Stokes number, equation defining

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

Typical cascade impactors consist of a series of nozzle plates, each followed by an impaction plate each set of nozzle plate plus impaction plate is termed a stage. The sizing characteristics of an inertial impactor stage are determined by the efficiency with which the stage collects particles of various sizes. Collection efficiency is a function of three dimensionless parameters the inertial parameter (Stokes number, Stk), the ratio of the jet-to-plate spacing to the jet width, and the jet Reynolds number. The most important of these is the inertial parameter, which is defined by Equation 2) as the ratio of the stopping distance to some characteristic dimension of the impaction stage (10), typically the radius of the nozzle or jet (Dj). 

This is known as Stokes law. Experimentally, Stokes law is found to hold almost exactly for single particle Reynolds number, Rcp <0.1, within 9% for Rep < 0.3, within 3% for Rep < 0.5 and within 9 % for Rep < 1.0, where the single particle Reynolds number is defined in Equation (2.4). 

This equation incorporates inertial effects on diffusional transport for dilute suspensions and defines the diffusion tensor, D(Sf), dependent on both particle inertia and the shear rate through the Stokes number St. 

The behavior of continuum fluid flow is governed by the well-known Navier-Stokes equations. In the low-Reynolds-number regime, where the kinematic fluid viscosity V is large compared with the characteristic velocity of the flow u and inertia terms can safely be neglected, the Stokes equations can be deduced from the Navier-Stokes equations. The particle Reynolds number is defined for the particle radius a as ... 

For Rein < 2 x 10, Stkinso can be seen to increase, showing that small cyclones are less efficient than one would expect from Stokesian scaling. Our defining equation, (8.1.4), for the Stokes number. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

In the first two cases the Navier-Stokes equation can be applied, in the second case with modified boundary conditions. The computationally most difficult case is the transition flow regime, which, however, might be encountered in micro-reactor systems. Clearly, the defined ranges of Knudsen numbers are not rigid rather they vary from case to case. However, the numbers given above are guidelines applicable to many situations encoimtered in practice. 

It is also possible to derive the Reynolds number by dimensional analysis. This represents a more analytical, but less intuitive, approach to defining the condition of similar fluid flow and is essentially independent of particular shape. In this approach, variables in the Navier-Stokes equation (relative particle-fluid velocity, a characteristic dimension of the particle, fluid density, and fluid viscosity) are combined to yield a dimensionless expression. Thus... 

To specify the velocity field u, we must solve the Navier-Stokes equations subject to the boundary condition (9-160) at infinity. For present purposes, we follow the example of Section C and assume that the Reynolds number, defined here as Re = a2yp/ii, is very small so that the creeping-flow solution for a sphere in shear flow (obtained in Chap. 8, Section B) can be applied throughout the domain in which 6 differs significantly from unity. Hence, from (8-51) and (8-57), we have... 

The main parameter determining the gas rarefaction is the Knudsen number Kn = Ha, where I is the mean free path of fluid molecules and a is a typical dimension of gas flow. If the Knudsen number is sufficiently small, say Kn < 10 , the Navier-Stokes equations are applied to calculate gas flows. For intermediate and high values of the Knudsen number, the Navier-Stokes equations break down, and the implementation of rarefied gas dynamics methods is necessary. In practical calculations usually the rarefaction parameter defined as the inverse Knudsen number, i.e.. 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pf/L/p, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro-/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term (pu Vu) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro-/ nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity... 

A fluid can consider like a continuous medium when the fluid is dense, this meaning that the particles are behind the other particles and there isn t space between they, in this case the fundamental equations that conduct the fluid s evolution are of kind Navier-Stokes (NE). One form of determinate if the continuous medium is acceptable, is through of Knudsen s number (Kn=/1/1), which is defining like the relation between the free mean trajectory A (mean distance that a molecule travels through before collisions with other molecule) and the characteristic length 1 the model continuous medium is acceptable for a rank of 0.01 < Kn < 1. In other words the Knudsen s number should be less that unit, in this form the continuous hypothesis will be valid. 

A similar case may be made for the use of density in Stokes law, the buoyancy of particles in the separation zone must be taken into account. The fine particles displace the continuous phase and hence it is the density of the liquid that is used in the model. In any case, the suspension density in the zone is not known but is likely to be much less than that of the feed. The second use of fluid density is in the resistance coefficient, Eu. The density to be used there depends on how we define the Euler number the dynamic pressure in the denominator (equation 6.9) is simply a yardstick against which we measure the pressure loss through a cyclone. We have used the clean liquid density in the dynamic pressure alternatively, the feed suspension density may be used. It is immaterial which of the two densities is used (they are both equally unrealistic) provided the case is clearly defined conversion from one to the other is a simple matter. 

The impaction characteristics of the cascade impactors are governed by the fluid s velocity flow field, which in turn is specified by the Navier-Stokes equations. A solution of the Navier-Stokes equations (Marple et al 1974) reveals that the velocity flow field is a function of the physical configuration of the impactor and the Reynolds number. Re, of the flow passing through the nozzle. The Reynolds number, based on the hydraulic diameter of the nozzle, is defined as... 

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