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

## Authors

- First Online:

- Received:
- Accepted:

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

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## 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

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 (

*f*Re) 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.

## 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.

*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):

*k*is the specific heat ratio (

*c*

_{p}/

*c*

_{v}) for the considered gas.

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 N_{2}, H_{2} 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.

- 1.
The transport processes are steady-state and 2-D.

- 2.
The gas is Newtonian, incompressible, isothermal and with a fully developed velocity profile.

- 3.
All channel walls are rigid and non-porous.

- 4.
The physical properties of the fluid are constant.

*V*is the dimensionless velocity (

*u*/

*W*).

*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), σ

_{v}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.

*b*=0 and using the modified Knudsen number (see Yu and Ameel 2001), defined as follows:

*f*) times the Reynolds number (Re), has been calculated through the following relation (Shah and London 1978):

*D*

_{h}

^{2},

*n**=

*n*/

*D*

_{h}and Γ*=Γ/

*D*

_{h}.

*h*and the maximum width

*a*).

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 |

| 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 |

| 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 |

| 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 | ||||||||||||

| 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

*W*is the average velocity,

*L*is the microchannel length and

*K*

_{i}represents the minor losses due to the inlet, exit and hydrodynamic development length. The above expression can be solved for

*f*Re, thus:

*Q*is the volumetric flow rate and Ω

^{*}is the dimensionless cross-sectional area (Ω/

*D*

_{h}

^{2}). The relative uncertainty in measuring

*f*Re, δ

_{fRe}, can be estimated according to the following formula (Moffat 1988):

*F*

_{II}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 (δ

_{Δp}, δ

_{Dh}, δ

_{Q}, ...):

*f*Re can thus be obtained:

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 *f*Re.

_{min}for a given experimental facility (i.e., for a fixed value of the uncertainty on

*f*Re).

*f*Re for the three different shapes of silicon KOH-etched microchannels shown in Fig. 2. For a fixed value of the uncertainty on

*f*Re, 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.

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 δ_{fRe} equal to 20% covers up the effects of the slip-flow up to Kn equal to 0.03 for a rectangular cross-section.

*f*Re 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

*f*Re 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).

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 *f*Re 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.

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

The actual experimental uncertainty on

*f*Re (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*f*Re.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.