Smooth Micro-Channels

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

3 Velocity Field and Pressure Drop in Single-Phase Flows 

Pfund et al. (2000) studied the friction factor and Poiseuille number for 128-521 pm rectangular channels with smooth bottom plate. Water moved in the channels at Re = 60—3,450. In all cases corresponding to Re 2,000 the friction factor was inversely proportional to the Reynolds number. A deviation of Poiseuille number from the value corresponding to theoretical prediction was observed. The deviation increased with a decrease in the channel depth. The ratio of experimental to theoretical Poiseuille number was 1.08 0.06 and 1.12 zb 0.12 for micro-channels with depths 531 and 263 pm, respectively. 

Xu et al. (2000) investigated de-ionized water flow in micro-channels with hy-draulie diameter ranging from 30 to 344 pm at Reynolds numbers ranging from 20 to 4,000. Two test modules were used. The first test module consisted of a cover and an aluminum plate, into which a micro-channel, inlet and outlet sumps were machined. A Plexiglas plate was used to cover the channel. The second module was fabricated from a silicon wafer, and a 5 mm thick Pyrex glass was utilized to 

The transition to turbulent flow occurred at Re of about 1,500. The authors noted that for smaller micro-channels, the flow transition would occur at lower Re. The early transition phenomenon might be affected by surface roughness and other factors. 

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Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

Table 3.3 Experimental results of single-phase fluid flow in smooth micro-channels... 

For Re 2 X 10, the relative roughness that corresponds to the boundary between the smooth and rough channels is k /rQ 0.08. The latter shows that the hydraulic characteristics of the present micro-channels (Table 3.4) are close to characteristics of smooth micro-channels. 

There is a significant scatter between the values of the Poiseuille number in micro-channel flows of fluids with different physical properties. The results presented in Table 3.1 for de-ionized water flow, in smooth micro-channels, are very close to the values predicted by the conventional theory. Significant discrepancy between the theory and experiment was observed in the cases when fluid with unknown physical properties was used (tap water, etc.). If the liquid contains even a very small amount of ions, the electrostatic charges on the solid surface will attract the counter-ions in the liquid to establish an electric field. Fluid-surface interaction can be put forward as an explanation of the Poiseuille number increase by the fluid ionic coupling with the surface (Brutin and Tadrist 2003 Ren et al. 2001 Papautsky et al. 1999). 

For single-phase fluid flow in smooth micro-channels of hydraulic diameter from 15 to 4,010 pm, in the range of the Reynolds numbers Re < Recr, the Poiseuille number, Po, is independent of the Reynolds number. Re. 

For smooth micro-channels the transition from laminar to turbulent flow occurs at Re = 1,500—2,200. For turbulent flows the friction factor maybe calculated as... 

In Chap. 3 the problems of single-phase flow are considered. Detailed data on flows of incompressible fluid and gas in smooth and rough micro-channels are presented. The chapter focuses on the transition from laminar to turbulent flow, and the thermal effects that cause oscillatory regimes. 

A micro-channel heat sink can be classified as single-phase or two-phase according to the state of the coolant inside it. For single-phase fluid flow in smooth... 

Turner et al. (2004) studied the independent variables relative surface roughness, Knudsen number and Mach number and their influence on the friction factor. The micro-channels were etched into silicon wafers, capped with glass, with hydraulic diameters between 5 and 96 pm. Their surface roughness was 0.002 < ks< 0.06 pm for the smooth channels, and 0.33 < / < 1 -6 pm for the glass-capped ones. The surface roughness of the glass micro-channels was measured to be in the range 0.0014 [Pg.39]

This chapter has the following structure in Sect. 3.2 the common characteristics of experiments are discussed. Conditions that are needed for proper comparison of experimental and theoretical results are formulated in Sect. 3.3. In Sect. 3.4 the data of flow of incompressible fluids in smooth and rough micro-channels are discussed. Section 3.5 deals with gas flows. The data on transition from laminar to turbulent flow are presented in Sect. 3.6. Effect of measurement accuracy is estimated in Sect. 3.7. A discussion on the flow in capillary tubes is given in Sect. 3.8. 

Several investigators obtained friction factors in micro-channels with rough walls that were greater than those in smooth wall channels. These observations should be considered taking into account the entrance effects, losses from change in channel size, etc. 

The existence of roughness leads also to decreasing the value of the critical Reynolds number, at which transition from laminar to turbulent flow occurs. The character of the dependence of the friction factor on the Reynolds number in laminar flow remains the same for both smooth and rough micro-channels, i.e., X = const/Re. 

The deviation of the data related to flow in smooth (Table 3.3) and rough (Table 3.4) micro-channels may be a result of heterogeneity of the actual roughness, where height of some roughness peaks signiflcantly exceeds its mean value. For example, under conditions of experiment by Pfund et al. (2000) the mean heigh of... 

Author Smooth/ rough Micro-channel d [ im] L [mm] L/d RCcr Remarks considered characteristic... 

The behavior of the flow in micro-channels, at least down to 50 pm in diameter, shows no difference with macro-scale flow. For smooth and rough micro-channels with relative roughness 0.32% turbulent flow occurs between 1,800 < Recr < 2,200, in full agreement with flow visualization and flow resistance data. In the articles used for the present study there was no evidence of transition below these results. 

Table 4.2 Smooth circular micro-channels experimental conditions ...

Mata, A., Boehm, C., Fleischman, A.J., Muschler, G., Roy, S., Analysis of connective tissue progenitor cell behavior on polydimethylsiloxane smooth and channel micro-textures. Biomed. Microdevices 2002, 4(4), 267-275. 

Bavifere, R., Ayela, F., Le Person, S. and Favre-Marinet, M., An experimental study of water flow in smooth and rectangular micro-channels. To be published. 

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