Channelled laminarly flowing liquid

CONVECTIVE DIFFUSION FROM A CHANNELLED LAMINARLY FLOWING LIQUID... 

Because most applications for micro-channel heat sinks deal with liquids, most of the former studies were focused on micro-channel laminar flows. Several investigators obtained friction factors that were greater than those predicted by the standard theory for conventional size channels, and, as the diameter of the channels decreased, the deviation of the friction factor measurements from theory increased. The early transition to turbulence was also reported. These observations may have been due to the fact that the entrance effects were not appropriately accounted for. Losses from change in tube diameter, bends and tees must be determined and must be considered for any piping between the channel plenums and the pressure transducers. It is necessary to account for the loss coefficients associated with singlephase flow in micro-channels, which are comparable to those for large channels with the same area ratio. 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

Two cases are considered. The first, the laminar flow of a thin film down an inclined surface, is important in the heat transfer from a condensing vapour where the main resistance to transfer lies in the condensate film, as discussed in Chapter 9 (Section 9.6.1). The second is the flow in open channels which are frequently used for transporting liquids down a slope on an industrial site. 

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. 

The quasi-one-dimensional model of laminar flow in a heated capillary is presented. In the frame of this model the effect of channel size, initial temperature of the working fluid, wall heat flux and gravity on two-phase capillary flow is studied. It is shown that hydrodynamical and thermal characteristics of laminar flow in a heated capillary are determined by the physical properties of the liquid and its vapor, as well as the heat flux on the wall. 

Peles el al. (2000) elaborated on a quasi-one-dimensional model of two-phase laminar flow in a heated capillary slot due to liquid evaporation from the meniscus. Subsequently this model was used for analysis of steady and unsteady flow in heated micro-channels (Peles et al. 2001 Yarin et al. 2002), as well as the study of the onset of flow instability in heated capillary flow (Hetsroni et al. 2004). 

Experimental measurements of the wall shear stress exerted by a falling liquid film have been reported for the cases of film flow outside a vertical tube (B14) and in a channel of variable slope (F7). In both cases the experimental results in the zone of smooth laminar flow were in agreement... 

The liquid pumping in the microfluidic chip is mostly achieved by using electro-osmotic flow (EOF) [324]. Other liquid pumping methods have also been employed for microfluidic flow. Flow has been employed for fraction collection and generation of concentration gradient. Laminar flow in the microfluidic channel allows liquid-liquid extraction and microfabrication to occur within the channels. Moreover, valving and mixing are needed in order to achieve a better flow control. All these microfluidic flow operations are further described in subsequent sections. 

Magnetic resonance imaging permitted direct observation of the liquid hold-up in monolith channels in a noninvasive manner. As shown in Fig. 8.14, the film thickness - and therefore the wetting of the channel wall and the liquid hold-up -increase nonlinearly with the flow rate. This is in agreement with a hydrodynamic model, based on the Navier-Stokes equations for laminar flow and full-slip assumption at the gas-liquid interface. Even at superficial velocities of 4 cm s-1, the liquid occupies not more than 15 % of the free channel cross-sectional area. This relates to about 10 % of the total reactor volume. Van Baten, Ellenberger and Krishna [21] measured the liquid hold-up of katapak-S . Due to the capillary forces, the liquid almost completely fills the volume between the catalyst particles in the tea bags (about 20 % of the total reactor volume) even at liquid flow rates of 0.2 cm s-1 (Fig. 8.15). The formation of films and rivulets in the open channels of the structure cause the further slight increase of the hold-up. 

Fig. 8.23. Experimental gas-liquid mass transfer coefficients for fully developed laminar flow (monolith section from 80—500 mm) = 25 cpsi square channels A = 50 cpsi square channels ...

Figure 4.10 Schematic illustration of the liquid flow in a channel shown by arrows (a) Laminar flow at low flow rate and (b) turbulent flow at high flow rate.

The velocity profile of the flow in a channel varies across the diameter of the channel regardless of the flow rate. It has a minimum value ( 0) near the channel walls and a maximum value at the center of the flow. This variation in velocities arises from the adhesive forces between the channel walls and the liquid, causing the liquid layers nearest to the walls to be slower than those in the center. Consequently, the layer region nearest to the wall always exhibits a laminar flow even at high Reynolds numbers as the viscous forces dominate [200,201] (Figure 4.11). 

In microscale channels, the viscous forces dominate the inertial effect resulting in a low Reynolds numbers. Hence, laminar flow behavior is dominant and mixing occurs via diffusion. However, in a liquid-liquid system, the interfacial forces acting on the interface add complexity to the laminar flow as the relationship between interfacial forces and other forces of inertia and viscous results in a variety of interface and flow patterns. Gunther and Jensen [202] illustrated this relationship as a function of the channel dimension and velocity as shown in Figure 4.12. The most regularly shaped flow pattern is achieved when interfacial forces dominate over inertia and viscous forces at low Reynolds numbers, as represented in Figure 4.12 by the area below the yellow plane [202,203]. 

---