical kinetic theory (Morini et al. 2003):
$$ \operatorname{Re} = \frac{{{\text{Ma}}}} {{{\text{Kn}}}}\sqrt {\frac{{\pi k}} {2}} $$
(1)
where k is the specific heat ratio (cp/cv) for the considered gas.
With Eq. 1, for an imposed rarefaction degree (Kn), one can calculate the maximum value of the Reynolds number for which the Mach number is less than 0.3. For Knudsen numbers in the slip-flow region, Fig. 1 shows the range of Reynolds numbers for which the flow can be considered incompressible. It is evident that the compressibility effects can be neglected for high Knudsen numbers only when the Reynolds number is very low.
Fig. 1

Limit of validity of the incompressibility assumption for different gases in terms of Reynolds number as a function of the Knudsen number.

Bearing in mind Fig. 1, it is possible to analyse critically some published results on the friction factors for gas flow through microchannels. As an example, let us consider the first work on this topic by Wu and Little (1983). They tested rectangular and trapezoidal channels with a hydraulic diameter ranging between 45.5 and 83.1 μm. The working fluids were N2, H2 and Ar. They measured the friction factors while varying Reynolds number between 100 and 10,000. They compared the measured friction factors with those predicted for smooth pipes by the Moody chart. Their experimental values were higher than conventional numbers for both the laminar and turbulent regimes. They explained this fact by invoking the effects of the large relative roughness and its non-uniformity along the wetted perimeter of the microchannel. By observing Fig. 1, it is well evident that for Reynolds numbers greater than 100 the compressibility effects cannot be ignored. This fact was however ignored by the authors which could justify the higher values of friction factor observed.

There are many theoretical works in which the rarefaction effects for incompressible flow are studied in the slip-flow regime by solving the Navier–Stokes equation with a modified boundary condition on the velocity. For a microchannel having an axially uniform cross-section of area Ω and perimeter Γ, a two-dimensional (2-D) analysis of the steady-state flow field can be made under the following assumptions:
  1. 1.

    The transport processes are steady-state and 2-D.

     
  2. 2.

    The gas is Newtonian, incompressible, isothermal and with a fully developed velocity profile.

     
  3. 3.

    All channel walls are rigid and non-porous.

     
  4. 4.

    The physical properties of the fluid are constant.

     
The Navier-Stokes momentum equation for the fluid can be written in a dimensionless form as follows:
$$ \nabla ^{*2} V + p^* = 0 $$
(2)
where V is the dimensionless velocity (u/W).
Beskok et al. (1996) and Beskok and Karniadakis (1999) proposed an isothermal incompressible slip-flow model in which the velocity slip to the channel walls satisfies the equation:
$$ \left[ {V + \kappa \frac{{2 - \sigma _v }} {{\sigma _v }}\frac{{{\text{Kn}}}} {{1 - b{\text{Kn}}}}\frac{{\partial V}} {{\partial n}}} \right]_w = 0 $$
(3)
where the coefficient κ can be rigorously determined by means of the Boltzmann equation (see Sharipov 2003) and assumes values near to 1 (1.1 for hard spheres), σv is the accommodation coefficient of the solid surface (Saxena and Joshi 1981), b is a high-order slip coefficient, and ∂/∂n denotes the normal derivative at the wall. For the sake of completeness, it is important to point out that different velocity slip boundary conditions have been defined in the open literature. Recently, Aubert and Colin (2001) proposed a high-order boundary condition to take into account the slip-flow effects at the walls, where the accommodation factors are treated as empirical constants of the model.
Morini et al. (2003) numerically solved the differential problem defined by Eqs. 2 and 3 by assuming b=0 and using the modified Knudsen number (see Yu and Ameel 2001), defined as follows:
$$ {\text{Kn}}_v = \kappa \frac{{2 - \sigma _v }} {{\sigma _v }}{\text{Kn}} $$
(4)
in which the accommodation factor and the Knudsen number are taken into account together. Morini et al. (2003) studied the velocity distribution in microchannels with trapezoidal (ϕ=54.74°), rectangular (ϕ=90°) and hexagonal (double trapezoidal) cross-sections typical of KOH-etched microchannels (see Fig. 2).
Fig. 2

Rectangular, trapezoidal and double-trapezoidal KOH-etched microchannels.

The Poiseuille number, defined as the Fanning friction factor for the fully developed flow (f) times the Reynolds number (Re), has been calculated through the following relation (Shah and London 1978):
$$ \left( {f\operatorname{Re} } \right)_{{\text{Kn}}_v } = - \frac{1} {{2\Omega ^* }}\int\limits_{\Gamma ^* } {\left. {\frac{{\partial V}} {{\partial n^* }}} \right|_{\Gamma ^* } {\text{d}}\Gamma ^* } $$
(5)
with Ω*=Ω/Dh2, n*=n/Dh and Γ*=Γ/Dh.
For any considered cross-section, the reduction of the friction factor due to rarefaction effects was calculated by comparing the Poiseuille number for an assigned value of the modified Knudsen number with the value that the Poiseuille number assumes for Kn=0 (i.e., continuum flow):
$$ \Phi = \frac{{\left( {f\operatorname{Re} } \right)_{{\text{Kn}}_v } }} {{\left( {f\operatorname{Re} } \right)_{{\text{Kn}} = 0} }} $$
(6)
The friction factor reduction coefficient Φ depends on the geometry of the cross-section and on the modified Knudsen number. Morini et al. (2003) showed that the friction factor reduction coefficient can be expressed as follows:
$$ \Phi = \frac{1} {{1 + \alpha {\text{Kn}}_v }} $$
(7)
The appropriate value of α for rectangular, trapezoidal and double-trapezoidal microchannels was numerically determined as a function of the cross-section aspect ratio. The values of α for rectangular, trapezoidal and double-trapezoidal cross-sections are reported in Table 1 as a function of the aspect ratio β of the channel (defined as the ratio between the depth h and the maximum width a).
Table 1

Values of the coefficient α defined by Eq. 7 and Poiseuille number fRe (Eq. 5) for different cross-sections.

Rectangular

β

0

0.01

0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α

6.00

6.02

6.01

6.05

6.12

6.18

6.32

6.44

6.61

6.83

7.12

7.51

7.88

fReKn=0

24

23.68

22.48

21.17

19.07

17.51

16.37

15.55

14.98

14.61

14.38

14.26

14.23

Trapezoidal

β

0

0.05

0.16

0.21

0.29

0.35

0.41

0.45

0.52

0.55

0.62

0.66

0.707

α

6.00

6.05

6.19

6.25

6.40

6.61

6.90

7.18

7.88

8.29

9.30

10.04

10.76

fReKn=0

24

22.17

18.65

17.24

15.57

14.69

14.06

13.83

13.65

13.66

13.69

13.62

13.31

Double-trapezoidal

β

0

0.02

0.1

0.2

0.4

0.5

0.6

0.8

0.9

1

1.2

1.3

1.414

α

6.00

6.00

6.05

6.09

6.26

6.42

6.67

7.73

8.43

9.27

11.29

12.41

13.51

fReKn=0

24

23.46

21.51

19.50

16.75

15.92

15.40

15.03

15.04

15.11

15.08

14.79

14.06

Circular

α

8.00

            

fReKn=0

16

            

By means of the data from Table 1 and Eqs. 6 and 7, it is possible to calculate the theoretical Poiseuille number for a gas through a KOH-etched microchannel as a function of the channel aspect ratio and of the modified Knudsen number.

3 The role of experimental uncertainty on the Poiseuille number

In general, the experimental determination of the friction factor in a microchannel is achieved through measurement of the pressure difference between two manifolds. For incompressible flow, this pressure differential can be expressed as follows:
$$ \Delta p = \frac{{\rho W^2 }} {2}\left[ {f\frac{L} {{D_h }} + \sum\limits_i {K_i } } \right] $$
(8)
where W is the average velocity, L is the microchannel length and Ki represents the minor losses due to the inlet, exit and hydrodynamic development length. The above expression can be solved for fRe, thus:
$$ f\operatorname{Re} = 2\Omega ^* \frac{{\Delta pD_h^4 }} {{Q\mu L}} - \frac{1} {{\Omega ^* }}\frac{{\rho Q}} {{L\mu }}\sum\limits_i {K_i } = F_I - F_{II} $$
(9)
where Q is the volumetric flow rate and Ω * is the dimensionless cross-sectional area (Ω/Dh2). The relative uncertainty in measuring fRe, δfRe, can be estimated according to the following formula (Moffat 1988):
$$ U_{f\operatorname{Re} } = \pm \sqrt {\left( {\frac{{\partial f\operatorname{Re} }} {{\partial F_{\text{I}} }}U_{F_{\text{I}} } } \right)^2 + \left( {\frac{{\partial f\operatorname{Re} }} {{\partial F_{{\text{II}}} }}U_{F_{{\text{II}}} } } \right)^2 } = \pm \sqrt {U_{F_{\text{I}} }^2 + U_{F_{{\text{II}}} }^2 } $$
(10)
The contribution of FII can be ruled out with appropriate experiment design, as reported by Mala and Li (1999) and Celata et al. (2002); in this case, the uncertainty of the Poiseuille number can be expressed through the relative uncertainty in the measured quantities (δΔp, δDh, δQ, ...):
$$ U_{f\operatorname{Re} } = U_{F_{\text{I}} } = \pm F_{\text{I}} \sqrt {\left( {\frac{{U_{D_h } }} {{D_h }}} \right)^2 + \left( {\frac{{U_{\Delta p} }} {{\Delta p}}} \right)^2 + \left( {\frac{{U_{\Omega ^* } }} {{\Omega ^* }}} \right)^2 + \cdots } = \pm F_{\text{I}} \cdot 4\delta _{D_h } \sqrt {1 + \left( {\frac{{\delta _{\Delta p} }} {{4\delta _{D_h } }}} \right)^2 + \left( {\frac{{\delta _{\Omega ^* } }} {{4\delta _{D_h } }}} \right)^2 + \cdots } $$
(11)
From Eq. 11 the relative uncertainty on fRe can thus be obtained:
$$ \delta _{f\operatorname{Re} } = \frac{{U_{F_{\text{I}} } }} {{F_{\text{I}} }} = \pm 4\delta _{D_h } \sqrt {1 + \left( {\frac{{\delta _{\Delta p} }} {{4\delta _{D_h } }}} \right)^2 + \left( {\frac{{\delta _{\Omega ^* } }} {{4\delta _{D_h } }}} \right)^2 + \cdots } $$
(12)
We shall restrict our analysis to the latter form, which is more amenable to algebraic manipulation, but it is to be remarked that the criterion laid out applies regardless of the expression of both the experimental uncertainty and the friction factor reduction. Moreover, we shall consider only positive values of the experimental uncertainty, as these are critical to obtain valid experimental results.

It is interesting to note that measurement of the hydraulic diameter is the most critical to the overall measurement uncertainty due to the factor of 4 in Eq. 12. This means that even if the relative uncertainty on the microchannel diameter is comparatively low, (1–3% by using an SEM), this uncertainty contributes alone to a 4–12% uncertainty on fRe.

If the gas flowing through a microchannel can be considered incompressible, it is clear that the effect of gas rarefaction cannot be evidenced if:
$$ \left( {f\operatorname{Re} } \right)_{{\text{Kn}}} (1 + \delta _{f\operatorname{Re} } ) \geq \left( {f\operatorname{Re} } \right)_{{\text{Kn}} = 0} $$
(13)
By using Eq. 7 it is possible to conclude that Eq. 13 is satisfied only if:
$$ \frac{{1 + \delta _{f\operatorname{Re} } }} {{1 + \alpha Kn}} \geq 1 $$
(14)
Equation 14 allows calculation of the lowest Knudsen number for which it is possible to experimentally evidentiate the effects on friction factor due to rarefaction:
$$ {\text{Kn}}_{{\text{min}}} = \frac{{\delta _{f\operatorname{Re} } }} {\alpha } $$
(15)
Since α depends on the geometry of the cross-section and on the aspect ratio of the channel, it is possible to calculate for different shapes the value assumed by Knmin for a given experimental facility (i.e., for a fixed value of the uncertainty on fRe).
In Fig. 3 the minimum Knudsen number is given as a function of the aspect ratio and of the uncertainty on fRe for the three different shapes of silicon KOH-etched microchannels shown in Fig. 2. For a fixed value of the uncertainty on fRe, the minimum Knudsen number decreases when the microchannel aspect ratio increases. This fact is due to the definition adopted in this paper for the Knudsen number. Since the channel depth (h) has been used as the characteristic length in Knudsen number definition, by increasing the microchannel aspect ratio for a fixed depth the wetted perimeter is smaller and hence the effects due to the slip-flow at the walls become less evident. It is to be noticed that the role of the microchannel aspect ratio is different for the three cross-sections considered. For a fixed value of the uncertainty, the minimum Knudsen number depends strongly on the aspect ratio for a double-trapezoidal microchannel; on the contrary, for a rectangular microchannel the minimum Knudsen number depends on the channel aspect ratio weakly.
Fig. 3

Minimum Knudsen number as a function of the experimental uncertainty on fRe and the aspect ratio for rectangular (a), trapezoidal (b), and double-trapezoidal (c) KOH-etched microchannels.

The behaviour of the three cross-sections considered here tends to become similar when their aspect ratio tends to zero. By observing Fig. 3, it is evident how the experimental uncertainty on the Poiseuille number can partially mask the effects due to the slip-flow at the walls. For example, a value of δfRe equal to 20% covers up the effects of the slip-flow up to Kn equal to 0.03 for a rectangular cross-section.

Let us consider an experimental facility designed to measure the friction factor of helium at atmospheric conditions (300 K) through a silicon microchannel. We suppose that, by considering the characteristics of each component of the measurement chain, the uncertainty on fRe can be estimated to be 10%. The microchannels to be tested have a width of 200, 100, 50 and 25 μm. By using the data shown in Fig. 3 for rectangular, trapezoidal and double trapezoidal cross-sections, the minimum Knudsen number for which it is possible to experience the effect of the slip-flow can be determined. For a fixed value of the uncertainty on fRe equal to 10% the minimum Knudsen number for each cross-section is shown in Fig. 4 as a function of the channel aspect ratio. If the mean free path for the gas at atmospheric conditions is known (λ=0.194 μm), one can calculate the value of the Knudsen number (λ/h) varying the channel depth for a fixed width. By superimposing these curves to the three trends of the minimum Knudsen number as a function of the aspect ratio, one can calculate the range of the channel aspect ratio for which it is possible to experience the effects of the rarefaction with the considered experimental apparatus (Fig. 4).
Fig. 4

A comparison of the minimum Knudsen number as a function of the aspect ratio of rectangular, trapezoidal and double-trapezoidal KOH-etched microchannels.

From Fig. 4 it can be noticed that for a microchannel having a width equal to 200 μm the effect of the slip-flow can be evidenced only for 0<β<0.05, which means that the channel depth has to be less than 10 μm. This result is independent of the channel geometry. For a microchannel having a width equal to 50 μm, the minimum value of the Knudsen number is reached for β=0.25; this underlines the fact that the slip-flow effects can be detected only when using microchannels having a depth smaller than 12.5 μm. For a microchannel having a width of 25 μm, it is interesting to note that for trapezoidal microchannels any value of the aspect ratio can be considered able to evidence the effects of the rarefaction; on the contrary, only the rectangular and double-trapezoidal microchannels having an aspect ratio less than 0.5 (h=12.5 μm) can be used to single out the slip-flow effects. It is to be remarked that, in the example considered here, the minimum value of the Knudsen number for which the rarefaction effects are not masked by the experimental uncertainty on fRe ranges from 0.014 to 0.017. By introducing the minimum Knudsen number into Eq. 1, the maximum value of the Reynolds number (based on the channel depth) is obtained for which the compressibility effects can be neglected. By imposing an average Mach number of 0.3 in Eq. 1, the maximum value of the Reynolds number ranges between 28.5 and 34.6.

4 Discussion of the results

The results obtained in this paper have been employed to review some experimental analysis quoted in the open literature on the frictional characteristics of gases through microchannels.

An in-depth analysis of the gas flow in microchannels was conducted by Harley et al. (1995). In this case the experimental investigation was made for low Reynolds numbers ranging between 0.43 and 0.012. The exit Mach number ranged from 0.002 to 0.0189 which means that the rarefaction effects can be considered uncoupled from those due to compressibility. The average Knudsen number ranged between 0.002 and 0.129; the authors declared an uncertainty on the pressure drop of 2.5%, on the volumetric flow rate 2.2%, and on the hydraulic diameter 2%. The overall uncertainty on the Poiseuille number was 12%. They tested eight different trapezoidal microchannels etched on 〈100〉 silicon. By using the data reported in Table 1 and Eq. 9, it is possible to calculate the minimum value of the Knudsen number for each microchannel for which the rarefaction effects can be observed. In Fig. 5 the operative conditions of the shallower channels tested (named JH10, JP9, V3 and JH6) are shown and compared with the theoretical value of the minimum Knudsen for which the rarefaction effects can be experienced. It is interesting to note that only the channel JH6 having a depth of 0.51 μm can be correctly used in order to test the rarefaction effects. This fact means that much of the experimental data from Harley et al. (1995) cannot be regarded as meaningful in demonstrating the influence of rarefaction on the reduction of the friction factor observed experimentally.
Fig. 5

Comparison between the experimental conditions of Araki et al.(2000) and Harley et al.(1995) and the theoretical limit on the Knudsen number.

Araki et al. (2000) performed an experimental investigation of nitrogen and helium flow through silicon KOH-etched trapezoidal microchannels having hydraulic diameters of 10.3 μm (channel A, β=0.707), 9.41 μm (channel B, β=0.134) and 3.92 μm (channel C, β=0.05). At very low Reynolds numbers (0.042<Re<4.19) the frictional resistance was found to be lower than that in channels of conventional size. This effect has been ascribed by the authors to rarefaction. The experimental uncertainty on the friction factor was estimated at 10.9%. By applying the approach described here to the experimental data of Araki et al. (2000), only in the experiments conducted with helium in channels A and B and with nitrogen in channel C was the average Knudsen number higher than the theoretical minimum value, as shown in Fig. 5. This fact can be explained by observing that helium has a mean free path larger than nitrogen at the same operative conditions; as a consequence, for a fixed microchannel, the Knudsen numbers that can be obtained with helium flow are higher that those obtained by using nitrogen.

5 Conclusions

This paper is devoted to analysing the friction factor of gas flow through microchannels; in particular, the conditions for experimentally evidencing rarefaction effects on the pressure drop have been investigated theoretically. The main conclusions that have been drawn can be summarised as follows:
  • The actual experimental uncertainty on fRe (in general equal to 8–14%) masks the effects due to rarefaction for small Knudsen numbers. It has been demonstrated that in the range 0.001<Kn<0.01 the effects due to the slip-flow cannot be investigated with the present values of uncertainty on fRe.

  • To obtain experimental data able to validate the classical slip-flow models, only microchannels having a depth of 1–20 μm can be employed. If microchannels having larger depths are to be used, the test fluids must be gases with a large mean free path (for instance, helium is better than hydrogen and nitrogen).

  • To consider the gas as incompressible, the pressure drop cannot be higher than 5% of the initial static pressure and the Mach number cannot be larger than 0.3. These conditions impose a severe limitation on the Reynolds numbers, that cannot be larger than 10–40.

The theoretical data reported in this paper can be useful in the design phase because they allow the designer to select appropriate geometrical characteristics of the microchannels and the range of the Reynolds numbers to use in an experimental campaign focused on evidencing the role of gas rarefaction on the frictional resistance through microchannels.

Acknowledgements

This work has been funded through grant I/R/266/02 by ASI (Italian Space Agency) and COFIN 03 by MIUR-URST.

Copyright information

© Springer-Verlag 2004