Reynolds Number system application

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

Starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (u, = /xRe/dpp). The following is an interpolation formula that can be applied over all three settling regimes ... 

A second hold-up correlation reported by T. Otake and K, Okada [55] represents a survey of considerable literature, and is applicable to aqueous and non-aqueous systems for Reynolds numbers from 10, - 20,000 [40]. 

The dynamical regimes that may be explored using this method have been described by considering the range of dimensionless numbers, such as the Reynolds number, Schmidt number, Peclet number, and the dimensionless mean free path, which are accessible in simulations. With such knowledge one may map MPC dynamics onto the dynamics of real systems or explore systems with similar characteristics. The applications of MPC dynamics to studies of fluid flow and polymeric, colloidal, and reacting systems have confirmed its utility. 

As noted in Section 6.1.3 of Volume 2, the Carman-Kozeny equation applies only to conditions of laminar flow and hence to low values of the Reynolds number for flow in the bed. In practice, this restricts its application to fine particles. Approaches based on both the Carman-Kozeny and the Ergun equations are very sensitive to the value of the voidage and it seems likely that both equations overpredict the pressure drop for fluidised systems. 

Because equation 4.18 is applicable to low Reynolds numbers at which the flow is streamline, it appears that the flow of fluid at high concentrations of particles in a sedimenting or fluidised system is also streamline. The resistance to flow in the latter case appears to be about 30 per cent lower, presumably because the particles are free to move relative to one another. 

In broad terms scale-up is an engineering technique for translating performance in a small system to performance in a large system. A useful review of the formal material available on theories of models, similitude, dimensional analysis, etc., written from the chemical engineering point of view, is available in a recent book (J4). The practical applications of these theories involve the use of dimensionless groups, such as Reynolds number, in correlations which describe the performance of a system in terms equally applicable to large or small systems. This method of scale-up is familiar to all engineers. 

A useful generalization noted in the previous section is the widespread applicability of impeller Reynolds number for correlating performance data from different-scale operations in geometrically similar systems. In some heterogeneous systems, it may be necessary to modify the definitions of density and viscosity for use in this Reynolds number, and to introduce groups like the Weber number to account for interfacial forces (see Section V). The main point is that it requires experiment to establish finally the form of the controlling groups. 

Microfluidics is the manipulation of fluids in channels, with at least two dimensions at the micrometer or submicrometer scale. This is a core technology in a number of miniaturized systems developed for chemical, biological, and medical applications. Both gases and liquids are used in micro-/nanofluidic applications, ° and generally, low-Reynolds-number hydrodynamics is relevant to bioMEMS applications. Typical Reynolds numbers for biofluids flowing in microchannels with linear velocity up to 10 cm/s are less than Therefore, viscous forces dominate the response and the flow remains laminar. 

The use of tubular Donnan dialysis systems has stimulated attempts to provide predictive models, principally for industrially oriented applications. Ng and Snyder (39) have recently published one such attempt applied to dialysis of nickel(II) into a sulfuric acid receiver solution. Correlations between mass transport coefficient and Reynolds number are reported, and the factors controlling transport over a range of nickel concentrations are discussed. 

Liquid flow is incompressible, so, in micrometer-scale channels, the flow has a small Reynolds number (Re), usually less than 1, and the flow in simple microchannels is laminar, thus chaotic or turbulent flows are not observed [1]. Many types of microfluidic device have been developed on the basis of this flow behavior. Functional flow control methods based on laminar flow profiles have been proposed and applied in microflow devices and systems. Passive and active flow control methods and their applications are introduced in this section. 

Equation (7,17) is called the Kozeny-Carman equation and is applicable for flow through beds at particle Reynolds numbers up to about 1.0. There is no sharp transition to turbulent flow at this Reynolds number, but the frequent changes in shape and direction of the channels in the bed lead to significant kinetic energy losses at higher Reynolds numbers. The constant 150 corresponds to = 2.1, which is a reasonable value for the tortuosity factor. For a given system, Eq. (7,17) indicates that the flow is proportional to the pressure drop and inversely proportional to the fluid viscosity. This statement is also known as Darcy s law, which is often used to describe flow of liquids through porous media. 

It is possible to extend the procedure developed above to certain multiphase applications in which the power requirement has to be estimated. In the case of low-viscosity liquid/liquid systems, as encountered in solvent extraction, and for coarse solids suspended in low-viscosity liquids at low concentrations, the operation is likely to be carried out in the turbulent region. The single-phase power curves can be used in such instances with the mean density being used in both the power number and Reynolds number. However, such an approach must not be used for gas/liquid systems where predictions based on average density values can lead to gross over-estimates of the power requirement. This is considered in detail in Chapter IS. 

---