Parameter dimensionless

The basic dimensionless parameters encountered in metallurgical engineering are briefly described below. Additional dimensionless parameters are listed in Appendix 1 together with their physical meanings. 

This parameter represents the ratio of inertial force due to temporal acceleration of fluid motion to inertial force due to spatial acceleration. It is conveniently used to describe the shedding of vortices from solid bodies immersed in fluids. 

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The potential has a spurious maximum at r where the r ° tenn again starts to dominate. The dimensionless parameter a is a measure of the steepness of the repulsion and is often assigned a value of 14 or 15. The ideas... 

The dimensionless parameters Ot, , Ctc appearing in the last expression are connected with the sums and differences of the adiabatic potentials as shown elsewhere [149,150]. This effective Hamiltonian acts onto the basis functions (A.l) with A = 2. 

Let us now assemble the complete set of dimensionless parameters for the problem. These are set out in Table 11.1, where the last column indicates the nature of their dependence on the external pressure p, the mean pore diameter and the pellet radius a. Symbols ft and 0... 

The set of parameters given in Table 11.1 is by no means unique, since any independent set of combinations of the given parameters will serve equally well. The particular set adopted will depend primarily on the purpose for which they are to be used. Thus, if we are interested in the dependence of the effectiveness factor on one particular physical variable, it is obviously convenient to choose the dimensionless parameters in such a way that all but one are independent of this variable. A plot of the effectiveness factor against one dimensionless parameter will then summarize the desired information. 

These examples are sufficient to illustrate one of the principal factors which should influence the choice of dimensionless parameters. Neglect of these considerations may lead, and indeed sometimes has led, to graphical presentations which are at best clumsy and at worst valueless. 

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

Stress relaxation time, obtained from rheograms based on viscometric flows, is used to define a dimensionless parameter called the Deborah number , which quantifies the elastic character of a fluid... 

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

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. 

The separation parameters have been calculated for a centrifuge in which the behavior of the circulating gas is described by Martin s equation. The flow pattern efficiency is shown in Figure 15(b) as a function of the dimensionless parameter M, where M is equal to (ME /2RT). In this case the maximum flow pattern efficiency attainable is 0.956. 

In this work, we determine constraints on the dimensionless parameters of the system (dimensionless electrode widths, gap size and Peclet number), first qualitatively and then quantitatively, which ensure that the proposed flow reconstmction approach is sufficiently sensitive to the shape of the flow profile. The results can be readily applied for identification of hydrodynamic regimes or electrode geometries that provide best performance of our flow reconstmction method. 

The relative contribution of over-barrier ( > Vq) transitions and tunneling ( < Vq) to the integral (2.1) is governed by the dimensionless parameter... 

The velocity ratio is the single most important factor in determining the performance of an expander. The velocity ratio, p, is a dimensionless parameter, relating the physical size of the expander to the gas conditions being considered. It is defined as ... 

If using dimensionless parameters obtained through dimensional analysis, or similitude, the data will eollapse into a single eurve. In this ease, the useful parameters are redueed flow, q, redueed power. 

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... 

The previous equations are some of the major dimensionless parameters. For the flow to remain dynamically similar, all the parameters must remain constant however, constancy is not possible in a practical sense, so one must make choices. 

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

The dimensionless parameter B increases with volatility and typically lies between 1.2 and about 8 as shown in the following table [64] ... 

There was some argument in the literature over the relative merits and demerits of the JKR and the DMT theories [23-26], but the controversy has now been satisfactorily resolved. A critical comparison of the JKR and DMT theories can be obtained from the literature [23-30]. According to Tabor [23], JKR theory is valid when the dimensionless parameter given by Eq. 25 exceeds a value of about five. 

Note that this list of dimensionless parameters is by no means unique. A set of variables in which each variable in the new set is a combination of the abovementioned set is also permissible and is in principle completely equivalent to the original set. In fact, an infinite number of sets of the dimensionless parameters exist, each of which could be justified as the "original" set. The normal approach at this point is to find an explicit functional relation among one set of variables... 

F = dimensionless parameter depending on limiting conditions noted below... 

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). 

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

The following reviews seale-up of ehemieal reaetors, eonsiders the dimensionless parameters, mathematieal modeling in seale-up, and seale-up of a bateh system. 

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