Friction Factor in Microchannels

It is possible to sum up the main results quoted in the open literatiue on the friction factor in microchannels highlighting the peculiarities, proposed by different authors, with respect to the conventional macrochannels ... 

Friction factor in microchannels Pressure drop in microchannels Transition in microchannels... 

Hetsroni et al. [6] also reexamined previous studies of friction factor in microchannels and drew the same conclusions that they did for transition in microchannels. They found that the anomalous results reported in some studies could be explained by the same factors that contributed to the observation of anomalous transitional behavior. Indeed, in the only study performed to date combining both microPIV and extensive pressure drop measurements. Sharp and Adrian [8] found that transition as measured by microPIV agreed well transition as inferred from friction factor data and also found that their measured friction factors agreed well with macroscale results. As with transition to turbulence, the experimental evidence on friction factors in turbulent microchannel flow shows that microscale flow exhibits the same behavior as macroscale flows. 

The friction factor can thus be determined without measuring the pressure drop along the microchannel but by means of temperature and flow rate measurements. This kind of measurement is unsuitable when macrochannels are tested. For this reason Eq. 25 can be considered as an example of the role of scaling effects and to suggest new measurement procedures at the microscale. This relation has been used by Celata et al. [4] in order to determine the friction factor in microchannels. In addition, since Eq. 25 is valid only in the laminar regime, one can use it to individuate the laminar-to-turbulent transition in tnicrochannels. This has been experimentally demonstrated by Celata et al. [4] and Rands et al. [5]. 

Celata GP, Morini GL, Marconi V, McPhail S, Zummo G (2006) Using viscous heating to determine the friction factor in microchannels - an experimental validation. Exp Fluid Therm Sci 30(8) 725-731... 

Wu HY, Cheng P (2003) Friction factors in smooth trapezoidal silicon microchannels with different aspect ratios. Int J Heat Mass Trans 46(14) 2519-2525... 

For long microchannels for which L Z)h (this condition is very common in microchannels), one can use the fully developed value of the friction factor in the calculation of the pressure drop ... 

Finally, it must be noted that Eq. 17 can be used to determine experimentally the value assumed by the friction factor in an adiabatic microchannel for incompressible liquids ... 

Discrepancy between the conventional theory and the microchannel measurements of friction factor/ in gaseous flow has been attributed to compressibility [6]. In general it appears that/Re for compressible slip flow is less than that for the incompressible case [3]. In a comprehensive study of results of microscale single-phase internal flows [1], it has been found that the only definitive conclusion that can be reached from the currently available data is that gaseous slip flow data indicate an approximate 60% reduction in / compared to macroscale theory at the same Re (while for laminar non-slip water flow/ appears to be approximately 20% higher than the theoretical predictions). The finding for compressible slip flow seems to be supported by a simple analytical correlation with the factor c 0.6 0.05 [3] ... 

Many correlations have been proposed in literature for the friction factor and heat transfer, based on experimental investigations on liquid and gas flow in microchannels. Garimella and Sobhan (2003) presented a comprehensive review of these investigations conducted over the past decade. 

Toh et al. [45] investigated numerically three-dimensional fluid flow and heat transfer phenomena inside heated microchannels. The steady, laminar flow and heat transfer equations were solved using a finite-volume method. The numerical procedure was validated by comparing the predicted local thermal resistances with available experimental data. The friction factor was also predicted in this study. It was found that the heat input lowers the frictional losses, particularly at lower Reynolds numbers. Also, at lower Reynolds numbers the temperature of the water increases, leading to a decrease in the viscosity and hence smaller frictional losses. 

If the characteristic linear dimension of the flow field is small enough, then the measured hydrodynamic data differ from those predicted by the Navier-Stokes equations [79]. With respect to the value in macrocharmels, in microchannels (around 50 microns of section) (i) the friction factor is about 20-30% lower, (ii) the critical Reynolds number below which the flow remains laminar is lower (e.g., the change to turbulent flow occurs at lower linear velocities) and (iii) the Nusselt number, for example, heat transfer characteristics, is quite different [80]. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value for the fully developed flow in the conventional channel. These effects have been investigated extensively in relation to the development of more efficient cooling devices for electronic applications, but have clear implications also for chemical applications. 

The earliest studies related to thermophysieal property variation in tube flow conducted by Deissler [51] and Oskay and Kakac [52], who studied the variation of viscosity with temperature in a tube in macroscale flow. The concept seems to be well-understood for the macroscale heat transfer problem, but how it affects microscale heat transfer is an ongoing research area. Experimental and numerical studies point out to the non-negligible effects of the variation of especially viscosity with temperature. For example, Nusselt numbers may differ up to 30% as a result of thermophysieal property variation in microchannels [53]. Variable property effects have been analyzed with the traditional no-slip/no-temperature jump boundary conditions in microchannels for three-dimensional thermally-developing flow [22] and two-dimensional simultaneously developing flow [23, 26], where the effect of viscous dissipation was neglected. Another study includes the viscous dissipation effect and suggests a correlation for the Nusselt number and the variation of properties [24]. In contrast to the abovementioned studies, the slip velocity boundary condition was considered only recently, where variable viscosity and viscous dissipation effects on pressure drop and the friction factor were analyzed in microchannels [25]. 

One aspect which has been evidenced in numerical and experimental works is the influence of electrostatic interaction between fluid and channel walls on the friction factor and on the Nusselt number. In particular, it has been observed that in the presence of an electric double layer, the Nusselt number increases because the temperature and the velocity gradients at the walls tend to be increased. The electroosmotic interactions between the fluid and the walls could explain, for microchannels with a hydraulic diameter less than 20 pm, the dependence of the Nusselt number on the Reynolds number, even in the laminar regime. 

Since data cover only a few simple entrance geometries, the designer must exercise judgment in the application of the correlations proposed to calculate the friction factor and the convective heat transfer coefficient in the entrance region of a microchannel. 

To eliminate the tmcertainties caused by estimation of minor losses K, K, many authors have suggested performing tests with microchannels of different lengths in this manner it is possible to even out the inlet and outlet minor losses by subtracting the pressure drop of the shorter tube from that of the longer one for a fixed value of the average Reynolds number. By following this method, the value of the friction factor would be calculated as follows ... 

On the contrary, as evidenced in [10], many experimental works published in the last decade seem to be partially in disagreement with the conventional theory, and if these results are used in order to establish the validity of the conventional theory for microchannels in laminar and turbulent regime, the answer obtained is not unequivocal. In fact, some authors found that the predictions of the conventional theory agree with the experimental results on the friction factor however, for the same range of hydraulic diameters, some other authors found the opposite result. 

The fully developed turbulent friction factor seems to be in disagreement with the Blasius equation for smooth microcharmels and with the Colebrook correlation for rough microchannels. 

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

Discrepancy between the conventional theory and the microchannel measurements of friction factor f in gaseous flow has been attributed to compressibility [6]. In general it appears that e for compressible slip flow is less than that for the incompressible case [3]. In a comprehensive study of results of microscale... 

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