Reynolds numbers, flow systems

In some cases, in particular for rarefied systems and/or low Reynolds number flows a velocity discontinuity at wall can be observed. This is the shp flow regime. The Reynolds number can be low enough for the classical boundary layer theory not to be completely valid but not low enough for the inertia terms to be neglected. The fluid velocity at the wall is not zero and... 

Unlike the flows considered in Chapter 3 which were essentially imidirectional, the fluid flows in particulate systems are either two- or three-dimensional and hence are inherently more difficult to analyse theoretically, even in the creeping (small Reynolds number) flow regime. Secondly, the results are often dependent on the rheological model appropriate to the fluid and a more generalised treatment is not possible. For instance, there is no standard non-Newtonian drag curve for spheres, and the relevant dimensionless groups depend on the fluid model which is used. Most of the information in this chapter relates to time-independent fluids, with occasional reference to visco-elastic fluids. 

Another metric for propulsion systems is the specific impulse, 7sp, which represents the thrust per unit mass flow rate of propellant. Ideally, 7sp is not a function of scale, but depends mostly on the type of propellant used. In practice, aerodynamic losses due to lower Reynolds number flows in the nozzle and nonideal nozzle... 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

High-Viscosity Systems A axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in... 

A criterion for the namre of flow, either laminar or mrbulent flow in fluid systems is the value of the Reynolds number. This number, Rg is defined by the equation... 

The value of the Reynolds number which approximately separates laminar from turbulent flow depends, as previously mentioned, on the particular conhg-uration of the system. Thus the critical value is around 50 for a him of liquid or gas howing down a hat plate, around 500 for how around a sphere, and around 2500 for how tlrrough a pipe. The characterishc length in the dehnition of the Reynolds number is, for example, tire diameter of the sphere or of the pipe in two of these examples. 

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 ... 

Impeller Reynolds Number a dimensionless number used to characterize the flow regime of a mixing system and which is given by the relation Re = pNDV/r where p = fluid density, N = impeller rotational speed, D = impeller diameter, and /r = fluid viscosity. The flow is normally laminar for Re <10, and turbulent for Re >3000. 

Darcy s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and conecting the velocity with porosity yields a Reynolds number in the form ... 

Scope, 52 Basis, 52 Compressible Flow Vapors and Gases, 54 Factors of Safety for Design Basis, 56 Pipe, Fittings, and Valves, 56 Pipe, 56 Usual Industry Pipe Sizes and Classes Practice, 59 Total Line Pressure Drop, 64 Background Information, 64 Reynolds Number, R,. (Sometimes used Nr ), 67 Friction Factor, f, 68 Pipe—Relative Roughness, 68 Pressure Drop in Fittings, Valves, Connections Incompressible Fluid, 71 Common Denominator for Use of K Factors in a System of Varying Sizes of Internal Dimensions, 72 Validity of K Values,... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

This equation should generally valid for all particle systems and working conditions with (9p-9)/9 1, dp/qL< 6 and A/qL> 125... 250. The last condition of fully developed turbulent flow is very important. To small values A/qp which mostly corresponds to Reynolds numbers Re <10 (small reactors, higher viscosity s of media and small power input) leads to an distinct reduction of stress. That was observed by the investigations in 166] which were carried out with the... 

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