Dimensionless characteristic number

With the average elongational strain rate of the flow field between the eddies and the relaxation time of the polymer molecules, one can define a dimensionless characteristic number, the Deborah number, which represents the ratio of a characteristic time of flow and a characteristic time of the polymer molecule, and thus one can transfer considerations in porous media flow to the turbulent flow region. 

Consideration of length and time scales is fundamental as they provide an indication of the main mechanisms at work. The combination of length and time scales with material parameters such as molecular diffusivity and viscosity leads to dimensionless characteristic numbers that provide guides to the relative importance of competing mechanisms. 

Technically relevant catalysts undergo during their operational lifetime millions of cycles therefore, a large number of product molecules can be formed with a small number of catalytic centers. The number of cycles that a catalyst can undergo prior to its deactivation is a very important performance criterion in catalysis. This dimensionless characteristic number for each catalyst is called the turnover number (TON) and describes the catalyst s stability and lifetime. The TON, which originates from the field of enzymatic catalysis, is defined as the maximum amount of reactant (in moles) that a certain number of catalytically active centers (in moles) can convert into a certain product. For a A —> B reaction we obtain ... 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

First the dimensionless characteristics such as Re and Pr in forced convection, or Gr and Pr in free convection, have to be determined. Depending on the range of validity of the equations, an appropriate correlation is chosen and the Nu value calculated. The equation defining the Nusselt number is... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

A characteristic quantity describing the viscous flow state is the dimensionless Reynolds number Re. 

It is obvious that we could name all the geometric parameters indicated in the sketch. They were all the geometric parameters of the stirrer and of the vessel, especially its diameter D and the liquid height H. In cases of complex geometry, such a procedure would compulsorily deflect from the problem. It is therefore advisable to introduce only one characteristic geometric parameter, knowing that all the others can be transformed into dimensionless geometric numbers by division with this one. 

The gas throughput characteristic of a hollow stirrer generally has the form /VA = /(Fr dy H, Ga, dr/dt, HJdf), where NA = qt/(Ndf) is the dimensionless flowrate number, Fr s Ndjg the Froude number, and Ga = dfg/v2 the Galileo number. For liquids with viscosities close to that of water and for HJd = 1, the gas throughput characteristics for the tube stirrer shown in Fig. 9 are as follows ... 

Axial Convective Diffusion. The variation in width, length and direction of individual channels formed by the interstices of the packing give rise to a dispersion which can be characterized by the dimensionless Bodenstein number. Bo, which is a similar number as the Peclet number but with the particle diameter as characteristic dimension... 

Whether a polymer exhibits elastic as well as viscous behavior depends in part on the time scale of the imposition of a load or deformation compared to the characteristic response time of the matei ial. This concept is expressed in the dimensionless Deborah number ... 

The density based characteristic number K and other such numbers Kgl, Kg2,... characterise the dimensionless temperature dependence of the density ... 

What are the conditions for flow to become turbulent This depends on the preponderance of inertial stresses—proportional to pv2—over frictional or viscous stresses. The latter are equal to j/ P in laminar flow is proportional to v/L, where L is a characteristic length perpendicular to the direction of flow. The ratio is proportional to the dimensionless Reynolds number, given by... 

Certain quantities are defined as the ratios of two quantities of the same kind, and thus have a dimension which may be expressed by the number one. The unit of such quantities is necessarily a derived unit coherent with the other units of the SI and, since it is formed as the ratio of two identical SI units, the unit also may be expressed by the number one. Thus the SI unit of all quantities having the dimensional product one is the number one. Examples of such quantities are refractive index, relative permeability, and friction factor. Other quantities having the unit 1 include characteristic numbers like the Prandtl number and numbers which represent a count, such as a number of molecules, degeneracy (number of energy levels), and partition function in statistical thermodynamics. AU of these quantities are described as being dimensionless, or of dimension one, and have the coherent SI unit 1. Their values are simply expressed as numbers and, in general, the unit 1 is not explicitly shown. In a few cases, however, a special name is given to this unit, mainly to avoid confusion between some compound derived units. This is the case for the radian, steradian and neper. 

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