Application to laminar flows

The development of microfluidics makes the study of the movement of a small particle in a laminar flow an especially dynamic topic nowadays. The eontents of this ehapter, without having gone into the detail of theoretioal developments, show the eomplexity and difficulty of the subject. A pertinent modeling of the movement of a particle, in order to encompass phenomena such as the lesuspension of particles from walls or the migration of particles in a direction perpendicnlar to the flow streamlines, should incorporate the lift force exerted on a partiele, even though it is much weaker than the drag force. 

For a particle in a turbulent flow whose eharacteristie seales of veloeity and length are denoted respeetively by and, the results described in this ehapter mean that the movement of the partiele ahgns with the direetion of the turbulent fluid flow if the eharacteristie time tp of the transient regime, during whieh the 

13 See the books by Nielsen (Coastal Bottom Boundary Layers and Sediment Transport, World Scientific, 1992) and Fredsoe and Deigaard (Meehanics of Coastal Sediment Transport, World Scientific, 1992). 

Centrifugal separation of solid particles in a fluid, but also of non-miscible droplets in another liquid or of gas bubbles in a liquid, is a frequently employed process. Its principle has already been described in the previous chapter on the basis of the BBOT equations. 

1 This chapter was partly inspired by the handouts of the course taught at ENSIC by N. Midoux (Les operations unitaires mecaniques du genie chimique, ENSIC-CRIFIC Document). 

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Practical applications to laminar flow reactors are still mainly in the research literature. The first good treatment of a variable-viscosity reactor is... 

Wilck M. and Stratmann F. (1997) A 2-D multicomponent modal aerosol model and its application to laminar flow reactors. J. Aerosol. Sci. 28, 959-972. 

The semi-empirical global transport correlation proposed by Hawthorn [15] is the most commonly used for the definition of local Nusselt and Sherwood numbers, applicable to laminar flows in square ducts ... 

If the pressure difference between 1 cm in porous media is 1 atm, viscosity of fluid is 1 cp, and volume flow rate is 1 cc. s for the cross section with 1 cm, is 1 darcy. This Darcy s law is applicable to laminar flow with Reynolds number (Re) less than 1-10. 

The effective diffusivities determined from limiting-current measurements appear at first applicable only to the particular flow cell in which they were measured. However, it can be argued plausibly that, for example, rotating-disk effective diffusivities are also applicable to laminar forced-convection mass transfer in general, provided the same bulk electrolyte composition is used (H8). Furthermore, the effective diffusivities characteristic for laminar free convection at vertical or inclined electrodes are presumably not significantly different from the forced-convection diffusivities. 

When calculating pressure relationships and types of gas (viscosity) it is necessary to keep in mind that different equations are applicable to laminar and molecular flow the boundary between these areas is very difficult to ascertain. As a guideline one may assume that laminar flow is present at leak rates where Q > 10 mbar l/s and molecular flow at leak rates where Q < 10 mbar l/s. In the intermediate range the manufacturer (who is liable under the guarantee terms) must assume values on the safe side. The equations are listed in Table 5.2. 

This is termed the boundary layer momentum integral equation. As previously mentioned, it is equally applicable to laminar and turbulent flow. In laminar flow, u is the actual steady velocity while in turbulent flow it is the time averaged value. 

It is noted that Equations (10-34a) and (l0-34b) can be applied to laminar flows. Other expressions are available for turbulent flows (e.g.. Bird et al., [I960]). Equation (10-34c) is applicable over a wide range of flow rates. 

This shows that the forward velocity is a parabolic function of position in the tube varying from zero at the wall where r is equal to Tq, to the maximum value at the center of the cylindrical tube, where is equal to —ro(dP/dx)/(4r ). It is emphasized that equation (6.3.4) is only applicable to laminar or non-turbulent flow in a cylindrical tube. When a liquid flows under different geometrical boundary conditions, the relationship between the flux and the force is not the same. 

The Peclet number reflects the ratio of heat transferred by convection to that transferred by conduction and is most commonly found in applications in laminar flow or with liquid metals. 

These types of three-dimensional electrodes on microchips could also be used for a wider variety of biomethods, including electroporation, drug delivery, and electrostimulated cell culturing. Three-dimensional electrodes will also have applicability in laminar flow-based fuel cells and biofuel cells due to the increase in roughness factor. Three-dimensional electrode... 

This friction factor is applicable to gas flow only and incoiporates both turbulent and laminar contributions. Since the shapes of the flow channels of the structured paclrings are geometrically similar, one would expect that a single set of constants for Eq. (5.8-18) would cover all packing sizes. 

Where H is the charmel height (the smaller dimension in a rectangular channel), tw,av the average wall shear stress, V the kinematic viscosity, and p the density of the fluid. In internal flows, the laminar to turbulent transition in abrupt entrance rectangular ducts was found to occur at a transition Reynolds number Ret = 2200 for an aspect ratio ac = I (square ducts), to Ret = 2500 for flow between parallel planes with = 0 [4]. For intermediate channel aspect ratios, a linear interpolation is recommended. For circular tubes. Ret = 2300 is suggested. These transition Reynolds number values are obtained from experimental observations in smooth channels in macroscale applications of 3 mm or larger hydraulic diameters. Their applicability to microchannel flows is still an open question. 

In section 16.5, we discuss the lift force applied on a particle in a unidirectional flow. This force is not taken into account by the BBOT equations. Finally, we conclude with the application of the results presented in this chapter to laminar flows, and then to turbulent flows (section 16.7). 

The most important engineering application of laminar flow at low Reynolds Number is hydrodynamic lubrication that is discussed next. Other applications, such as flow in capillaries of small diameter, are less important in engineering, and therefore, consideration of this type of flow is postponed to the end of this chapter. 

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. 

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