Reynolds number dimensionless parameters

The Weber number becomes important at conditions of high relative velocity between the injected Hquid and surrounding gas. Other dimensionless parameters, such as the Ohnesorge ((We /Re), Euler (AP/Pj y i)y and Taylor (Re/ We) numbers, have also been used to correlate spray characteristics. These parameters, however, are not used as often as the Reynolds and Weber numbers. 

Dimensional analysis leads to various dimensionless parameters, wliieli are based on the dimension s mass (M), length (L), and time T). Based on these elements, one ean obtain various independent parameters sueh as density (p), viseosity (/i), speed (A ), diameter ( )), and veloeity (V). The independent parameters lead to forming various dimensionless groups, whieh are used in fluid meehanies of turbomaehines. Reynolds number is the ratio of the inertia forees to the viseous forees... 

An impeller designed for air ean be tested using water if the dimensionless parameters, Reynolds number, and speeifie speed are held eonstant... 

This chapter reviews the various types of impellers, die flow patterns generated by diese agitators, correlation of die dimensionless parameters (i.e., Reynolds number, Froude number, and Power number), scale-up of mixers, heat transfer coefficients of jacketed agitated vessels, and die time required for heating or cooling diese vessels. 

Hicks et al. [8] developed a correlation involving the Pumping number and impeller Reynolds number for several ratios of impeller diameter to tank diameter (D /D ) for pitched-blade turbines. From this coiTclation, Qp can be determined, and thus the bulk fluid velocity from the cross-sectional area of the tank. The procedure for determining the parameters is iterative because the impeller diameter and rotational speed N appear in both dimensionless parameters (i.e., Npe and Nq). 

Other dimensionless parameters, namely the Reynolds number, Prandtl number, and Nusselt number ean be represented as follows ... 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

Reynolds number A dimensionless parameter that represents the ratio of the inertia forces to the viscous forces in a flow. Its magnitude denotes the actual flow regime, such as streamline (laminar), transitional, or turbulent. 

The dependence of the measured rise in fluid mixed-cup temperature on Reynolds number is illustrated in Fig. 3.12. The difference between outlet and inlet temperatures increases monotonically with increasing Re at laminar and turbulent flows. Under conditions of the given experiments, the temperature rise due to energy dissipation is very significant AT = 15—35 K at L/ i = 900—1,470 and Re = 2,500. The data on rising temperature in long micro-tubes can be presented in the form of the dependence of dimensionless viscous heating parameter Re/[Ec(L/(i)] on Reynolds number (Fig. 3.13). 

The behavior of the gas as it flows down the tube is controlled by fluid mechanics and a complete investigation wouldbe lengthy and outside the scope of this book. It is enough to say that the Reynolds number, which is a dimensionless parameter that characterizes the... 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

The flow domain of TCP can be described by two dimensionless hydrodynamic parameters, corresponding to the rotational speed of the inner cylinder and the imposed axial flow rate the Taylor number, To, and the axial Reynolds number, Re, respectively ... 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

Since the same simplified set of dimensionless parameters holds exactly at both high and low Reynolds numbers, it is reasonable to expect that it will hold, at least approximately, over the entire range of conditions for which the drag coefficient can be determined by the Ergun equation or an equation of similar form. 

Using these Reynolds number scale factors, the errors in the dimensionless drag coefficient j3L/psua using the simplified scaling models are shown in Figs. 24 and 25 for u0/umf of 10 and 1000, respectively, plotted as a function of Rep, based on parameters for the exact scaled bed. For a particle Reynolds number of 1000 or less, which corresponds to... 

Figure 33. Dimensionless spoutdiametersasafunctionof dimensionless height for small columns. Case A test case Case B all dimensionless parameters matched, bed diameter halved Case C particle Reynolds number mismatched Case D Froude number mismatched Case E density ratio, Reynolds number mismatched Case F bed Reynolds number mismatched Case G internal friction angle, loose packed voidage mismatched. (From He et al., 1995.)...

From equation (4.26), it is seen that the important dimensionless parameters driving momentum transport are the Reynolds number, the Froude number, the Euler number, and the length and velocity ratios in the flow field. The dimensionless variables all vary between zero and a value close to one, so they are not significant in determining which terms in the governing equation are important. 

The second method uses dimensionless numbers to predict scale-up parameters. The use of dimensionless numbers simplifies design calculations by reducing the number of variables to consider. The dimensionless number approach has been used with good success in heat transfer calculations and to some extent in gas dispersion (mass transfer) for mixer scale-up. Usually, the primary independent variable in a dimensionless number correlation is Reynolds number ... 

Figure 1 Various dimensionless parameters [dimensionless velocity, v = v/ND pumping number, Nq = Q/ND power number, Np=[Pgc/pN D ) and dimensionless mixing time, f = as a function of the Reynolds number for the analysis of turbine-agitator systems. Source Adapted from Ref. 22.

Flexibility in the choice of parameters and their reliable extrapolation within the range covered by the dimensionless numbers. These advantages become clear if one considers the well-known Reynolds number. Re = vL/v, which can be varied by altering the characteristic velocity V or a characteristic length L or the kinematic viscosity v. By choosing... 

Engineers commonly use dimensionless ratios such as the Reynolds number and the lift coefficient to help understand complex experimental data, organize equations and model building, and relate model testing in a wind tunnel to that of a prototype flight. This kind of analysis is called dimensional analysis because it uses the dimensional nature of important variables to derive dimensionless parameters that determine the scaling properties of a physical system. 

Nb h Critical Reynolds number for onset of turbulence i Re, Reynolds number of gas stream N-Rei Reynolds number at onset of instability Nr Nusselt dimensionless film thickness parameter, defined by Eq. (97)... 

Often it is useful to combine variables that affect physical phenomenon into dimensionless parameters. For example, the transition from laminar to turbulent flow in a pipe depends on the Reynolds number, Re = pLv/p, where p is the fluid density, I is a characteristic dimension of the pipe, v is the velocity of flow, and // is the viscosity of the fluid. Experiments show that the transition from laminar to turbulent flow occurs at the same value of Re for different fluids, flow velocities, and pipe sizes. Analyzing dimensions is made easier if we designate mass as M, length as L, time as t, and force as F. With this notation, the dimensions of the variables in Re are ML 3 for p, (L) for L, (L/t) for v, and (FL 2t) for //. Combining these it is apparent that Re = pLu/p, is dimensionless. 

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