Poiseuille number

The data on pressure drop in irregular channels are presented by Shah and London (1978) and White (1994). Analytical solutions for the drag in micro-channels with a wide variety of shapes of the duct cross-section were obtained by Ma and Peterson (1997). Numerical values of the Poiseuille number for irregular microchannels are tabulated by Sharp et al. (2001). It is possible to formulate the general features of Poiseuille flow as follows ... 

Glass and silicon tubes with diameters of 79.9-166.3 iim, and 100.25-205.3 am, respectively, were employed by Li et al. (2003) to study the characteristics of friction factors for de-ionized water flow in micro-tubes in the Re range of 350 to 2,300. Figure 3.1 shows that for fully developed water flow in smooth glass and silicon micro-tubes, the Poiseuille number remained approximately 64, which is consistent with the results in macro-tubes. The Reynolds number corresponding to the transition from laminar to turbulent flow was Re = 1,700—2,000. 

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. 

Fig. 3.1 Dependence of the Poiseuille number on the Reynolds number. Reprinted from Li et al. (2003) with permission...

Maynes and Webb (2002) presented pressure drop, velocity and rms profile data for water flowing in a tube 0.705 mm in diameter, in the range of Re = 500-5,000. The velocity distribution in the cross-section of the tube was obtained using the molecular tagging velocimetry technique. The profiles for Re = 550,700,1,240, and 1,600 showed excellent agreement with laminar flow theory, as presented in Fig. 3.2. The profiles showed transitional behavior at Re > 2,100. In the range Re = 550-2,100 the Poiseuille number was Po = 64. 

Lelea et al. (2004) investigated experimentally fluid flow in stainless steel microtubes with diameter of 100-500 pm at Re = 50-800. The obtained results for the Poiseuille number are in good agreement with the conventional theoretical value Po = 64. Early transition from laminar to turbulent flow was not observed within the studied range of Reynolds numbers. 

Fig. 3.3 The normalized Poiseuille number as a function of pressure for carbon tetrachloride in 10 pm micro-tube. Reprinted from Cui et al. (2004) with permission...

Author Micro-channel Fluid Re Poiseuille number RCcr... 

Author Micro-channel Shape 4i [mm] Fluid Re P exp/POtheor Poiseuille number Dependence on Re Relative roughness % fe/rol RCcr... 

A study of forced convection characteristics in rectangular channels with hydraulic diameter of 133-367 pm was performed by Peng and Peterson (1996). In their experiments the liquid velocity varied from 0.2 to 12m/s and the Reynolds number was in the range 50, 000. The main results of this study (and subsequent works, e.g., Peng and Wang 1998) may be summarized as follows (1) friction factors for laminar and turbulent flows are inversely proportional to Re and Re ", respectively (2) the Poiseuille number is not constant, i.e., for laminar flow it depends on Re as PoRe ° (3) the transition from laminar to turbulent flow occurs at Re about 300-700. These results do not agree with those reported by other investigators and are probably incorrect. 

Li et al. (2003) studied the flow in a stainless steel micro-tube with the diameter of 128.76-179.8 jm and relative roughness of about 3-4%. The Poiseuille number for tubes with diameter 128.76 and 171.8 jm exceeded the value of Po corresponding to conventional theory by 37 and 15%, respectively. The critical value of the Reynolds number was close to 2,000 for 136.5 and 179.8 pm micro-tubes and about 1,700 for micro-tube with diameter 128.76 pm. 

It appears that the Poiseuille number in a rectangular channel depends on the aspect ratio, e. In order to reveal an explicit form of the dependence (p e), it is necessary to solve the problem defined by Eqs. (3.8) and (3.9) to obtain... 

The data on the drag for micro-tube diameters of 16.6, 19.7, 26.3 and 32.2 pm are presented in Fig. 3.11 in the form of the dependence of the Poiseuille number on Re. The latter was determined by an average of the mixed-mean temperature at the inlet and outlet of the micro-tube. The data of Pig. 3.11 show that the Poiseuille number practically shows no dependence on Re in the range 500 < Re < 2,000. The... 

The data presented in the previous chapters, as well as the data from investigations of single-phase forced convection heat transfer in micro-channels (e.g., Bailey et al. 1995 Guo and Li 2002, 2003 Celata et al. 2004) show that there exist a number of principal problems related to micro-channel flows. Among them there are (1) the dependence of pressure drop on Reynolds number, (2) value of the Poiseuille number and its consistency with prediction of conventional theory, and (3) the value of the critical Reynolds number and its dependence on roughness, fluid properties, etc. 

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). 

The results obtained by Brutin and Tadrist (2003) showed a clear effect of the fluid on the Poiseuille number. Figure 3.14 shows results of experiments that were done in the same experimental set-up for hydraulic diameters of 152 and 262 pm, using distilled water and tap water. The ion interactions with the surface can perhaps explain such differences. Tap water contains more ions such as Ca +, Mg +, which are 100 to 1,000 times more concentrated than H3O+ or OH . In distilled water only H30 and OH exist in equal low concentrations. The anion and cation interactions with the polarized surface could modify the friction factor. This is valid only in the case of a non-conducting surface. 

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. 

The uncertainty of calculating the Poiseuille number from the measurements must be taken into account. The viscosity-pressure relationship of certain liquids (e.g., isopropanol, carbon tetrachloride) must be kept in mind to obtain the revised theoretical flow rate. The effect of evaporation from the collection dish during the mass flow rate measurement must be taken into consideration. The effect of evaporation of collected water into the room air may not be negligible, and due to the extremely low mass flow rates through the micro-channel this effect can become significant. 

The values of the Nusselt and the Poiseuille numbers for heat transfer and friction for fully developed laminar flows through specifled channels are presented in Table 7.1 (Shah and London 1978). 

Table 7.1 The Nusselt and the Poiseuille numbers for fuUy developed laminar flow...

Figure 2.35 Cross-section through a staggered arrangement of micro fins designed for heat transfer enhancement in a micro channel (above) and ratio of Nusselt and Poiseuille numbers as a function of air flow per unit area for different total fin lengths (below), taken from [127].

For fully developed laminar flows, it is obvious that the product of the friction factor and the Reynolds number is constant. This product is called the Poiseuille number. 

Scaling the lengths with D, x = x / A, it is seen that the Poiseuille number represents a non-dimensional pressure loss... 

An error of 2% on a channel dimension can lead to a 14 % error in the Darcy coefficient determination. It is essential to use an adapted instrumentation to measure the geometrical characteristics of a channel. Sometimes, the cross section may not be the same from one end of a channel to the other and, if necessary, the manufacturer s data must be verified carefully. An uncertainty analysis on the Poiseuille number determination is given by Celata [8] following the work by Holman [19]... 

From the velocity expression the Poiseuille number is calculated as... 

In a simplified approach, it is supposed that a gas blanket of thickness 8 completely separates the liquid from the solid wall (Figure 9). For a liquid flowing between two plates, Sabry gave the Poiseuille number as ... 

The evolution of the Poiseuille number f Re) as a function of the Reynolds number is shown on figure 12. It is observed that the classical value for the laminar regime is obtained if the Reynolds number is less than 2000. The laminar turbulent transition occurs for the conventional value. The authors [22] investigated the entrance effects. They conclude that the friction factor is insensitive to the channel height and that there was no sign of a faster transition to turbulence compared to conventional channel flows. 

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